Computational Nanotechnology Modeling and Applications with MATLAB®
Nano and Energy Series Series Editor: Sohail Anwar Computational Nanotechnology: Modeling and Applications with MATLAB® Sarhan M. Musa Forthcoming titles Sustainable Manufacturing Arun N. Nambiar and Adnar H. Sabuwala Nanotechnology: Business Applications and Commercialization Sherron Sparks Green Building: Applied Nanotechnology and Renewable Energy Sohail Anwar Advanced Nanoelectronics Razali Bin Ismail, Mohammad Taghi Ahmadi, and Sohail Anwar Nanotechnology: Social and Ethical Issues Ahmed S. Khan Introduction to Renewable Energy Systems Sohail Anwar
Edited by
Sarhan M. Musa
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110808 International Standard Book Number-13: 978-1-4398-4177-8 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
To my father Mahmoud, my mother Fatmeh, and my wife Lama
Contents Preface..............................................................................................................................................ix Acknowledgments ..................................................................................................................... xvii Editor............................................................................................................................................. xix Contributors................................................................................................................................. xxi 1. Introduction to Computational Methods in Nanotechnology ......................................1 Orion Ciftja and Sarhan M. Musa 2. Computational Modeling of Nanoparticles .................................................................... 29 Ufana Riaz and S.M. Ashraf 3. Micromagnetics: Finite Element Analysis of Nano- Sized Magnetic Materials Using MATLAB® .................................................................................................................. 75 Shin-Liang Chin and Timothy Flack 4. System-Level Modeling of N/MEMS ............................................................................... 97 Jason Vaughn Clark 5. Numerical Integrator for Continuum Equations of Surface Growth and Erosion ......................................................................................................... 189 Adrian Keller, Stefan Facsko, and Rodolfo Cuerno 6. Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes of Carbon, Silicon-Carbide, and Carbon-Nitride ................................... 217 W.-D. Cheng, C.-S. Lin, G.-L. Chai, and S.-P. Huang 7. MATLAB® Applications in Behavior Analysis of Systems Consisting of Carbon Nanotubes through Molecular Dynamics Simulation............................. 251 Masumeh Foroutan and Sepideh Khoee 8. Device and Circuit Modeling of Nano-CMOS............................................................. 301 Michael L.P. Tan, Desmond C.Y. Chek, and Vijay K. Arora 9. Computational and Experimental Approaches to Cellular and Subcellular Tracking at the Nanoscale................................................................................................. 333 Zeinab Al-Rekabi, Dominique Tremblay, Kristina Haase, Richard L. Leask, and Andrew E. Pelling 10. Computational Simulations of Nanoindentation and Nanoscratch ........................ 363 Cheng-Da Wu, Te-Hua Fang, and Jen-Fin Lin 11. Modeling of Reversible Protein Conjugation on Nanoscale Surface ...................... 381 Kazushige Yokoyama vii
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12. Computational Technology in Nanomedicine ............................................................. 411 Viroj Wiwanitkit 13. Future Directions: Opportunities and Challenges...................................................... 429 George C. Giakos Appendix A: Material and Physical Constants .................................................................. 451 Appendix B: Symbols and Formulas .................................................................................... 455 Appendix C: MATLAB® .......................................................................................................... 471
Preface Nanotechnology and nanoscience have now emerged as the leading fields to a technological revolution in the new millennium. Nanotechnology involves research across disciplines such as engineering, physics, chemistry, biology, and medicine. It engages in the science of matter that is smaller than 100 nm. Therefore, the ability to manipulate atoms at the nanoscale helps scientists create tiny devices the size of a few atoms for many applications such as consumer supplies, electronics, and computers in the development of molecular-sized transistors for the memory of microcomputers, as well as in the fields of information and biotechnology, aerospace, energy, and environment, and in the improvement of drug delivery in identifying cancer cells in medicine, in energy productions, etc. Using nanotechnology brings many benefits to industry as it requires little labor and low cost. This book can be used as a reference by engineers, scientists, and biologists who are involved in the computational techniques of nanotechnology. It has been written for professionals, researchers, and students who need to discover the challenges and the opportunities concerning the development of the next generation of nanoscale computational nanotechnology: modeling and applications with MATLAB®. It provides a broad spectrum of technical information, research ideas, and practical knowledge regarding nanotechnology applications, and intends to provide a thought-provoking perspective on how nanotechnology is poised to revolutionize the field of computational nanotechnology. It also describes emerging nanotechnology applications as well as nanotechnology applications under development that hold promise for significant innovations in engineering, physics, chemistry, biology, and medicine. The book has been divided into 13 chapters. Chapter 1 presents an introduction to computational methods in nanotechnology. It shows that nanoscale structures are quite intriguing because in the nanometer region almost all physical and chemical properties of systems are generally size dependent. The dependence on size of properties of such materials makes them fascinating from a technological and theoretical point of view. At the same time, nanoscale science and engineering offers an entirely new design motif for developing advanced materials and new devices. The nanoscience revolution has created an urgent need for a more robust quantitative understanding of matter at the nanoscale through modeling and simulation. Realizing this potential, however, will require a concerted effort in theoretical and modeling research to overcome some of the computational challenges in the field. Because of the variety and large number of nanoscale building blocks, it is impossible to cover all of them here. However, we will try to give a brief overview of two very important nanoscale systems—molecular magnets and semiconductor quantum dots. The objective of Chapter 1 is to look closely at the theory and modeling of these two nanoscale systems and also to briefly address some of the critical research issues in the fast-evolving field of nano-magnetism and nano-electronics, where molecular magnets and quantum dots are the respective building blocks. Studies of molecular magnets and semiconductor quantum dots offer great promise for the future technologies, but demand computationally intensive work. In order to get the fullest understanding of the basic properties of various nanoscale systems, a wide range of multidisciplinary scientists is required in order to utilize various modeling approaches and tools including MATLAB. While focusing on specific examples of ultrasmall systems, we attempt to give a brief and basic understanding of various computational methods that ix
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help us to understand the key properties of various nanoscale molecular magnetic and semiconductor quantum dot systems. Chapter 2 presents computational modeling of nanoparticles. Nanomaterials constitute an emerging subdiscipline in the chemical and materials sciences. All conventional materials like metals, semiconductors, glass, ceramic, or polymers can in principle be obtained with a nanoscale dimension. Nanomaterials have various microstructural features such as nanodiscs, quantum dots, nanowires, nanotubes, nanocoatings, and nanocomposites. The unique functionality of nanoparticle-based materials and devices depends directly on size- and structure-dependent properties. Nanoparticle size must be tightly controlled to take full advantage of quantum size effects in technological applications, and agglomeration must be prevented. This can only be done if the concentration is controlled, which requires that the rate of new particle formation be quantitatively determined. Real-time measurements of particle size distributions and particle structure are, thus, enabling techniques for the advancement of nanotechnology. Accurate and reliable models for simulating transport, deposition, coagulation, and dispersion of nanoparticles and their aggregates are needed for the development of design tools for technological applications, including nanoparticle instrumentation, sampling, sensing, dilution, and focusing nanoparticle behavior in chemically reactive systems. Computational techniques allow us to explore and validate hypotheses about experimentally observed behavior that may otherwise not be accessible through traditional experimental techniques. Additionally, computer simulations allow for theory to propose areas of interest to which experimental techniques may be applied. Two of the most common computer simulation methods, Monte Carlo (MC) and molecular dynamics (MD), have been widely applied to a variety of model systems ranging from large proteins to nanoclusters. Computer science offers more opportunities for nanotechnology. Soft computing techniques such as swarm intelligence, genetic algorithms, and cellular automata can impart desirable emergent properties like growth, self-repair, and complex networks to systems. Many researchers have successfully applied such techniques to real-world problems, including complex control systems in manufacturing plants and air traffic control. Research in intelligent systems involves the understanding and development of intelligent computing techniques as well as the application of these techniques for real-world tasks, often to the problems of other research areas. The techniques in intelligent systems comprise methods or algorithms in artificial intelligence (AI), knowledge representation/ reasoning, machine learning, and natural computing or soft computing. With some modifications in nanotechnology characteristics, these techniques can be applied to control a swarm of a trillion nanoassemblers. It is anticipated that soft computing methods such as these will overcome concerns about harmful implications of nanotechnology and prevent the notorious scenario of self-replicating nanorobots multiplying uncontrollably. A preliminary framework has been developed to help determine what research is needed and how it can be integrated into the designing of computer-simulated models of nanoparticles. The focus of Chapter 2 is to highlight the significance of nanoparticles and the challenges in the integration of these nanomaterials for their potential applications in various fields. Chapter 3 presents micromagnetics: finite element analysis of nano-sized magnetic materials using MATLAB. Fabrication techniques developed over the past decades have allowed engineers and scientists to form magnetic materials in the nanometer regime. Such materials exhibit very interesting magnetic properties that are significantly different from bulk materials. At such scales, the behaviors of the magnetic materials are inadequately described by Maxwell’s equations alone. The theory of micromagnetics deals with
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the behaviors of these nano-sized magnetic materials by including the quantum mechanical effects that are significant at this scale. Micromagnetics theory has been successful in predicting the formation of domain walls in magnetic materials and also the formation of interesting magnetic states such as the vortex and leaf states. It also finds its application in many engineering aspects such as the digital data storage technology. Chapter 3 will introduce the basic theory of micromagnetics, show how the finite element method can be applied to the governing equation of micromagnetics (Landau–Lifshitz–Gilbert equation), and finally show how all these can be done using MATLAB. Chapter 4 presents system-level modeling of N/MEMS; it discusses some of the latest advances in designing and modeling complex nano- and microelectromechanical (N/MEMS) systems at the system level. These advances have been applied to an open source tool called PSugar at Purdue University. For design, the author discusses a graphical user interface (GUI) that allows users to quickly configure complex systems in 3D using a computer mouse or pen at a rate faster than what might be drawn with pencil and paper. This GUI is coupled to a powerful net list language for design flexibility. For modeling, the chapter applies recent advances in analytical system dynamics and differential–algebraic equations (DAEs) into a framework that facilitates the systematic modeling of multidisciplinary systems that may comprise static or dynamic constraints. Where appropriate, we verify lumped models against their distributed model counterparts. Several test cases are discussed that demonstrate a wide range of utility of system-level analysis for N/MEMS. The test cases include simple to complex MEMS, and NEMS using molecular dynamics. Chapter 5 explains numerical integrators for continuum equations of surface growth and erosion. Growth of thin films by physical vapor deposition and erosion of solid surfaces by ion beam sputtering have high relevance in the fields of micro- and nanotechnology. Therefore, the theoretical modeling of the surface morphology and its evolution during growth and erosion have received considerable attention. Compared to atomistic models, continuum equations are able to cover a much larger spatial and temporal scale accessing the macroscopic scales at which nontrivial features of the morphological dynamics occur. This is especially relevant for surface erosion, which can induce a linear instability of the surface, resulting in the formation of periodic nanoscale patterns. In Chapter 5, the software package “Ripples and Dots” is presented, which is designed for the numerical integration of continuum equations modeling the surface evolution during ion beam erosion in (2 + 1) dimensions. These equations are based on the Bradley– Harper model and often feature, in addition to the linear instability, nonlinear and noise terms, for example, the prominent Kuramoto–Sivashinsky equation. However, due to the similarity with continuum equations for interface growth, the software package can be used as well for studying continuum equations describing thin film growth, like the wellknown Kadar–Parisi–Zhang equation, the Edwards–Wilkinson equation, and the linear molecular beam epitaxy equation. Numerical data obtained by integrating various continuum equations for surface erosion are presented and compared to analytical predictions and experimental observations. It is demonstrated that the evolution of the surface morphology observed in the numerical integrations is in good qualitative agreement with experimental results both for normal and oblique ion incidence. In addition, special cases of erosion at oblique ion incidence with simultaneous sample rotation and erosion with multiple ion beams are presented. Chapter 6 presents configuration optimizations and photophysics simulations of singlewall nanotubes of carbon, silicon-carbide, and carbon-nitride. First, the authors optimize the structures of ground state for finite open single-wall carbon nanotubes (CNTs)
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of (n, n) type (n = 3–6) by B3LYP/6-31G* level. They then calculate the electronic structures and simulate the electronic absorption, third-order polarizability, and two-photon absorption spectra by TDB3LYP/6-31G* level combining with sum-over-states method based on the optimized configurations. The obtained results show that the tube diameter increases, the gap between LUMO–HOMO reduces, electronic absorption spectrum is redshift, and two-photon absorption cross section enhances as chiral vector length increases at constant axis length for (n, n) tubes. It is shown that the third-order polarizability along with the tube-axis direction decreases, as the bisector direction between the parallel and perpendicular to the tube axis increases while chiral vector length of (n, n) tube increases. That is, the smaller diameter and the larger the anisotropy of polarizabilities are for finite open single-wall CNTs with the same length of tube axes. The π–π charge transfers within (n, n) tubes are the electronic originations of nonlinear polarizability and two-photon absorption spectra at low-energy regions, in view of state-dependent third-order polarizability and configuration interaction contributions. Second, the authors employ periodical density functional theory to investigate geometrical structures and linear optical properties of SiC and CNTs with different types. The obtained results indicate that the optical properties of the zigzag, armchair, and chiral SiC nanotubes are dependent on the diameter and chirality. It is indicated that the armchair CNTs appear as polygons while zigzag ones are circles. Both the armchair and zigzag CNTs are direct band gap semiconductors and their band gaps are dependent on tube size and chirality. The calculated dielectric functions of CNTs are dependent on tube size and the directions of polarization. Chapter 7 explains MATLAB applications in behavior analysis of systems consisting of CNTs through molecular dynamics simulation. This chapter presents the applications of MATLAB in calculations for nano-systems, which mostly contain CNTs and consist of five sections. In Section 7.2, the molecular dynamic simulation and analytical studies of nano-oscillators are reviewed. Also, some specialized applications of MATLAB for studying the behavior of nano-oscillators are introduced. In Section 7.3, the authors study the behavior of (8, 2) single-walled CNTs and functionalized CNTs (FCNTs) with four functional groups in water using the molecular dynamic simulation method. Glutamine, as a long chain functional group, and carboxyl, as a short chain functional group, have been used as functional groups in FCNTs. Four functional groups in each FCNT were localized at two positions: (i) all four functional groups were in the sidewalls of the nanotube and (ii) two functional groups were at the ends and two functional groups were in the sidewalls of the nanotube. The intermolecular interaction energies between CNTs or FCNTs and water molecules, the plots of radial distribution function, and the diffusion coefficients of CNTs and FCNTs in water were computed for investigating the effects of type and position of functional groups on the behavior of FCNTs in water. The obtained results from three methods are consistent with each other. Results showed that the position of the functional groups in FCNTs has an important role in the interaction of hydrophilic groups of FCNTs with water molecules. Furthermore, the authors also investigated the behavior of FCNTs with 16 carboxyl functional groups in water. The presence of these large numbers of carboxyl functional groups on CNTs prevents water molecules from moving toward hydrophilic carboxyl functional groups. This well demonstrates the advantage of using a lower number of functional groups, each containing many hydrophilic groups like the glutamine functional group. In Section 7.4, the molecular dynamics simulations were presented to investigate the interfacial binding between the single-walled CNTs and conjugated polymers including
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polythiophene (PT), polypyrrole (PP), poly (2,6-pyridinylenevinylene-co-2,5-dioctyloxy-pphenylenevinylene) (PPyPV), and poly(m-phenylenevinylene-co-2,5-dioctyloxy-p-phenylenevinylene) (PmPV). The intermolecular interaction energy between single-walled CNTs and polymer molecules was computed and the morphology of polymers physisorbed to the surface of nanotubes was investigated by the radius of gyration and the alignment angle. The influence of nanotube radius and temperature on the interfacial adhesion of nanotube–polymer and Rg of polymers was explored in detail. Our simulation results showed that the strongest interaction between the single-walled CNTs and these conjugated polymers was observed first for PT, then for PPy and PmPV, and finally for PPyPV. Our results showed that the intermolecular interaction in our systems is strongly influenced by the specific monomer structure of polymer and nanotube radius, but the influence of temperature could be negligible. The high values of intermolecular interaction energy of such composites suggest that an efficient load transfer exists in the interface between nanotube and heterocyclic conjugated polymer, which plays a key role in composite reinforcement practical applications. In Section 7.5, using molecular dynamics simulations, neon adsorption on an openended (10, 10) single-walled CNT was investigated at supercritical temperatures of neon, that is, T = 50, 70, and 90 K. Adsorption isotherms, heat of adsorption, self-diffusion coefficients, activation energy, and structural properties of neon gas were computed and analyzed in detail. All adsorption isotherms are predicted to be of Langmuir shape type I at this range of temperature. The results showed that the temperature is in direct correlation with adsorption capacity, that is, increasing the temperature causes lower adsorption of neon gas by single-walled CNTs. All aforementioned quantities confirm this fact and are in good agreement with previous experimental and theoretical works. Chapter 8 discusses device and circuit modeling of nano-CMOS. The mobility and saturation velocity are the two important parameters that control the charge transport in a conducting MOSFET channel. The mobility is degraded both by the gate electric field as well as by the channel electric field, the former due to quantum-confinement and the latter due to streamlining of velocity vectors in the intense driving electric field. The saturation velocity in the channel is ballistic, that is, limited to the thermal velocity for nondegenerate carriers and to the Fermi velocity for degenerate carriers in the inversion regime. The calculated current–voltage characteristics of a MOSFET channel in the inversion regime using MATLAB are in agreement with the experimental data on a 45 and 80 nm channel. Additional modeling steps can be implemented in MATLAB environment to simulate drain current versus gate voltage and extract the drain-induced barrier lowering (DIBL), substhreshold swing (SS), and on–off ratio. Voltage transfer curve (VTC) of a complementary transistor (p- and n-type) to extract the DC gain, noise margin, rising time, falling time, and propagation delay via RC delay model are all easily implementable in MATLAB. This could be an easy alternative to HSPICE where the design of a new formulation of a transistor model is done using SPICE’s voltage-controlled current source (VCCS) subcircuit. Moreover, simulation programs are unable to keep pace with the changes that are needed as devices are scaled down to nanometer scale. Chapter 9 covers computational and experimental approaches to cellular and subcellular tracking at the nanoscale. It has been shown that the cell maintains control over its mechanical properties and that these control mechanisms are highly influenced by the mechanical microenvironment. Many elements of cellular control are found in the cytoarchitecture (cytoskeleton, focal adhesions, contractility, and the extracellular matrix). Each of these elements contributes in some way to the transduction of mechanical information into a biochemical signal, which allows the cell to perform the necessary physiological
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functions. Live cell imaging involves noninvasive analysis of dynamical processes using confocal, epi-fluorescence, atomic force, and traction force microscopies, and the use of computational techniques for further quantification of cellular dynamics. In this chapter, we present two MATLAB codes based on particle tracking to compute traction force fields: spatial tracking of cellular structures and cellular deformation. The chapter begins with an overview of the main structural components of the cytoskeletons that play a primary role in the mechanical behavior of eukaryotic cells. This is followed by a description of two types of mechanically tunable substrates available to investigate cellular traction forces and the effect of mechanical forces on cellular behavior. We will also show how confocal and epi-fluorescent microscopy techniques can be used to image cellular responses and provide images that can be analyzed using computational approaches in MATLAB. Chapter 10 discusses computational simulations of nanoindentation and nanoscratch. It describes molecular dynamics (MD) simulation and its uses in investigating the atomistic mechanisms of nanoindentation and nanoscratch processes under different indentation loads, various contact interferences, temperatures, and loading rates. The results showed that when the loads and the loading rates increased, both Young’s modulus and the hardness of the films increased. When the indentation was operating under high temperatures, the thermal softness behavior took place causing a reduction in Young’s modulus. The frictional and indented behavior of a diamond asperity on a diamond plate was carried out using molecular dynamics and experiments. The contact load, contact area, dynamic frictional force, and dynamic frictional coefficient increased as the contact interference increased at a constant loading velocity. The contact and frictional behavior can be evaluated between a rigid smooth hemisphere and a deformable rough flat plane by combining the deformed behavior of the asperity obtained from MD results with the fractal and statistic parameters. The comparison and the discrepancy of simulated results and nanoindentation and scratching experimental results have also been discussed. Chapter 11 discusses the modeling of reversible protein conjugation on nanoscale surface. The protein-coated gold nanoparticles can be used to investigate the conformational change of proteins at a solid–liquid interface. This approach was applied to investigate the conjugation of amyloid β protein (Aβ) over gold nanocolloid in conjunction with a mechanism of fibrillogenesis, which is associated with Alzheimer’s disease. Among tested Aβ proteins and various sizes of gold colloids, only Aβ1–40-coated 20 ± 1 nm gold colloidal nanoparticles exhibited a reversible color change as the pH was externally altered between pH 4 and 10. This size-selective reversibility is an important implication for the initial reversible step reported for the fibrillogenesis of Aβ1–40. When the process was repeated with ovalbumin, all tested sizes of gold colloidal nanoparticles (10, 20, 30, 40, 50, 60, 80, and 100 nm) showed a quasi-reversible color change, implying that the conjugation process is partially dependent on the protein. Temperature-dependent features were observed in the reversibility of Aβ1–40 conjugated to 20, 30, and 40 nm gold colloidal suspension. While Aβ1–40-coated 20 nm gold colloid nanoparticles exhibited a reversible color change under 50°C except for 5°C ± 0.2°C and lower, Aβ1–40-coated 30 and 40 nm colloids exhibited reversible color change when the temperature was lowered to 18°C ± 0.2°C and 6°C ± 0.2°C, respectively. This specific and unique size and temperature dependence in reversible color change strongly suggest that the noncovalent intrinsic intermolecular potential formed between the nanocolloidal surface and each Aβ1–40 monomer conjugated at the surface drives the process. Our series of studies explored conformational changes unique to the surface size of the gold colloidal nanoparticles and opened new insights for proteinassociated nanomaterials.
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Chapter 12 deals with computational technology in nanomedicine. It shows that, at present, nanomedicine is an important branch of nanotechnology. It is a very useful novelty in medicine and holds promise in the diagnosis and treatment of several diseases. The application of computational technology plays an important role in the development of nanomedicine research. Several computer programs can be useful in conducting research in this field. However, an important and easy-to-use program is MATLAB, which can help medical scientists answer complex questions. Simulation of complex biological processes can be easily performed based on MATLAB. The author, therefore, elaborates on how MATLAB can be useful in nanomedicine. Also, a summary of important reports on the application of MATLAB in nanomedicine is given in this chapter. Chapter 13 presents future directions: opportunities and challenges for computational nanotechnology. It shows that the major research objectives in molecular nanotechnology are the design, modeling, and fabrication of molecular microchips, instrumentation, machines, and molecular devices. The design and modeling of molecular machines requires, however, intensive computational efforts. Efficient nanoscale systems algorithms, molecular modeling software, and efficient mathematical computational tools are of primary importance. In this chapter, the technical challenges of mathematical modeling in the nanotechnology arena are highlighted by introducing future nanoscale systems research directions, nanophotonics technological advances, reconfigurable and adaptable biomolecular structures, and novel nanoscience applications that may have a significant technological and societal impact on humanity. Finally, the book concludes with three appendices: Appendix A shows common material and physical constants, with the considerations that the materials constants values varied from one published source to another due to many varieties of most materials and also because conductivity is sensitive to temperature, impurities, and moisture content, as well as the dependence of relative permittivity and permeability on temperature, humidity, and the like; Appendix B shows common symbols and useful mathematical formulas; Appendix C covers basic tutorial on MATLAB. Sarhan M. Musa Editor For MATLAB® and Simulink® product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail:
[email protected] Web: www.mathworks.com
Acknowledgments I would like to express my sincere appreciation and gratitude to all the book’s contributors. Thank you to Brain Gaskin for his wonderful heart and for being a great American neighbor. I would also like to thank Kerry L. Madole for her support and understanding. It is my great pleasure to acknowledge the outstanding help of the team at Taylor & Francis/ CRC Press in preparing this book, especially from Nora Konopka, Kari Budyk, Glenon C. Butler, Jr., and Brittany Gilbert. Thanks also to Arunkumar Aranganathan from SPi Global for his outstanding help. Many thanks to Professor Sohail Anwar for his useful comments and Naomi Fernande for her great suggestions. I would also like to thank Dr. Kendall Harris, my college dean, for his constant support. Finally, the book would never have seen the light of day if not for the constant support, love, and patience of my family.
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Editor Sarhan M. Musa received his PhD in electrical engineering from City University of New York, New York. He is currently an associate professor in the Department of Engineering Technology at Prairie View A&M University, Texas. He has been director of Prairie View Networking Academy, Texas, since 2004. From 2009 to 2010, Dr. Musa was a visiting professor in the Department of Electrical and Computer Engineering and also worked in the Nanoelectronic Systems Laboratory at Rice University, Texas. His research interests include computational methods in nanotechnology, computer communication networks, and numerical modeling of electromagnetic systems. He has published more than 100 papers in peer-reviewed journals and conferences. He currently serves on the editorial board of the Journal of Modern Applied Science. Dr. Musa is a senior member of the Institute of Electrical and Electronics Engineers. He is also a 2010 Boeing Welliver Fellow.
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Contributors Zeinab Al-Rekabi Department of Physics University of Ottawa Ottawa, Ontario, Canada
Shin-Liang Chin Cavendish Laboratory University of Cambridge Cambridge, United Kingdom
Vijay K. Arora Faculty of Electrical Engineering Universiti Teknologi Malaysia Skudai, Malaysia
Orion Ciftja Department of Physics Prairie View A&M University Prairie View, Texas
and Division of Engineering and Physics Wilkes University Wilkes-Barre, Pennsylvania S.M. Ashraf Materials Research Laboratory Department of Chemistry Jamia Millia Islamia New Delhi, India G.-L. Chai State Key Laboratory of Structural Chemistry Fujian Institute of Research on the Structure of Matter Chinese Academy of Sciences Fuzhou, Fujian, China Desmond C.Y. Chek Faculty of Electrical Engineering Universiti Teknologi Malaysia Skudai, Malaysia W.-D. Cheng State Key Laboratory of Structural Chemistry Fujian Institute of Research on the Structure of Matter Chinese Academy of Sciences Fuzhou, Fujian, China
Jason Vaughn Clark School of Electrical and Computer Engineering School of Mechanical Engineering Purdue University West Lafayette, Indiana Rodolfo Cuerno Departamento de Matemáticas Grupo Interdisciplinar de Sistemas Complejos Universidad Carlos III de Madrid Leganés, Spain Stefan Facsko Institute of Ion Beam Physics and Materials Research Helmholtz-Zentrum Dresden-Rossendorf Dresden, Germany Te-Hua Fang Department of Mechanical Engineering National Kaohsiung University of Applied Sciences Kaohsiung, Taiwan Timothy Flack Department of Engineering University of Cambridge Cambridge, United Kingdom xxi
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Masumeh Foroutan Department of Physical Chemistry School of Chemistry College of Science University of Tehran Tehran, Iran George C. Giakos Department of Electrical and Computer Engineering The University of Akron Akron, Ohio Kristina Haase Department of Physics University of Ottawa Ottawa, Ontario, Canada S.-P. Huang State Key Laboratory of Structural Chemistry Fujian Institute of Research on the Structure of Matter Chinese Academy of Sciences Fuzhou, Fujian, China Adrian Keller Institute of Ion Beam Physics and Materials Research Helmholtz-Zentrum Dresden-Rossendorf Dresden, Germany and Interdisciplinary Nanoscience Center Aarhus University Aarhus, Denmark Sepideh Khoee Department of Polymer Chemistry School of Chemistry College of Science University of Tehran Tehran, Iran
Contributors
Richard L. Leask Department of Chemical Engineering McGill University Montreal, Quebec, Canada C.-S. Lin State Key Laboratory of Structural Chemistry Fujian Institute of Research on the Structure of Matter Chinese Academy of Sciences Fuzhou, Fujian, China Jen-Fin Lin Department of Mechanical Engineering National Cheng Kung University Tainan, Taiwan Sarhan M. Musa Department of Engineering Technology Prairie View A&M University Prairie View, Texas Andrew E. Pelling Department of Physics and Department of Biology University of Ottawa Ottawa, Ontario, Canada Ufana Riaz Materials Research Laboratory Department of Chemistry Jamia Millia Islamia New Delhi, India Michael L.P. Tan Faculty of Electrical Engineering Universiti Teknologi Malaysia Skudai, Malaysia and Department of Engineering University of Cambridge Cambridge, United Kingdom
Contributors
Dominique Tremblay Department of Physics University of Ottawa Ottawa, Ontario, Canada Viroj Wiwanitkit Hainan Medical University Haikou, Hainan, China and Wiwanitkit House Bang Khae, Bangkok, Thailand
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Cheng-Da Wu Department of Mechanical Engineering National Kaohsiung University of Applied Sciences Kaohsiung, Taiwan Kazushige Yokoyama Department of Chemistry The State University of New York College at Geneseo Geneseo, New York
1 Introduction to Computational Methods in Nanotechnology Orion Ciftja and Sarhan M. Musa CONTENTS 1.1 Introduction............................................................................................................................1 1.2 Molecular Magnetism ...........................................................................................................4 1.3 Semiconductor Quantum Dots .......................................................................................... 12 1.4 Conclusion ............................................................................................................................22 References........................................................................................................................................ 24
1.1 Introduction Nanoscale science and engineering offer the possibility for revolutionary advances in both fundamental science and technology and may have an impact on our economy and society that is comparable in scale and scope to the transistor electronics and the Internet. At its basic level, nanoscale science is the study of novel phenomena and properties of materials that occur at extremely small length scales, typically on the nanoscale that is the size of atoms and molecules [1]. On a similar path, nanotechnology is the application of nanoscience and engineering methods to produce novel materials and devices [2]. The terms nanostructure, nanoscience, and nanotechnology are currently quite popular in both the scientific and the general press. Nanoscale structures are quite intriguing because generally in the nanometer region, almost all physical and chemical properties of systems become size dependent. For example, although the color of a piece of gold remains golden as it reduces from centimeters to microns, the color changes substantially in the regime of nanometers. Similarly, the melting temperatures of small systems change as they enter the nanoscale region, where the surface energies become comparable to the bulk energies. Because properties at the nanoscale are size dependent, nanoscale science and engineering offer an entirely new design motif for developing advanced materials and new devices. The nanoscience revolution has created an urgent need for a more robust quantitative understanding of matter at the nanoscale by modeling and simulation since the absence of quantitative models that describe newly observed phenomena increasingly limits the progress in the field. While the fundamental techniques of modeling such as density functional theory, quantum and classical Monte Carlo (MC) methods, molecular dynamics, and fast multigrid algorithms are well known, new insights come from the application of these methods in the field of nanoscience. Advances in computer technology and increase in computational capabilities have made possible the modeling and simulation of complex systems with millions of degrees of freedom. However, the full potential of novel 1
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theoretical and modeling tools has not been reached yet. Realizing this potential, however, will require a concerted effort in theoretical and modeling research to overcome some of the computational challenges in the field. For example, although our ability to synthesize and fabricate various nano-building blocks (nanotubes, quantum dots, clusters, and nanoparticles) with novel and useful properties is steadily improving, we lack the ability to incorporate them into complex, functional systems, and we do not fully understand the theoretical intricacies of such systems. There are many theoretical and modeling challenges to overcome in nanoscale studies. Just to mention a few of such challenges: (i) to bridge electronic through macroscopic length scale and timescale, (ii) to determine the transport mechanisms at the nanoscale, (iii) to devise theoretical and simulation approaches to study nano interfaces, (iv) to describe with reasonable accuracy the response of nanoscale structures to external probes such as electric/magnetic fields, and so on. With each fundamental theoretical and computational challenge that must be met in nanoscience come opportunities for research and discovery. New tools and techniques (e.g., scanning tunneling microscopes and atomic force microscopes) are giving us the ability to put atoms and molecules where we want them. Researchers are discovering new properties that emerge at nanometer length scales that are different from the properties of both individual atoms/molecules and bulk materials. Scientists and engineers have successfully synthesized and characterized a broad range of nano-building blocks with novel and potentially useful properties. Thus, nanoscale science and technology raise research issues that are fundamental to a growing number of disciplines and also display the potential for applications of enormous economic and social significance. In recent years, interest in the area of nanotechnology has exploded nationwide with many universities and groups of researchers working in different aspects of nanotechnology. This work has allowed scientists a share in the nanotechnology research and has created strong educational and research programs focusing on nanotechnology and institute interdisciplinary partnerships across field and disciplinary boundaries. Because of the variety and large number of nano-building blocks and structures, it is impossible to cover all of them. Therefore, in this work we will try to give a brief overview of only two of such nanoscale systems, namely, molecular magnets and semiconductor quantum dots. While we will look more closely into the theory and modeling of these two nanoscale systems, we will also briefly address some of the critical research issues in the fast-evolving field of nanomagnetism and nanoelectronics where molecular magnets and quantum dots are the respective building blocks. Clearly, a full understanding of the properties of these two very important characteristics of nano-building blocks is imperative for the whole field of nanoscience for a number of reasons. Just as knowledge of the atom allows us to make and manipulate larger structures, knowledge and accurate modeling of such nano-building blocks will allow us to reliably manufacture larger artificial structures with prescribed properties. This is the centerpiece of new functional nanomechanical, nanoelectronic, and nanomagnetic devices. Molecular magnets and semiconductor quantum dots are well defined by experiment and are tractable using standard theoretical and computational tools. Moreover, they have been demonstrated to hold promise in future technologies. To this effect any complete theoretical and modeling initiative aiming to understand such systems should focus on (i) creating a unique theoretical and modeling expertise, (ii) exploring novel theories and models to predict behavior and reliability of functional nano-size sensors and devices, and (iii) understanding classical transport mechanisms at nanoscales. Last but not least, all these efforts should contribute to educate and train the first generation of truly
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multidisciplinary nanoscientists and engineers for positions of leadership in academia and industry, as well, maximize the benefits to society by stimulating the creation of new technologies and developing innovative solutions. On these lines, there are great potentials for nanotechnology applications in the field of biology and medical sciences since there is a convergence in length scales between inorganic nanostructures and biomolecules such as DNA and proteins. First steps are very promising, for example, nanostructures such as quantum dots are already being used as biosensors. For a long time, magnets have found a wide range of applications in science and technology; however, molecular magnets belong to a field that is still at an early stage of development. Research interest in this field is motivated by the need for a better understanding of the fundamental principles that govern magnetic behavior, in particular, when moving from isolated molecules to bulk solids, as well as the need for new improved magnetic materials. The trend toward devices to the ultimate scale—the molecular one—is a force driving the expansion of this field. Molecular chemistry has already shown the capability to synthesize new magnetic systems in mild conditions (from solutions, at room temperature) with specific magnetic and electronic properties. Among the many advances in the field of nanotechnology, invention of sophisticated experimental tools has made possible the fabrication of various nanoscale semiconductor structures in a precise and controlled way. In such semiconductor devices, electron’s quantum mechanical nature dominates. The payoff of this behavior is that electronic devices built on nanoscale not only can pack more densely on a chip but also can operate far faster than conventional transistors. With the shrinking size of these devices, electrons manifest pronounced quantum behavior and their motion becomes confined with confinement that can happen in one, two, or three dimensions. The great progress achieved on fabricating nanoscale molecular magnets and nanoscale semiconductor quantum dot devices has opened up a new frontier, whose aim is the ultimate miniaturization of such nanoscale systems. The current–voltage characteristics of such atomic and molecular systems hold the promise of revolutionary new devices for ultrasensitive probes and detectors, very high-speed and ultra-large density electronic components, and new electrochemical energy storage devices and the possibilities of novel logic layouts. Indeed, recent demonstrations of molecular-based logic gates and nonvolatile random access memory have already been realized and point to the exciting possibilities of molecular devices, and perhaps even quantum computing. However, from a theoretical point of view, the accurate prediction of the properties of various nanodevices, such as the calculation of current–voltage characteristics from first principles, remains a serious challenge. Clearly, given the increased scientific and technological interest in this field of research, there is a need for new computational methods to investigate the fundamental properties of nanoscale systems. New computational methods definitely will impact further advances in physics, chemistry, and engineering. A better quantitative understanding of the properties of such nanoscale systems at different conditions is critical to achieve a quick commercialization of potentially groundbreaking novel nanoelectronic and nanomagnetic devices. Studies of the above-mentioned two nano-building blocks, molecular magnets and semiconductor quantum dots, offer great promise for the future but will require new theoretical approaches and computationally intensive studies. Current algorithms and numerical methods must be made more efficient and new ones should be invented. Currently, molecular dynamics simulation methods can handle systems with tens of thousands of atoms; however, to fully exploit their power, algorithms need to be made scalable and fully parallelized. Computational methods are especially useful in providing benchmarks for
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Computational Nanotechnology: Modeling and Applications with MATLAB®
nano-building blocks, where experimental data are unreliable or hard to reproduce. Lack of clear prescriptions for obtaining reliable results in nano blocks is another challenging problem for experiment and theory. While new experiments will need to be designed to ensure reproducibility and the validity of the measurements, the theoretical challenge is to construct new theories that would cross-check such conclusions. While great strides have been made in various simulation methods, a number of fundamental issues remain. In particular, the diversity of time and length scales remains a great challenge at the nanoscale. Intrinsic quantum attributes like transport and charge transfer remain a challenge to incorporate into classical description. For example, at best, quantum Monte Carlo (QMC) simulation methods are effective and easy to implement only at zero or very low temperatures. Even though the QMC method and its many variants are currently the most accurate methods that can be extended to systems in the nanoscience range, major improvements are needed. To conclude, there is a compelling effort being made to bring a wide range of multidisciplinary scientists to work together to understand phenomena in nanoscience, utilizing various modeling approaches and tools. One of such powerful tools that can be used successfully to study the basic properties of various nanoscale systems is MATLAB®. Modeling with MATLAB attempts to give a brief and basic understanding of a key and versatile tool that can be used effectively in many computational settings.
1.2 Molecular Magnetism Research into molecular magnetism and the closely related field of molecular electronics is quite new but very active. The interest in the magnetic properties of synthesized molecular clusters [3,4] containing relatively small numbers of paramagnetic ions has only increased in the recent years. With the ability to control the placement of magnetic moments of diverse species within stable molecular structures, one can test basic theories of magnetism and explore the design of novel systems that offer the prospect of useful applications [5,6]. Recent successful efforts in synthesizing solid lattices of weakly coupled molecular clusters containing few strongly interacting spins has opened up the possibility of experimentally studying magnetism at the nanoscale. Due to the presence of organic ligands, which wrap the molecular clusters, the intercluster magnetic interaction is vanishingly small when compared to intracluster interactions; therefore, the properties of the bulk sample reflect the properties of independent individual nanoscale molecular clusters. The magnetic ions in each molecular cluster can be generally arranged in different ways, giving rise to structures of very high symmetry (e.g., rings) and/or of lower symmetry presenting other important features. A common feature of these organic-based molecular magnets is that intermolecular magnetic interactions are extremely weak compared to intramolecular interactions (within individual molecules). As a result, a bulk sample can be described in terms of independent individual molecular magnets. As examples of molecular magnets with ultrasmall numbers of embedded paramagnetic ions, we mention two dimer systems, one [7] consisting of V 4+ ions (spin 1/2) and the other [8] Fe3+ ions (spin 5/2): a nonregular triangular array [9] of Fe3+ ions (spin 5/2), a nearly equilateral triangular array [10] of V 4+ ions, a nearly square array [11] of Nd 3+ (spin 9/2), a regular tetrahedron array [12] of Cr 3+ (spin 3/2), and a non-regular tetrahedron array [13] of Fe3+ (spin 5/2) ions.
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Also noteworthy is the pyrochlore antiferromagnet Tb2Ti2O 7, although distinct from the class of organic molecules yet sharing the feature that the Tb3+ ions (spin 6) reside on a network of very weakly coupled tetrahedra [14]. Early work on molecular magnetic clusters [15] recognized only antiferromagnetic (AF) coupling, but later it has been realized that ferromagnetic (F) coupling is also possible [16]. In many of these clusters, the individual molecules appear to make independent contributions to the bulk magnetic susceptibility, that is, the intermolecular magnetic interactions between different molecular clusters are negligible so that the magnetic properties are solely determined by the intramolecular couplings of spins within a given molecular cluster. A very common choice of paramagnetic ion is Fe3+, which has a large spin (spin 5/2) and for which classical spin formalism is directly applicable, except at sufficiently low temperatures [17–19]. There are indications that even a classical Heisenberg spin model provides a very satisfactory description of a ring of eight Cr 3+ ions (spin 3/2), again except at very low temperatures. The complex known as Fe4 is well described [13] by such a classical spin model. For AF exchange it turns out that the magnetic frustration of this system is a very intricate function of temperature, magnetic field, and the ratio of two coupling constants. In particular, the well-known molecular magnets, Fe8 and Mn12, have attracted considerable interest in recent years. The Fe8 cluster is formed of 8 Fe3+ ions (spin 5/2). The ground state spin is large, S = 10 and the system has competing AF interactions. The eight iron atoms are split in two sets: six with moments parallel to the applied field and two with moments antiparallel to the applied field [20]. The Mn12 cluster is formed of 6 Mn2+ ions (spin 5/2) and 4 Mn3+ ions (spin 2). The ground state spin is again large, S = 12 and competing AF interactions do occur. It is now well understood that magnetization happens only at the Mn sites and not on the ligands. In both cases, the magnetic moment of each molecule sees a barrier characterized by 2S + 1 energy levels and a magneto-crystalline anisotropy. For some values of the applied field, the levels from one side of the barrier “see” the levels from the other side, and the phenomena of quantum tunneling of the magnetization occurs, leading to rapid jumps in the magnetic moment in the sample. Magnetic clusters, such as Fe8 and Mn12, with high spins in the ground state and Ising-type anisotropy show slow relaxation of the magnetization at low temperature, which eventually relaxes with a tunneling mechanism. Because of the relatively large value of spin it turns out that many molecular magnets can be described to very high accuracy [21] at sufficiently high temperatures, by using the classical Heisenberg spin model, with exchange interaction between classical unit vectors. Only for sufficiently low temperatures, one needs to incorporate the quantum character of the spins. This underscores the potential utility of a classical spin treatment in its ability to provide useful results. It is perhaps surprising that even this route is fraught with difficulties. One might expect that the determination of the partition function for a few-spin system would be a relatively simple task. However, to put this matter in perspective, it should be recalled that for a finite open chain of classical spins, which interact with nearestneighbor isotropic Heisenberg exchange, the partition function has been evaluated only in the absence of an external magnetic field [22]. For the related system, where the linear chain is closed so as to form a Heisenberg ring, the calculation of the partition function and related magnetic properties is extremely involved. Exact, unwieldy infinite series expansions of these quantities were successfully derived by Joyce [23] many years ago, but only for zero magnetic fields. For more complicated models where we have at least two different exchange interactions, no solutions can be obtained following Joyce’s method. With the introduction of an external magnetic field the analytic calculation of the partition function has been an intractable problem even for small numbers of interacting spin moments [24,25].
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Computational Nanotechnology: Modeling and Applications with MATLAB®
From the theoretical point of view, there is a great interest to study the exact nature of the ground state of molecular magnets and explain the observed properties given that one of the great attractions of the whole field is the potential to build compounds that behave as classical solid state magnets. This way we would be able to fabricate structures that exhibit completely new properties, such as wide range of magnetic ordering temperatures, so that the relationship between magnetism and temperature can be precisely controlled. The other principal focus is to provide the theoretical underpinnings for designing such structures. The challenging theoretical goal is to obtain a deeper understanding of how electronic and molecular structures (spin density, band structure and anisotropy) relate to macroscopic physical properties (magnetic and electrical). Thus, there is a great need for a much stronger fundamental science base to underpin the developments of novel magnetic materials for useful applications in the future. To reach this point, several fundamental problems are still to be solved, such as understanding the details of molecular forces, the nature of interactions between molecules, the properties of the molecule itself, and the way to bridge systems from a singlemolecule level to the macroscopic world. Of concern is also the interaction between spin carriers. Several properties are relevant here: overlap and orthogonality, spin delocalization, spin polarization effects, which are probably relevant in cases involving mobile electrons, etc. Obviously, the initial task from the perspective of theory and modeling of nanoscale molecular magnetic systems is to construct a spin Hamiltonian from a given set of experimental data. For the sake of simplicity, in this review, we assume that the spin Hamiltonian can be quite arbitrarily written in the form H=
∑
J ijSi ⋅ Sj − μ
∑
B ⋅ Si ,
(1.1)
where S⃗ i and S⃗ j are the respective magnetic spins at site i and j Jij is the coupling term between a pair of spins B⃗ is the applied magnetic field μ is the gyromagnetic factor and in a short hand notation the first term incorporates all types of spin–spin interactions (exchange and magneto-crystalline anisotropy). The relative order of magnitude of various spin Hamiltonian terms should be in agreement with the experimental results. Generally, such type of information can be fairly easily extracted for all previously mentioned cases of molecular magnets. If we claim that we know the Hamiltonian, then the difficult task to carry out is the analytical/computational calculation of the properties of the system using any desirable level of theoretical and/or numerical sophistication: from semiempirical treatments to ab initio all particle computations. Since analyzing even relatively small assemblies of molecules in real time and space involves a large amount of computing power, MC methods, in which the overall properties of a molecular cluster are analyzed by considering just a sample of them, can be used successfully. Despite known shortcomings, modeling efforts of this nature have been successful to shed light and achieve a better understanding of key phenomena in the field of nanoscale magnetism: (i) tuning of the Curie temperatures and/or coercive fields, (ii) understanding bistable systems, where the electronic spin can change direction under influence of external factors such as pressure or
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thermal energy with obvious potential for future molecular electronic switching or display devices, (iii) understanding how magnetism is modified by conducting and superconducting materials and bistable materials where magnetic properties change abruptly, and (iv) understanding the transitional range between quantum and classical systems, just to mention a few. Since our objective is not to give a broad overview of the subject of nanoscale molecular magnetism, in the following, we focus our attention on low nuclearity complexes consisting of clusters of few spins. These structures, while easier to handle, are also ideal prototypes to illustrate the application of various computational and modeling tools. At the same time few-spin molecular magnets are likely to represent the “molecular” nanoscale building bricks for the formation of high-nuclearity larger molecular clusters. Therefore, their characterization is an essential step for broader studies targeting larger magnetic systems. The spin dynamics of these nanoscale magnetic clusters is of particular interest since it can directly be probed by different experimental methods such as nuclear magnetic resonance (NMR) [26]. In view of the importance of knowing the dynamical behavior of spin–spin correlation functions, it is most desirable to find model systems, which can be solved exactly. This way one can test the regimes of validity of various experimental results and theoretical approximation schemes. Among the variety of spin–spin correlation functions, the time-dependent spin autocorrelation function is closely linked with spin dynamics; therefore, it is natural to focus on this quantity. Earlier studies have numerically investigated the time-dependent spin autocorrelation function of many-spin systems, for instance, a classical Heisenberg model with nearest-neighbor exchange interaction between spins [27,28]. The goal of these simulations was the study of the expected power-law decay of the long-time spin autocorrelation function for many-spin systems at infinite temperature [29]. In this work, we focus on the spin dynamics of ultrasmall, nanoscale molecular magnets consisting of coupled classical (high) Heisenberg spins. In particular, we give exact expressions (in integral form) for the time-dependent spin autocorrelation function at arbitrary temperatures for a dimer system of classical (high) spins that interact with both exchange and biquadratic exchange interactions. The mathematical difficulty to solve exactly the equations of motion and to perform the phase-space average for interacting spins makes an exact analytical calculation of the time-dependent spin autocorrelation function very challenging, even for the ultrasmall system considered here. To overcome these mathematical difficulties, we introduce a method, which simplifies the calculation of various quantities through the introduction of suitably chosen auxiliary time-independent variables into an extended phase-space integration. The present analytic results, although derived for the dimer system of spins [30], can provide useful benchmarks for assessing numerical methods that calculate the time-dependent spin dynamics of other more complex highspin magnetic systems. The Hamiltonian of a dimer system of spins with exchange and biquadratic interaction is written as follows: 2 H (t) = JS1(t)S2 (t) + K ⎡S1(t)S2 (t)⎤ , ⎣ ⎦ where J and K represent, respectively, the exchange and biquadratic exchange interactions S⃗ i (t) are time-dependent classical spin vectors of unit length (i = 1, 2)
(1.2)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
The orientation of the classical unit vectors, S⃗ i (t), at a moment of time, t, is specified by polar and azimuthal angles, θi(t) and φi(t), which, respectively, extend from 0 to π and 0 to 2π. The exchange interaction between a pair of spins can be either AF, J = |J| > 0, or F, J = −|J| < 0. The biquadratic exchange interaction, K, can be positive, zero, or negative. At an arbitrary temperature, T, the time-dependent spin autocorrelation function, CT (t) = 〈S⃗ i (0)S⃗ i (t)〉, is evaluated as a phase space average over all possible initial time orientations of the spins:
CT (t) =
∫
dS1(0) dS2 (0)exp [ −βH (0)] Si (0)Si (t)
∫
Z(T )
,
(1.3)
where i = 1 or 2 is a selected spin index dS⃗ i (0) = dθi(0)sin[θi(0)]dφi(0) is the initial time solid angle element appropriate for the ith spin, β = 1/(kBT) kB is Boltzmann’s constant The denominator of Equation 1.3 represents the partition function Z(T ) = dS1(0) dS2 (0)exp [ −βH (0)] ,
∫
∫
(1.4)
where H(0) is the initial time Hamiltonian of the spin system. In order to evaluate the timedependent spin autocorrelation function, we need first to solve the equations of motions for the spins and then perform the angular average over all possible initial time spin orientations in the phase space. The dynamics (equations of motion) of classical spins is determined from d ∂H (t) Si (t) = −Si (t) × , dt ∂Si (t)
(1.5)
where the set of solutions, {S⃗ i (t)}, depends on the initial orientation of the spins, {S⃗ i (0)}. The calculation of CT(t) follows several steps: (i) solve the equations of motion for the spins to obtain S⃗ i (t), (ii) calculate the partition function Z(T), and (iii) compute the integrals appearing in the numerator of Equation 1.3. By applying Equation 1.5 to each spin of the dimer, it is not difficult to note that the total spin, S⃗ (t) = S⃗ 1(t) + S⃗ 2(t), is a constant of motion, S⃗ (t) = S⃗ (0) = S⃗ , and as a result we can rewrite Equation 1.5 as d Si (t) = − ⎡⎣ J + K (S2 − 2)⎤⎦ Si (t) × S , dt
(1.6)
where S⃗ represents the constant total spin. The above differential equations for spins can be exactly solved in a new coordinate system (x′y′z′) in which the constant vector S⃗ lies parallel to the z′ axis. Let us denote (αi, βi) to be the polar and azimuthal angles of spin S⃗ i (0) with respect to the new coordinate system in which the direction of S⃗ defines the z′ (polar) axis. It follows that S cos (α i) = S⃗ i (0)S⃗ . The solution
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of Equation 1.6 for each spin component of S⃗ i (t) depends on the sign of [J + K(S2 − 2)]. Irrespective of the sign of [J + K(S2 − 2)], we find that the quantity S⃗ i (0)S⃗ i (t) is given by the expression Si (0)Si (t) = sin 2 (α i )cos [ ω(S)t ] + cos 2 (α i ), (1.7) where ω(S) = |J + K(S2 − 2)|S denotes a precession frequency and 0 ≤ S = |S⃗ | ≤ 2. Note that S⃗ i (0)S⃗ i (t) does not depend on the ith spin azimuthal angle βi. In as much as the spins are equivalent, without loss of generality, we fix i = 1 and concentrate on the calculation of CT (t) = 〈S⃗ 1(0)S⃗ 1(t)〉. From the definition of the total spin variable, S⃗ = S⃗ 1(0) + S⃗ 2(0), recalling that S cos(α1) = S⃗ 1(0)S⃗ , we easily find that 1 + S⃗ 1(0)S⃗ 2(0) = S cos(α1). Since the product S⃗ 1(0)S⃗ 2(0) is expressible in terms of the total spin as S⃗ 1(0)S⃗ 2(0) = S2/2 − 1, it follows that cos(α 1) = S/2 and depends only on the total spin magnitude. Through these simple mathematical transformations, we obtain ⎛ S2 ⎞ S2 F(t , S) = S1(0)S1(t) = ⎜ 1 − ⎟ cos [ ω(S)t ] + . 4⎠ 4 ⎝
(1.8)
Using a similar procedure, the Hamiltonian can be written in terms of the total spin variable as H (t) = H (0) =
J 2 K (S − 2) + (S 2 − 2)2 , 2 4
(1.9)
and is a constant of motion. By expressing all relevant quantities in terms of the total spin variable, which is a constant of motion, we now apply our calculation method whose success is based on the observation that the values of all multidimensional integrals, for example Z(T), are left unchanged if multiplied by unity, which is written as follows:
∫ ∫ d 3S
d3q exp ⎡iq S − S1(0) − S2 (0) ⎤ = 1. 3 ⎥⎦ ⎢ ⎣ (2π)
(
)
(1.10)
∫
Note that the above identity originates from the well-known formula, d 3Sδ ( 3 ) ⎡S − S1(0) − S2 (0)⎤ = 1, that applies to a three-dimensional (3D) Dirac delta function. ⎣ ⎦ Subsequent calculations are straightforward given that both H(0) and S⃗ 1(0)S⃗ 1(t) appearing in Equation 1.3 can be expressed solely in terms of S. As a result, the integrations over individual spin variables pose no problems. For the partition function, we obtain 2
βK 2 ⎡ βJ ⎤ Z(T ) = ( 4π)2 dSD(S)exp ⎢ − (S 2 − 2) − (S − 2)2 ⎥ , 2 4 ⎣ ⎦ 0
∫
where D(S) = 4πS2
∫
(1.11)
d3 q exp(iqS )(sin q/q)2 can be calculated analytically and is 3 ( 2π ) ⎧S/2 ⎪ D(S) = ⎨S/4 ⎪0 ⎩
0<S<2 S=2 S > 2.
(1.12)
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as a magnetic field. Quantum confinement of electrons is just one of several ways quantum mechanics reveals itself. Another pure quantum phenomena associated with electrons is their spin. Generally, the Bohr radius (size) of quantum dots is much larger than the Bohr radius of real atoms. This implies that ordinary magnetic fields (on the order of 1 T) control magneto-transport properties of quantum dots. An external magnetic field affects both orbital and spin motion of electrons. External control of the full quantum wave function in a semiconductor quantum dot may lead to novel technological application involving both charge and spin. Thus, there is great interest on the properties of electronic materials, such as single and coupled two-dimensional (2D) semiconductor quantum dots in presence of a magnetic field. From the theoretical point of view, a magnetic field applied perpendicular to the 2D dot plane will change the nature of single electron levels from those of the harmonic oscillator to Landau levels. The electronic spectrum will consist of discrete energy levels but will strongly depend on the applied perpendicular magnetic field, which also affects the electron’s spin. The importance of semiconductor quantum dots lies primarily in their tunability and sensitivity to external parameters as electrons are confined in all dimensions. As a result semiconductor quantum dots represent a unique opportunity to study fundamental quantum theories in a tunable atomic-like set-up. From a practical point of view, semiconductor quantum dots have great potentials to realize novel nanoelectronic devices and new research directions on these lines are emerging every time. In this work, we want to give a brief review of some of the theoretical and modeling approaches used in studies of semiconductor quantum dots. We will specifically consider only 2D few-electron semiconductor quantum dots. The main emphasis will be to clarify the relations between different theories and numerical methods used to study the properties of few-electron 2D semiconductor quantum dots in an external tunable parameter, for example, a magnetic field. While the focus will be on isolated 2D semiconductor quantum dots containing few electrons, we point out that all adopted computational methods can be straightforwardly implemented to studies of small arrays of coupled quantum dots in which electrons in different quantum dots are weakly coupled to each other, for instance, as in a double quantum dot [32,33]. In arrays of coupled quantum dots, for instance two laterally coupled quantum dots [34,35], the inter-dot distance between nearest-neighbor quantum dots can be experimentally adjusted. A given energy barrier separates the electrons of one dot from tunneling to the other dot. A two-well potential (with finite range) can be used to model the coupling of any two harmonic wells centered at a given distance apart. It is important to remark that the study of weakly coupled arrays of semiconductor quantum dots is a subject of great fundamental and practical interest since the magneto-transport properties of arrays of quantum dots can be dramatically different from the properties of the individual quantum dots that constitute them. While each individual quantum dot in the system still retains the set of well-defined charge states like in an atom, tunneling of electrons between two quantum dots may fundamentally change the individual properties of each of the quantum dots including charging effects or Coulomb oscillations. In such coupled systems, we may expect the electrons to reach a state similar to the “resonant tunneling condition” state and thus freely transmit between the two quantum dots. A properly tuned external field may then dramatically affect the behavior of the system, and novel magneto-transport features are expected to arise in weakly coupled arrays of quantum dots. Under these circumstances, the interplay between quantum confinement, electronic correlation, inter-dot coupling, and magnetic field will be more subtle, and we expect to see many interesting physical phenomena with possible technological applications. Quantum chemistry
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methods of molecular physics, which rely on knowledge of individual quantum dots, can be used. While the study of arrays of coupled quantum dots is a difficult problem, the analogy between real atoms and quantum dots (“artificial atoms”) provides us with a powerful set of methods from molecular physics such as the valence bond Heitler–London (HL) approach or the linear combination of atomic orbitals (LCAO) approach. Since knowledge of a single isolated quantum dot is a key ingredient of all other more elaborate treatments, we will focus our attention only on isolated 2D semiconductor quantum dots containing a few confined electrons. Such nanoscale systems are a topic of intensive ongoing research since they represent a whole new class of semiconductor devices with promising properties to meet the new technological challenges of the twenty-first century [36–40]. 2D semiconductor quantum dots are expected to provide the basis for future generations of device technologies such as threshold-less lasers and ultradense memories. Generally speaking, 2D quantum dots can be viewed as artificial atoms. They hold only a few electrons in contrast to standard semiconductor devices; therefore, they represent the ultimate limit of the semiconductor device scaling. As the device sizes are reduced, the number of carriers involved in the operation of a single device is reduced as well. In fact, state-of-the-art semiconductor structures will soon be plagued by dopant fluctuation and particle noise problems. Quantum dot device concepts utilize the discreteness of the electron charge and they offer a possible breakthrough in device and circuit technology representing the ultimate limit of the semiconductor device scaling. The physics of 2D semiconductor dots [41–47] shows many parallels with the behavior of naturally occurring quantum systems in atomic and nuclear physics. The difference is that in 2D quantum dots the confinement of electrons is due to an artificially created potential formed by the electrodes connected to layers of semiconductor, while in atomic systems, electrons are confined by the Coulomb attraction of the nucleus. As in an atom, the energy levels in a 2D semiconductor quantum dot become quantized due to the confinement of electrons. Unlike atoms, however, 2D semiconductor quantum dots can be easily connected to electrodes and are therefore excellent tools to study atomic-like properties in a controllable way. There is a wealth of interesting phenomena that have been seen in 2D quantum dot structures over the past decade. Initial studies of 2D quantum dots focused on their atomic-like properties [48–50] at low magnetic fields. In particular, the capacitance spectroscopy of 2D quantum dots in the low-magnetic-field regime shows that the ground states of parabolic 2D quantum dots exhibit shell structure and are believed to obey Hund’s first rule [48]. The shell structure is particularly evident in measurements of the change in electrochemical potential due to the addition of one extra electron. In this regime, parabolic 2D quantum dots exhibit shell structure manifesting their atomic-like nature. However, with the strengthening of the magnetic field, the structure of singleelectron levels changes to highly degenerate Landau levels and the electron correlations play a very important role, similar to the fractional quantum Hall effect (FQHE) case seen in bulk 2D electronic systems (2DES) [51–53]. Transport properties of 2D quantum dots in this regime are highly interesting because of possible applications of the single-electron tunneling in electronic devices [54] as well as use of such structures for quantum computation [55,56]. A standard theoretical model for 2D semiconductor quantum dots involves a number of approximations. The most common approximations are (i) to consider the motion of electrons exactly 2D, (ii) to consider a simplified confining potential (generally parabolic), and (iii) to consider the interaction potential between electrons of bare Coulomb form. A magnetic field perpendicular to the plane of the 2D quantum dot acts as an external tuning parameter for the confined system of electrons. The main effect of such a magnetic field, as
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it becomes stronger, is to change the nature of single-electron levels from those for the 2D harmonic oscillator to Landau levels. If the magnetic field is very strong, the energy levels with different angular momentum become degenerate or quasi-degenerate. In this regime, the electron correlations play a very important role, similar to what is seen in the FQHE in bulk 2D electron systems in high magnetic fields [57,58]. The properties of N-electron 2D quantum dots subject to a magnetic field, B⃗ = (0, 0, Bz), perpendicular to the dot plane are generally calculated by considering the following Hamiltonian: ˆ = H
N
∑ i 1
2 ⎧ 1 ⎡ ˆ ⎤ + V (r i )⎫⎬ + 1 p i + eA( ri ) ⎨ ⎢ ⎣ ⎦⎥ 2 m ⎩ ⎭ 4πε 0 ε r
N
∑ i> j
e2 + gμ BBzSz , ri − rj
(1.15)
where the first term is one-electron term the second term is the Coulomb potential energy the last term is the Zeeman energy V(r⃗ ) is the one-electron confinement potential The vector potential in a symmetric gauge is written as follows: B A(r ) = z (− y , x , 0), 2
(1.16)
where r⃗ = (x, y) is the 2D position vector −e (e > 0) is electron’s charge m is electron’s mass g is electron’s g-factor μB is Bohr’s magneton εr is the dielectric constant Sz is the z-component of the total spin To obtain the many-electron energy spectrum and wavefunctions one must solve the stationary Schrödinger equation for the Hamiltonian given earlier: ˆ Ψ(r 1 , … , r N ) = EΨ(r 1 , … , r N ). H
(1.17)
Clearly, this is a formidable task and this quantum problem cannot be solved exactly even for systems of as few as N = 2 electrons. The case of few-electron quantum dots [59–61] is of particular interest, since single-electron confinement energy, the cyclotron energy for ordinary magnetic fields, and electron–electron correlation are all on the same order of magnitude. As a result, a rich physics and a variety of complicated phenomena are manifested. For simplicity, we assume an isotropic parabolic confinement potential for the electrons, V (r ) = where ħω0 is the parabolic confinement energy r2 = x2 +y2
m 2 2 ω0r , 2
(1.18)
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One can write the parabolic confinement potential in dimensionless units for energy (potential) and length (distance) as follows: V (r ) 1 = (αr )2 , ω 0 2
(1.19)
where α=
mω 0 ,
(1.20)
is the inverse quantum oscillator length. A plot of the dimensionless potential, V(r)/ħω0, as a function of the dimensionless distance, αr, is given in Figure 1.5. The assumption of parabolic confinement is a common choice for the electron’s confinement in a 2D semiconductor quantum dot [62–65]. The parabolic confinement model explains reasonably well some of the main features associated with most common quantum dots. That said, it is important to remark that many other more realistic confinement potentials have also been considered such as anisotropic potentials [66–68] and/or non-parabolic ones [69–71]. If we neglect the Coulomb interaction between electrons, the Hamiltonian in Equation 1.15 (without the Zeeman term) is reduced to a sum of single particle Hamiltonians, H0(r⃗ ) written as follows: 2 ˆ 0 (r ) = 1 ⎡ p ˆ + eA(r )⎤ + m ω 02r 2 , H ⎦⎥ 2m ⎣⎢ 2
(1.21)
where ħω0 is the strength of the parabolic confinement potential. The quantum problem for such a Hamiltonian can be solved exactly and the eigenstates are routinely called the Fock–Darwin (FD) states since Fock [72] and Darwin [73] were the first to investigate this problem. The FD states are written as follows: ⎛ r2 ⎞ ⎛ r ⎞ Ψnmz (r , ϕ) = N nmz exp ⎜ − 2 ⎟ ⎜ ⎝ 4lΩ ⎠ ⎝ 2lΩ ⎟⎠
mz
⎛ r 2 ⎞ exp( −imzϕ) Lnmz ⎜ 2 ⎟ × , ⎝ 2lΩ ⎠ 2π
(1.22)
Dimensionless V ( r)/ћω0
10
FIGURE 1.5 Dimensionless 2D parabolic confining potential, V(r)/ħω0 as a function of the dimensionless distance, αr.
8 6 4 2 0 –6
–4
–2
0
2
Dimensionless length αr
4
6
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where is an effective magnetic length lΩ = 2mΩ n = 0, 1, … is the radial quantum number mz = 0, ±1, … is the z-angular momentum quantum number m Ln z ( z) are associated Laguerre’s polynomials [74] n! . The frequency Ω appearing in the lΩ2 (n +|mz|)! expression for the effective magnetic length is defined as follows: The normalization constant is, N nmz =
Ω 2 = ω 02 +
ω c2 , 4
(1.23)
where ωc = eBz/m is the well-known cyclotron frequency. The discrete allowed energies (without the Zeeman energy) are written as follows: Enmz = Ω ( 2n + 1 + mz ) −
ω c mz , 2
(1.24)
and depend on the magnetic field by means of the dependence of ωc on the magnetic field. The energy spectrum of few FD states in dimensionless units is plotted in Figure 1.6, which shows the dependence of ε = Enmz /( ω 0 ) as a function of the dimensionless magnetic field parameter, ωc/ω0. The energy expression in Equation 1.24 predicts that energy levels with positive mz shift downward and levels with negative mz shift upward as magnetic field increases. Thus, electrons may undergo transitions in their quantum numbers as a function of the magnetic field. When ω0 = 0 or in the limit of very large magnetic fields (ωc >> ω0), the effective magnetic length, lΩ becomes the electronic magnetic length: lΩ →
= mω c
; eBz
ωc → ∞. ω0
(1.25)
10
Energy (ε)
8 6 4 2 0
0.5
1
1.5
2 ωc/ω0
2.5
3
3.5
4
FIGURE 1.6 Single-particle energy levels, ε = Enmz / ( ω 0 ), as a function of the magnetic field parameter, ωc/ω0.
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If we assume ω0 ≠ 0, the effective magnetic length, lΩ can also be written as follows: 2
1 1⎛ω ⎞ = 2α 2 1 + ⎜ c ⎟ ; 4 ⎝ ω0 ⎠ lΩ2
ω 0 ≠ 0,
(1.26)
mω0 , is the inverse quantum oscillator length. Within this frame work, it is straightforward to obtain the 2D harmonic oscillator states starting from the FD states as the magnetic field approaches zero (ωc/ω0 → 0). As evidenced in Figure 1.6, the full energy spectrum of FD states is quite complicated by the presence of the magnetic field, which plays a very important role. In fact, it is the interplay and competition between confinement and magnetic field in semiconductor 2D quantum dots that creates a rich ground for novel phenomena and novel electronic phases generally not seen in naturally occurring atoms [75–77]. There are two dimensionless parameters that determine the behavior of 2D quantum dots in a magnetic field. One is the dimensionless Coulomb correlation parameter: where parameter, α =
λ=
e 2α , ( 4πε 0 ε r ω 0 )
(1.27)
and the other is the dimensionless magnetic field parameter: γ=
ωc . ω0
(1.28)
As clearly seen, the dimensionless parameter, λ, gauges the strength of the Coulomb correlation energy relative to the confinement energy while the other dimensionless parameter, γ, gauges the strength of the magnetic field relative to the confinement energy. Thus, one way to classify the various quantum regimes that determine the properties of the confined electrons in a 2D quantum dot is to look into the relative strengths of these two dimensionless parameters. The simplest, yet nontrivial, 2D quantum dot system consists of two electrons (N = 2) confined in a parabolic confinement potential and is often referred to as 2D quantum dot helium. Despite its small number of electrons, this 2D semiconductor quantum dot shows many characteristic features that mimic larger systems. In particular, the ground state energetics as a function of the magnetic field is highly nontrivial. With the increase of magnetic field, singlet to triplet spin state transitions have been theoretically predicted [78] and experimentally observed [79]. There have been several studies of few-electron 2D quantum dots with or without magnetic field employing a variety of techniques such as exact numerical diagonalization [80,81], Hartree–Fock (HF) theory [82–86], density functional theory [87–90], QMC methods [91–94], analytical treatments [95–98] as well as ab initio calculations [99,100]. Merkt et al. [80] studied the energy spectra of two interacting electrons in a parabolic potential in the absence and presence of a perpendicular magnetic field employing the exact numerical diagonalization technique. Wagner et al. [78] used the exact numerical diagonalization technique to study 2D quantum dot helium in a perpendicular magnetic field. They predicted oscillations between spin-singlet and spin-triplet ground states as a function of the magnetic field strength. They also proposed spin susceptibility or magnetization experiments at low temperature to verify this behavior. Ciftja et al. [101,102] performed exact numerical diagonalizations and studied 2D quantum dot helium in a perpendicular magnetic field and obtained the energy spectrum as well as other quantities of interest.
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Pfannkuche et al. [64] compared energies, pair correlation functions, and particle densities of the ground state of 2D quantum dot helium in a magnetic field obtained by Hartree, HF, and exact numerical diagonalization method. They found that the results of the Hartree approximation show strong deviations from exact numerical diagonalization results. They also found that the HF results for the singlet state differ markedly from the exact numerical diagonalization results; however, the HF results for the triplet state are in very good agreement with the exact results. Similarly, Harju et al. [103] studied 2D quantum dot helium in the presence of a perpendicular magnetic field using the variational theory and applying the variational Monte Carlo (VMC) simulation technique. A simple form of a trial wave function that gives results in good agreement with exact numerical diagonalization was presented. They also introduced a recipe of how to build a trial wave function, including a Jastrow function that takes into account the mixing of different Landau levels for the relative motion and as a result incorporates part of the electronic correlations. Since our main objective is not to give a definitive overview of all computational methods used in the field, in the following, we focus only on the so-called exact numerical diagonalization method that generally works fine for systems of few particles. The computational challenges and the increasingly large amount of computer power limit the application of this method to larger systems of particles. Exact numerical diagonalization results are also very important since they are a definitive benchmark to gauge the accuracy of other approximative methods and numerical recipes [104,105]. Thus, in the following, we give a brief general description of the exact numerical diagonalization method. To illustrate the numerical procedure of how to implement the exact numerical diagonalization method, let us consider the following general quantum eigenvalue equation: ˆ ψ =E ψ , H
(1.29)
ˆ is some Hamiltonian that does not allow an exact analytic solution of the problem. where H To search for a solution of the above problem, we expand the (unknown) wave function |ψ〉 as a linear combination of a certain number Nmax of linearly independent functions, |ϕ1〉, |ϕ2〉, …, |ϕN〉, which form the basis set: Nmax
ψ =
∑c
i
φi ,
(1.30)
i 1
where the constants ci are unknown. For simplicity, we can consider the functions |ϕi 〉 to be ortho-normalized, for example, exact ortho-normalized eigenstates of a simpler ˆ so that 〈ϕ |ϕ 〉 = δ . Hamiltonian H 0 i j ij Substituting Equation 1.30 into Equation 1.29 and performing the inner product of the resulting equation with bra-s, φ1 , φ2 , … , φNmax gives a system of N linear equations relative to the variables c1 , c2 , … , cNmax . Such system of equations can be written in matrix form as follows ⎛ H11 ⎜ H 21 ⎜ ⎜… ⎜… ⎜ ⎜⎝ H N 1 max
H12 H 22 … … …
… … … … …
… … … … …
H1Nmax
… H Nmax Nmax
⎞ ⎛ c1 ⎟ ⎜ c2 ⎟⎜ ⎟ ⎜ c3 ⎟ ⎜… ⎟⎜ ⎟⎠ ⎜⎝ cN
⎞ ⎛ c1 ⎞ ⎟ ⎜ c2 ⎟ ⎟ ⎟ ⎜ ⎟ = E ⎜ c3 ⎟ . ⎟ ⎜… ⎟ ⎟ ⎟ ⎜ ⎟ ⎜⎝ cN ⎟⎠ max ⎠ max
(1.31)
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where ˆ φj , H ij = φi H
(1.32)
denotes the so-called Hamiltonian matrix. This equation represents a matrix eigenvalue equation where the column of ci-s is the matrix eigenstate to be determined and E-s are the matrix eigenvalues. The essence is that the quantum problem of finding the exact energy eigenvalues of the stationary Schrödinger equation is now reduced to the numerical diagonalization of the resulting Hamiltonian matrix. At this phase, standard computational procedures and software tools such as MATLAB can be straightforwardly used. The only numerical error of the procedure originates from the truncation of the basis since in principle one has to choose a finite Nmax × Nmax matrix to diagonalize, where Nmax is the dimensionality of the truncated (finite) basis set. The diagonalization of such a Hamiltonian matrix will produce a set of energy eigenvalues and expansion coefficients (from where the eigenfunctions can be obtained). The energy eigenvalues obtained for a series of finite values of Nmax can be extrapolated to the Nmax → ∞ limit. When large basis sets are considered, a linear fit (in 1/Nmax) function best fits the data. The values extrapolated in the Nmax → ∞ limit represent the exact numerical solution of the energy eigenvalue problem (ground and excited state energies). A similar approach works very well for few-electron 2D semiconductor quantum dots of the nature considered here. In particular, if we focus on a 2D semiconductor quantum dot with only N = 2 electrons the process is rather streamlined. In this case, one can achieve a further degree of simplification by separating the Hamiltonian into two parts, one representing the center-of-mass (CM) motion and the other one representing the relative motion. The CM and relative coordinates/moments are defined as follows: r + r ˆ pˆ − pˆ 2 R = 1 2 ; P = pˆ 1 + pˆ 2 ; r = r1 − r2 ; pˆ = 1 . 2 2
(1.33)
Accordingly, the total mass of the system is M = 2m and the reduced mass is μ = m/2. In this representation, the Hamiltonian of the system of N = 2 electrons decouples and can be written as follows: ˆ (R, r ) = H ˆ R ( R) + H ˆ r (r ), H
(1.34)
ˆ (R ˆ ⃗ ) are, respectively, the CM and relative motion Hamiltonians given by ⃗ where H R ) and Hr( r ˆ2 ˆ R (R) = P + ω c Lˆ z + M Ω 2R2 , H 2M 2 2
(1.35)
2 2 ˆ r (r ) = pˆ + ω c lˆz + μ Ω 2r 2 + 1 e , H 2μ 2 2 4πε 0 ε r r
(1.36)
and
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where Lˆ z and ˆl z are, respectively, the CM and relative angular momentum operators. The CM wave function is known and so are the CM eigenenergies: ECM = Ω ( 2nCM + 1 + mCM ) −
ω c mCM , 2
(1.37)
where nCM = 0, 1, … and mCM = 0, ±1, … When looking for the ground state energy of the N = 2 semiconductor quantum dot system, one restricts the CM energy to the lowest energy level, ħΩ that corresponds to nCM = 0 and mCM = 0 and thus has to deal only with the relative motion Hamiltonian problem. If the electrons were not affected by the Coulomb interaction, the relative wave function would be of the same form as Equation 1.22 with r⃗ representing the relative coordinate, mass replaced by the reduced mass, and lΩ modified accordingly. As in any standard numerical diagonalization technique, we need to calculate the matrix elements of the Hamiltonian on the following basis: |nCMmCM; nmz 〉. The only nondiagonal terms arise from the Coulomb potential, which is diagonal with respect to nCM, mCM, and mz, but not n. As a result, the most general nonzero Hamiltonian matrix elements have the form nCM mCM ; nʹ mz|Hˆ |nCM mCM ; nmz = nCM mCM|Hˆ R| nCM mCM δ n ʹn + nʹ mz|Hˆ r|nmz .
(1.38)
For a given nCM, mCM, and mz, we have nCM mCM |Hˆ R|nCM mCM Ω = (2nCM + 1 + mCM ) − 12 ωω0c mCM . ω 0 ω0
(1.39)
Similarly, one can easily set up the Hamiltonian matrix for the other term, ˆ r nmz ( ω 0 ) = hn ʹn . The exact diagonalization procedure is then applied to solve nʹ mZ H the problem for various values of the dimensionless parameters, λ and γ. For any value of the quantum number, mz = 0, ±1, …, we build sufficiently large matrices with elements, hn’n and diagonalize them by using standard numerical packages (such as MATLAB). The smallest of the eigenvalues represents the ground state energy for the relative Hamiltonian. With the addition of the CM energy to the ground state energy of the relative motion we obtain the exact numerical diagonalization value of the total ground state energy of the system. Some of the energy results obtained are shown in Table 1.1. The values represent the exact numerical diagonalization ground state energies, ε = E/(ħω0) for the 2D semiconductor quantum dot system (N = 2) in a perpendicular magnetic field, for values of Coulomb correlation parameter, λ = e2α/(4πε0εr ħω0) = 0, 1, …, and values of magnetic field parameter, γ = ωc/ω0 = 0, 1, …, 6. The ground state angular momentum, mz is also specified. As expected, when ħω0 is kept constant, the ground state energy always shifts upward as magnetic field increases.
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TABLE 1.1 Exact Numerical Diagonalization Ground State Energies, ε = E/(ħω0), for a 2D Semiconductor Quantum Dot (N = 2) Subject to a Perpendicular Magnetic Field as a Function of Dimensionless Coulomb Coupling Parameter, λ = e2α/ (4πε0εrħω0) = 0, 1, … and Values of Magnetic Field, γ = ωc/ω0 = 0, 1, …. λ=0 mz λ=1 mz λ=2 mz λ=3 mz λ=4 mz λ=5 mz λ=6 mz
γ=0
γ=1
γ=2
γ=3
2.00000 0 3.00097 0 3.72143 0 4.31872 0 4.84780 0 5.33224 0 5.78429 0
2.23607 0 3.30508 0 4.06684 1 4.60594 1 5.11165 1 5.58995 1 6.04534 1
2.82843 0 3.95732 1 4.61879 1 5.23689 1 5.73642 2 6.21499 2 6.67999 2
3.60555 0 4.71894 1 5.43123 2 6.01256 2 6.53522 3 7.01716 3 7.46782 4
γ=4 4.47214 0 5.61430 1 6.30766 2 6.89002 3 7.41600 4 7.90109 4 8.34530 5
γ=5 5.38516 0 6.53067 2 7.22681 3 7.81384 4 8.33874 5 8.82281 6 9.27057 6
γ=6 6.32456 0 7.47400 2 8.17960 4 8.76220 5 9.28511 6 9.76657 7 10.21735 8
Note: The angular momentum quantum number, mz, of the ground state is also specified. The parameter α = mω 0/ has the dimensionality of an inverse length.
1.4 Conclusion Many great advances and a large number of experimental discoveries in the field of nanoscale devices and nanotechnology call for development of new theoretical concepts, models, and computational methods not only to study the properties of available systems but also to anticipate the properties of future nanoscale materials and devices that undoubtfully will be manufactured in the incoming years. Nanoscale molecular magnets and nanoscale 2D semiconductor quantum dots exemplify the physics of nanoscale systems and some of the challenging theoretical and computational problems faced in the field. Nanostructures of this nature are of great scientific and technological interest. They also represent a tremendous computational/simulation challenge to tackle for obtaining accurate numerical results. Nanoscale size critically impacts the properties of such systems. In this transitional regime between nano/mesoscopic and bulk regimes it is very hard to predict the properties of various systems and this poses one of the obstacles to overcome. By its very nature, the study of nanoscale systems, such as molecular magnets and semiconductor quantum dots, involves multiple length and scales as well as the combination of theories and modeling approaches that have been traditionally studied separately. This means that fundamental models and computational methods that were developed in separate contexts will have to be combined and eventually new ones invented. In this work, we try to give an introductory review of some of the properties of two prototype nanoscale systems, namely, few-particle molecular magnets and 2D semiconductor quantum dots. For brevity of treatment, we omit from our overview many other nanoscale structures as well as other phenomena that demand a more specialized treatment. Our objective is not
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23
to give a comprehensive review of the vast experimental and theoretical literature published over the last decade. The main emphasis is to clarify some physical properties of nanoscale molecular magnets and 2D semiconductor quantum dots from the modeling perspective. Clearly the interplay between small size, quantum confinement, correlation effects, external tuning parameters, etc., makes the properties of such systems different from extended bulk materials. We, intentionally, leave out many other subtle quantum effects that develop in such structures [106–112]. This was done with the only purpose of focusing the attention on the most basic general features without specifying many other details. With this goal in mind, we identify two simple nano-building blocks and focus on their properties. These two nanoscale prototypes represent the much larger family of molecular magnets and 2D semiconductor quantum dots. For the case studies under consideration, we gradually develop the theoretical background for the analysis of the problem, introduce the different quantum regimes and length scales that apply to such systems, and, finally, we focus on specific calculation methods to derive results of interest. With this approach in mind, we mention some computational methods using MATLAB that can be applied to calculate various quantities pertaining to such systems. We then attempt to illustrate key theoretical properties of molecular magnets and 2D semiconductor quantum dots in a pedagogical way by focusing on simple systems. When considering few-electron 2D semiconductor quantum dots, the main focus is on the exact numerical diagonalization method. Thus, we give a slightly more detailed description of the exact numerical diagonalization technique, while being very brief on all other computational methods including the QMC method [113–119]. Given the power of QMC methods, few words are in order regarding two of the most known of such methods, namely, VMC and diffusion Monte Carlo (DMC). These methods now provide a nearly exact description of the electronic structure of a variety of systems ranging from atoms/ molecules to strongly correlated ES with hundreds and thousands of particles. The VMC method, despite being approximate, is robust and yields very accurate results at a computational cost that grows relatively modestly with the number of particles and the statistical error can be made very small. The variational approach gives an insight into the physical processes related to the behavior of 2D semiconductor quantum dots in a magnetic field, provides very accurate results for a number of fundamental properties, and can be applied to study systems with a large number of electrons. The DMC simulation method [120–123] is computationally more demanding, but provides a “quasi-exact” solution of the quantum many-body problem within the fixed-node [124], released-node [125], fixed-phase [126], and/or released-phase approximations [127]. In this work, we have attempted to present an introductory review of the physics of small molecular magnets and few-electron 2D semiconductor quantum dots with particular focus on confinement effects, energy spectra, and modeling methods. We introduce two prototype nanoscale building blocks that serve as key elements to understand more general nanostructures with an arbitrary number of units. By developing an initial conceptual understanding of these nanoscale building blocks through modeling, we pave the way for more general methods that can be used to determine directly the magnitude, size, shape, and orientation of larger nanostructures that ultimately determine their optical, electronic, and vibrational properties. A complete theoretical understanding of the interplay between all these factors will put us in a position to attack a number of important fundamental problems that have already been identified and no doubt some that have not yet been recognized. It will also help us to develop computational methods that would directly determine the conditions under which various nanomagnetic and nanoelectronic devices can operate with the desired selectivity and sensitivity.
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Most of the results shown in figures and tables are our own calculations. A lot of work has been done over the last decade and obviously many phenomena shown here have been studied by other researchers. While we have tried to give credit to earlier work, certainly and inadvertently we might have overlooked other valuable work on the topic. While we have attempted to illustrate some relevant phenomena present in small molecular magnets and 2D semiconductor quantum dots, the intricate interplay between size, confinement, electronic correlations, magnetic field, spin effects, and low dimensionality certainly provides many other surprises that we did not discuss in this work. The same applies to many other interesting aspects of the physics of molecular magnets and 2D semiconductor quantum dots that we had to leave out of this work for brevity of treatment.
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Introduction to Computational Methods in Nanotechnology
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2 Computational Modeling of Nanoparticles Ufana Riaz and S.M. Ashraf CONTENTS 2.1 Introduction.......................................................................................................................... 29 2.2 Benefits of Computer Science for Nanotechnology ........................................................30 2.3 Modeling at Different Scales .............................................................................................. 31 2.3.1 Electronic Scale ........................................................................................................ 32 2.3.2 Atomistic Scale ......................................................................................................... 32 2.3.3 Mesoscale .................................................................................................................. 32 2.3.4 Continuum Scale...................................................................................................... 33 2.4 Concept of Computational Modeling of Nanostructures..............................................34 2.5 Computational Control of Matter through Modeling .................................................... 36 2.5.1 Empirical and Ab Initio Potentials........................................................................ 36 2.5.2 Molecular Dynamics Simulation........................................................................... 38 2.5.3 Monte Carlo Simulation .......................................................................................... 39 2.5.4 Advantages of Monte Carlo and Molecular Dynamics Simulation Techniques............................................................................................42 2.5.5 Limitations of Monte Carlo and Molecular Dynamics Simulation Techniques............................................................................................42 2.6 Modeling of Electronic Transport of Nanoparticles.......................................................43 2.7 Modeling of Mechanical Properties of Nanoparticles ................................................... 49 2.8 Modeling of Optical Properties of Nanoparticles........................................................... 55 2.9 Modeling of Bionanoparticles............................................................................................ 57 2.10 Modeling of Polymer Nanocomposites ............................................................................64 2.11 Opportunities and Challenges in Computer Modeling of Nanoparticles .................. 68 2.12 Conclusion ............................................................................................................................ 70 Acknowledgment.......................................................................................................................... 71 References........................................................................................................................................ 71
2.1 Introduction The role of nanoparticles, whether for improved materials, semiconductor fabrication, pharmaceuticals, environmental assessment, or evaluation of global climatic, depends on their chemistry as well as their physical characteristics [1,2]. The unique functionality of nanoparticle-based materials and devices depends directly on size- and structure-dependent properties. In industrial applications, often, chemistry is key to elucidating sources and formation mechanisms of nanoparticles. Physical characterization of nanoparticles is critical to the advancement of the underlying science and the development of practical nanotechnologies. Nanoparticle size must be tightly controlled to take full advantage of 29
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Computational Nanotechnology: Modeling and Applications with MATLAB®
quantum size effects in photonic applications, and agglomeration must be prevented [3]. Agglomeration can only be prevented if the number concentrations is tightly controlled, which requires that the rate of new particle formation be quantitatively determined. Realtime measurements of particle size distributions and particle structure are, thus, enabling technologies for the advancement of nanotechnology [4]. Key areas of research to improve our physical characterization capabilities include 1. 2. 3. 4. 5. 6. 7. 8. 9.
Rapid nanoparticle measurements Detection, characterization, and behavior in the low nanometer (<5 nm) size regime Particle standards for size, concentration, morphology, and structure Charging behavior and technology throughout the ultrafine and nanoparticle size regimes Distributed nanoparticle measurements Integral parameter measurement Off-line morphological, structural, and chemical characterization of nanoparticles Online morphological and structural characterization of nanoparticles Nanoparticle size distribution measurements with high detection efficiencies
Accurate and reliable models for simulating transport, deposition, coagulation, and dispersion of nanoparticles and their aggregates are needed for the development of design tools for technological nanoparticle applications, including • Nanoparticle instrumentation including sampling, sensing, dilution, focusing, and mixing • Nanoparticle behavior in complex geometry passages including the human respiratory tract, aerosol transport/delivery systems, energy systems, etc. • Nanoparticle behavior in chemically reactive systems including combustors and aerosol reactors • Development of efficient numerical schemes for simulating nanoparticle transport and deposition
2.2 Benefits of Computer Science for Nanotechnology Computation uses physical processes to realize abstract processes. For example, in manual computation, the beads on an abacus or the scales of a slide rule are manipulated to realize abstract mathematical operations such as addition and multiplication [5]. In analog computers, electrical processes realize a system of differential equations, and in digital computers, electrical switches implement binary logic in order to process numbers, matrices, characters, sequences, trees, and other abstract objects. What distinguishes these physical processes as computation is that their purpose is to realize an abstract (mathematical) process, which is in principle multiply realizable, that is, realizable by any physical process that has the same abstract structure. The idea of a general-purpose computation is especially intriguing, since it implies that a wide variety of physical processes could be programmed
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Computational Nanotechnology: Modeling and Applications with MATLAB®
containing a number of atoms; and (d) the continuum level, in which matter is regarded as a continuum, and the well-known macroscopic laws (equations of continuity and momentum conservation, constitutive equations such as Fourier’s law for heat flow, etc.) apply. 2.3.1 Electronic Scale The ab initio methods of description are the most rigorous and require no experimental input other than knowledge of the atomic species involved [7]. Such electronic level methods are required when significant rearrangement of electrons or other fundamental particles occur, such as in conductors and semiconductors, in photonic devices, or in chemical reactions where chemical bonds are broken and formed [8]. However, electronic structure methods are computationally demanding, and while they yield full electronic detail, their application at present is limited to systems having a relatively small number of atoms and to short timescales. For many applications, including the ground state properties of many materials and chemical reactions, ab initio density functional theory (DFT) is particularly useful, since it provides an effective compromise between accuracy (often of the order of 0.1–1 kcal/mol in energy) and computational tractability. The central idea of DFT is to describe the system of interacting electrons via their density [9]. The electron density, ρe(r), depends only on the three spatial variables x, y, z. Knowledge of the electron density is sufficient to calculate all of the ground state properties of the system, so that the energy E is a functional of ρe(r), that is, E[ρe]. This is the first theorem of DFT. The energy can be written as a sum of three terms: E[ρe ] = T[ρe ] + U[ρe ] + Exc [ρe ]
(2.1)
where T, U, and Exc are the kinetic energy of the (noninteracting) electrons of density ρe(r), the classical electrostatic energy due to Coulomb interactions among the particles, and the exchange and correlation energies due to many-body interactions, respectively, each being a functional of the electron density [10]. 2.3.2 Atomistic Scale The atomistic level methods such as Monte Carlo (MC) and molecular dynamics (MD) simulation are less demanding, enabling systems of thousands or millions of atoms to be studied over time intervals of 10 or 100 ns, depending on system size and complexity [11]. In using such methods, we lose the electronic detail, but for most physical processes where electron perturbation is small, this is not important. Examples include most phase changes and transport processes [12]. 2.3.3 Mesoscale In the fully atomistic classical treatment, molecules are modeled in atomistic detail and have access to all possible regions of phase spaces, the space (r1, r2, r3,…, r3N; p1, p2, p3,…, p3N) of all possible positions, r, and momenta, p, for the N atoms in the system [13]. At any instant the system will be at some point in this phase space, and its evolution in time is represented by a trajectory in phase space, which is determined by the intermolecular and intramolecular forces and any external fields. While such a treatment is desirable, it may not be feasible yet for systems of large molecules (polymers, proteins, etc.) or for
Computational Modeling of Nanoparticles
33
slow processes that occur over timescales greater than about a microsecond (e.g., many self-assembly processes, slow diffusion). For such systems, various mesoscale methods have been developed. Many applications of current interest require modeling at multiple scales. For example, at present many biological and polymer applications require mesoscale methods to achieve the timescales and length scales necessary, but atomistic level, and sometimes electronic level, calculations are also needed to investigate the effects of interatomic forces such as H-bonding, site–drug molecule interactions, and electron transfer [13]. Similarly, many materials applications will require electronic/atomistic and even mesoscale or continuum modeling. Investigation of heterogeneous reactions will require electronic level calculations to determine the potential energy surface, and hence the reaction mechanism and atomistic level simulations to determine yields and rates. For colloidal solution dynamics, a useful approach is to treat the solute–solute and solute–solvent interactions in a full MD description, while using a mesoscale treatment of the solvent– solvent interactions, the solvent dynamics is modeled by dividing the system into cells and updating the velocities of the solvent molecules by multiparticle collisions within each cell [14]. In fluid dynamics applications involving complex flows, and particularly for microfluidic flows, continuum treatments such as finite element analysis may not be accurate if the local mesh uses too fine a grid, since the continuum model does not account correctly for the atomistic fluctuations. In such cases, it is possible to use a particle-based method (e.g., MD) for such high-resolution regions, while using the continuum treatment for other regions where a coarser grid is possible. 2.3.4 Continuum Scale Continuum level methods are essentially unlimited in terms of length scale and timescale and are rigorously based on macroscopic conservation laws. They assume that matter is a continuum, that is, it can be subdivided without limit. Although this is not correct, it is a good approximation when dealing with length scales that are much greater than the mean free path of the molecules. In the continuum approximation, all physical properties (density, composition, pressure, etc.) have well-defined values at any given point in the system, that is, they are treated as fields. Continuum methods are based on numerical solution of the conservation equations for mass, momentum, and energy, subject to the constitutive equations governing diffusion (Fick’s Law), fluid flow (Newton’s Law), energy (Fourier’s Law), chemical reactions (the kinetic equations), etc., together with the thermodynamic equation of state. Commonly used numerical methods for solving these continuum equations include finite differences, finite control volume, and finite element methods (FEMs). A common approach is to discretize the space so that the unknown functions (e.g., temperature and pressure), which vary continuously in the system, are now represented by discrete values at lattice points, and the differential problem becomes an algebraic one that is amenable to solution on a computer [15]. A great many problems in civil, mechanical, aerospace, electrical, industrial, chemical, and materials engineering are appropriately solved using continuum methods. In these methods the electronic, atomic, and mesoscale detail is lost, but this is unimportant for many macroscopic scale problems. A disadvantage of continuum methods is that phenomenological coefficients that derive from atomic or subatomic properties (diffusion coefficients, thermal and electrical conductivities, chemical reaction rate constants, etc.) must be obtained separately, either from experiment or from a higher level calculation. In general, calculations at the continuum scale require extensive experimental input.
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Computational Nanotechnology: Modeling and Applications with MATLAB®
2.4 Concept of Computational Modeling of Nanostructures Computer modeling is an indispensable tool to interpret phenomena associated with the nanometer scale. Some of the most unexpected phenomena are displayed by carbon nanostructures. Describing correctly the different types of bonding, which carbon is known to exhibit, poses a challenge [16,17]. The Tersoff bond-order potential has been used widely to obtain reliable equilibrium geometries and deformation energies of graphitic sp2 as well as diamond-like sp3 bonding [17]. The continuum elasticity approach has shown unexpected accuracy when determining the strain energy in sp2-bonded single-wall carbon nanotubes and fullerenes. Somewhat surprisingly, this approach, which ignores atomic positions in the graphite plate, has been able to reproduce accurately the formation energies of fullerenes and nanotubes down to diameters as small as 0.5 nm. Computationally more demanding, but still widely used, is a parameterized version of a total energy functional based on the linear combination of atomic orbitals (LCAO) electronic Hamiltonian [18]. Often called the tight-binding approach, this energy functional has proven especially useful in the description of transition states during structural changes, since it correctly represents the hybridization between s and p orbitals of carbon, which underlies the sp, sp2, and sp3 bonding. This approach has gained wide popularity in MD simulations, since it yields semianalytical expressions for interatomic forces, thus speeding up the simulation. In very large systems, where the diagonalization of a large Hamilton matrix is time limiting, the recursive reconstruction of the local density of states has proven very powerful, in particular due to its natural suitability for implementation on massively parallel supercomputers [19]. Assigning a processor to each atom, with information about the atomic arrangement in the close vicinity available, the local electronic density of states, the binding energy, as well as the force acting on that particular atom can be determined with a satisfactory precision. The method of choice to obtain accurate information about the total energy and ground state equilibrium geometry of nanostructures from first principles is the DFT. Both the local density approximation (LDA) and the generalized gradient approximation (GGA) have been used successfully to describe fullerenes, nanotubes, polymer chains, diamond, and other carbon allotropes [20]. They are especially valuable in the description of hybrid structures and transition states, where a self-consistent determination of the charge distribution is essential. Since many interesting carbon nanostructures exhibit graphitic sp2 bonding, their electronic states near the Fermi level are dominated by ppπ hybridized atomic orbitals. The single-band Hückel Hamiltonian, which only considers ppπ interactions, has been widely used to rationalize the electronic structure near the Fermi level in fullerenes and nanotubes (Figure 2.2). Due to the stability of the sp2 bond, carbon nanotubes are thermally and mechanically extremely stable and chemically inert [21]. They contract rather than expand at high temperatures and are the ultimate thermal conductors. At the same time, nanotubes may be tuned into ballistic electron conductors or semiconductors. The sp2-bonded nanostructures may change their shape globally by a sequence of Stone–Wales transformations [22]. Specific nanotube assemblies may even acquire a permanent magnetic moment. In nanostructures that form during a hierarchical self-assembly process, even defects may play a different, often helpful, role. Efficient self-healing processes may convert less stable atomic assemblies into other, more perfect structures, thus answering
Computational Modeling of Nanoparticles
35
FIGURE 2.2 Structural model of carbon nanostructures. (From Park, N., Yoon, M., Berber, S., Ihm, J., Osawa, E., and Tománek, D., Phys. Rev. Lett., American Physical Society, 2003. With permission.)
Conduction band
2s
2p excitation (33 eV)
2p
Valence band 2s
Carbon nanotube
FIGURE 2.3 Photo-induced de-oxidation of a defective carbon nanotube. (From Miyamoto, Y., Jinbo, N., Nakamura, H., Rubio, A., and Tománek, D., Phys. Rev. Lett., American Physical Society, 2004. With permission.)
an important concern in molecular electronics. Defects may even be used in nanoscale engineering to form complex systems such as carbon foam or nanotube peapods. Recent computer simulations of defective nanotubes indicate an amazing self-healing ability of these nanostructures at elevated temperatures or in the electronically excited state (Figure 2.3). To a large degree, nanostructures owe this unexpected behavior to their larger structural flexibility in comparison to large systems [23]. Electronic excitations also last for a significant fraction of the atomic vibration period in nanostructures, thus capable of breaking or forming atomic bonds. This particular behavior is expected to prove very useful, when trying to establish selective purification techniques for nanostructures, including photoinduced deoxidation of defective nanotubes. Hierarchically, self-assembled nanostructures consisting of polyacetylene or polymerized diamond, enclosed in carbon nanotubes (Figure 2.4), may exhibit unusual transport properties and find use as functional building blocks of complex nano-electromechanical systems (NEMS) [24].
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Computational Nanotechnology: Modeling and Applications with MATLAB®
FIGURE 2.4 A fullerene molecule inside a nanotube as a prototype non-volatile memory element capable of storing 1 bit information. (From Kwon, Y.-K., Tománek, D., and Iijima, S., Phys. Rev. Lett., American Physical Society, 1999. With permission.)
2.5 Computational Control of Matter through Modeling Any material structure can be obtained from the quantum-mechanical Schrödinger equation. Solving the Schrödinger equation for an entire solid presents an enormous computational task and is not practical for our purposes: a number of approximations have to be made. Fortunately, the problem of modeling atomic structures is simplified because the mass of the electrons is much smaller than that of the nucleus, while the forces exerted on both are similar. As a result, the motion of the atomic nuclei is much slower than that of the electrons. This justifies the use of the Born–Oppenheimer approximation: within this approximation, the Hamiltonian is expressed as a function of the positions of the nuclei only, the rapid motion of the electrons having been averaged out. The potential energy term, which is a function of the positions of the particles, contains the interesting information regarding the atomic interactions. Assuming such a potential can be constructed, the computational techniques presented later on proceed in different ways, determining the underlying atomic structure. The potential energy function is thus a vital ingredient of every computer simulation. A great number of modeling techniques have been developed over the years, many of which have been optimized for a particular material or purpose. 2.5.1 Empirical and Ab Initio Potentials Empirical potentials are the simplest potentials around. These potentials are not designed to describe the atomic bonding process in all its details. Instead, they merely try to reproduce some of the bonding properties. These properties could be derived from experimental observations or stem from more advanced quantum-mechanical calculations. Empirical potentials are computationally easy to evaluate; this makes them attractive candidates for simulations involving many particles, such as the Keating potential [25] and the Stillinger– Weber (SW) [26] potential. In Keating potential, bonding geometries that deviate from these equilibrium crystal values are assigned an energy penalty E=
3 α 16 d 2
∑(
rij * rij − d 2
)
2
+
where r⃗ ij is the vector pointing from atom i to atom j α and β are constants d is the mean bond length
3 β 8 d2
⎛ 1 2⎞ ⎜⎝ rij * rik d ⎟⎠ 3 < jik >
∑
2
(2.2)
37
Computational Modeling of Nanoparticles
The symbol
denotes bonded pair of atoms i to j while <jik> denotes triple bonded atoms with i in the middle. The bond stretching and bond bending force constants α and β are usually obtained from a fit to the elastic properties of the material observed in experiments. The Keating potential is a single harmonic well, with only one minimum: the bottom of this well. The Keating potential allows only for a rather artificial kind of dynamics: a new energy minimum can be constructed by making explicit changes in the list of bonds. This limitation is serious for methods such as MD and the activation–relaxation technique. These methods ultimately depend on a potential energy landscape containing many local minima: the Keating potential will not work for these methods. In MC-based techniques, however, the Keating potential will prove to be extremely useful. A more realistic empirical potential for amorphous silicon, which does not require an explicit list of bonds, is the SW [26] potential given by ⎡ ⎤ λ V = εA ⎢ νij( 2) (rij ) + ν jik ( 3 ) ( rij , rik )⎥ A < jik > ⎥ ⎢ < ij > ⎦ ⎣
∑
∑
(2.3)
where V is the potential A, ε, and λ are lattice constants νij is the particle velocity moving from atom i to atom j rij⃗ and rik⃗ are the vectors pointing from atom i to atom j and from atom i to atom k, respectively For two-body part, the equation becomes
νij
( 2)
⎡ ⎛ ⎞ rij (rij ) = ⎢B ⎜ ⎟ ⎢ ⎜ σ⎟ ⎢⎣ ⎝ ⎠
−p
⎤ ⎛ rij ⎞ 1 ⎞ ⎛ Θ⎜a − ⎟ − 1⎥ exp ⎜ ⎟ ⎥ σ⎠ ⎝ rij /σ − a ⎠ ⎝ ⎦⎥
(2.4)
And for three-body part it is given by ⎡⎛ ⎞⎤ rij ⎞ ⎛ γ γ rik ⎞ ⎛ 2 + ν jik ( 3 ) (rij , rik ) = exp ⎢⎜ ⎟ ⎥ (cos θ jik − cos Θ 0 ) Θ ⎜⎝ a − σ ⎟⎠ Θ ⎜⎝ a − σ ⎟⎠ / σ / σ − − r a r a ⎠ ⎥⎦ ik ⎢⎣⎝ ij
(2.5)
where ⃗rij is the vector pointing from atom i to atom j with rij = |r⃗ ij| θjik is the angle between vectors r⃗ ij and r⃗ ik Θ0 is the tetrahedral angle given by Θ0 arc cos −1/3 Θ is the Heaviside step function σ, p, B, a, and γ are lattice constants The summations in Equation 2.5 include all pairs i – j and all triples j – i – k of atoms in the system. The problem associated with the SW potential outlined above is that most empirical potentials are fitted to a limited set of properties and usually give a poor description of
38
Computational Nanotechnology: Modeling and Applications with MATLAB®
the properties. Moreover, empirical potentials do not provide any electronic structure information. A possible way out of this dilemma is offered by the ab initio potentials. These potentials provide the most accurate way to calculate atomic forces, electronic properties, and vibrational properties. The majority of these methods make use of the LDA within the limits of the DFT. However, substantial computational effort is required, thus limiting the applicability of ab initio methods to relatively small systems, in practice up to 100 atoms. 2.5.2 Molecular Dynamics Simulation One of the principal tools in the study of materials is MD simulations. This computational method calculates the time-dependent behavior of a system. Consider a system of N particles. Let r⃗ i(t) and vi�(t) denote the position and the velocity of the ith particle at time t, respectively. Assume that the particles interact with each other via some potential energy function V(r⃗ 1,…, ⃗rN). The potential depends on the positions of the particles but contains no explicit time dependence. The classical equations of motion for such a system can be derived in the Lagrangian formalism and in this case reduce to Newton’s equation of motion: d2r⃗ i/dt2 = fi⃗ /mi, where mi is the mass of the ith particle and f⃗ i is the force acting on this particle. The forces are obtained from the gradient of the potential energy: f i = −∇i/V . The aim of MD is to solve Newton’s equation of motion. To start the simulation, initial positions and velocities are assigned to each particle. Next, the general scheme is as follows: 1. Calculate the positions of the particles at time t using the positions, velocities, and forces at previous times. 2. Calculate the forces and velocities of the particles at their new positions. 3. Return to (1) for the next step and repeat. One common choice to predict the positions of the particles at time t + Δt from the positions at previous times is through the velocity Verlet algorithm [27]. This is one of the simplest but also one of the most effective MD integration schemes. In this case, the positions at time t + Δt are obtained using f (t) ri (t + Δt) = ri (t) + νi (t)Δt + i ( Δt)2 2mi
(2.6)
where r⃗ i(t) and νi (t) denote the position and the velocity of the ith particle at time t mi is the mass of the ith particle fi⃗ is the force acting on the particle with velocities given by f i (t + Δt) + f i (t) νi (t + Δt) = νi (t) + Δt 2mi
(2.7)
39
Computational Modeling of Nanoparticles
At any time t, the instantaneous temperature T is determined by the velocities of the particles N
3( N − 1)kBT =
∑m ν i
2 i
(2.8)
i 1
where kB is the Boltzmann constant the factor N − 1 accounts for momentum conservation The initial velocity distribution will in general not be Maxwellian. The simulation must first undergo an equilibration phase before thermal equilibrium is reached. During equilibration, there is a significant drift in the macroscopic observables of the system, such as the total kinetic energy or pressure. By monitoring these observables as a function of time, one can determine when equilibrium has been reached. After equilibration, the macroscopic observables will fluctuate around their equilibrium values. At this point one usually begins to calculate variables of interest (such as pressure or energy) for the accumulation of time averages. The simulation is continued until the statistical error in the averages becomes small. MD enables us to simulate materials at the atomic level without the need for extensive theory. The only requirement is that a potential energy function must be provided. If the potential is realistic enough and if the pure classical approximation is valid, MD reproduces the dynamical processes that occur in the real substance. The problem of MD is mostly one of timescales. Atomic processes in solids, such as vibrations, typically occur on a timescale of a fraction of a picosecond. This puts an upper bound on the MD integration time step Δt in the order of a few femtoseconds. For processes that last longer than microseconds, MD is not practical. 2.5.3 Monte Carlo Simulation MC methods are traditionally used to solve problems in statistical mechanics although they can also be used to study amorphous materials. For a system in thermal equilibrium at temperature T, the probability pμ of finding the system in a state μ with energy Eμ is given by the Boltzmann distribution with normalization constant Z: pμ =
Z=
1 ⎛ E ⎞ exp ⎜ − μ ⎟ ⎝ kBT ⎠ Z ⎛
Eu ⎞
∑ exp ⎜⎝ − k T ⎟⎠
(2.9)
(2.10)
B
μ
where the sum is over all states μ accessible to the system. Within this framework, the expectation value 〈Q〉 of some observable Q can be written as a weighted average: Q ≈
∑Q P
μ μ
(2.11)
μ
where Qμ is the value of the observable in state μ. The quantity Z is called the partition function. It is a powerful quantity because virtually all macroscopic properties of
40
Computational Nanotechnology: Modeling and Applications with MATLAB®
a system, like pressure and specific heat, can be derived from it. For sufficiently simple systems, such as the ideal gas, the partition function can be determined exactly. Of course, in the case of a continuous system, the summation over states is replaced by integration over phase-space. In many cases, the partition function cannot be calculated exactly and other methods must be used. MC methods try to estimate the partition function by replacing the summation over all states with a summation over a subset of these states only. Suppose we pick M such states, {μ1, μ2, …, μM}, and in the picking procedure state μi has a probability g� of being selected. i Within this subset the best estimates for the partition function and the expectation values are given by M
Z=
1
∑g i −1
μi
⎛ E ⎞ exp ⎜ − ui ⎟ ⎜⎝ kBT ⎟⎠
(2.12)
where M are the states g�i is the probable state selected from {μ1, μ2, …, μM} Eui is given by the Boltzmann distribution Q
≈
1 Z
M
1
∑g i −1
⎛ E ⎞ Qui exp ⎜ − ui ⎟ ⎝ kBT ⎠ μi
(2.13)
The simplest approach is to pick them with equal probability. In this case the selection probabilities g�i cancel out and the estimate for the expectation value becomes M
∑ Q e − E /k T Q ≈ ∑ e − E /k T i 1 M
μi
μj
j 1
B
μi
(2.14)
B
In most cases, this turns out to be a poor choice. In many cases, only a small fraction of the states contribute significantly to the partition function. The total number of states is usually very high. If we select states with equal probability, the chance of mostly picking the states that contribute significantly is less. A much better choice is to select states with the Boltzmann probability g�i exp(−Ei = kBT). The estimate for the expectation value then becomes 1 Q = M
M
∑Q
μi
(2.15)
i 1
This is the common choice in MC methods. It is efficient because equilibrium systems often spend most of the time in only a relatively small number of states. By picking states with the Boltzmann probability, precisely, these states will be selected. Generating states such that each one appears with its Boltzmann probability can be accomplished using a
Computational Modeling of Nanoparticles
41
Markov process. Given an initial state μ the Markov process generates a new state ν. It will do so randomly: the probability of generating state ν from state μ is called the transition probability and is denoted by P(μ → ν)
(2.16)
These probabilities should depend neither on time nor on the history of the system. They should also obey the constraint
∑ P(μ → ν) = 1
(2.17)
ν
since the Markov process must always generate some new state ν. If the Markov process satisfies ergodicity and detailed balance, it will generate states according to the Boltzmann distribution. Ergodicity means that the Markov process should be able to reach any state of the system from any other state, provided it is allowed to run long enough. Detailed balance puts an additional constraint on the transition probabilities to ensure that, in equilibrium, the Markov process samples the Boltzmann distribution ⎡ −Eμ ⎤ ⎡ −Eν ⎤ exp ⎢ ⎥ P(μ → ν) = exp ⎢ (k T ) ⎥ P( ν → μ ) ( ) k T ⎣ B ⎦ ⎣ B ⎦
(2.18)
It is allowed for the Markov process to generate the same state ν, given a state μ. In other words, the transition probability P(μ → ν) does not have to be zero. This is extremely useful if we wish to avoid states in the Markov process with undesirable properties (for instance, states with high energy). If the Markov process would generate such a state ν from a state μ, we can immediately go back to state μ, forget state ν was ever generated, and try again. In this case, the transition probability becomes P(μ → ν) = g(μ → ν)A(μ → ν)
(2.19)
where g(μ → ν) is the probability of generating state ν from state μ A(μ → ν) is the acceptance probability If the Markov process generates state ν from state μ, the acceptance probability tells us we should accept this state ν, a fraction of the time A(μ → ν). The condition for detailed balance may be rewritten as g(μ → ν) A(μ → ν) ⎛ Eμ − Eν ⎞ = exp ⎜ ⎝ kBT ⎟⎠ g( ν → μ ) A( ν → μ )
(2.20)
A good MC algorithm is one in which the acceptance probability is of order one. As an example we consider a system of N particles, which interact via some potential energy function. We generate states by selecting one of the particles at random, followed by a random displacement of this particle. The probability of generating state ν from state μ is thus
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Computational Nanotechnology: Modeling and Applications with MATLAB®
determined by the probability of selecting a certain atom: g(μ → ν) = 1/N. The condition for detailed balance now reduces to the following: ⎛ Eμ − Eν ⎞ A(μ → ν) = exp ⎜ ⎟ A( ν → μ ) ⎝ kBT ⎠
(2.21)
Suppose we generate state ν from state μ and that state μ has the lower energy. As a result, the RHS of Equation 2.21 is smaller than 1 and hence A(ν → μ) > A(μ → ν). To maximize the acceptance probability, it would be ideal to set A(ν → μ) equal to unity and to adjust A(μ → ν) in order to satisfy Equation 2.21. In this case A(μ → ν) should be set to exp[(Eμ − Eν)/kBT]. The optimal algorithm is thus one in which ⎡ Eμ − Eν ⎤ A(μ → ν) = min ⎢1, exp ⎥ k B T ⎥⎦ ⎢⎣
(2.22)
This is the Metropolis probability, which is the method of choice in the majority of MC studies. 2.5.4 Advantages of Monte Carlo and Molecular Dynamics Simulation Techniques The MC method offers the advantage that it is readily applied to different sets of independent variables ((N, V, T), (N, P, T), (μ, V, T), etc.) since we know the probability distribution laws for all of these. Certain special techniques are available with MC to speed up equilibration for cases where MD cannot access the real times involved, for example, slow phase transitions, micelle formation, and polymer folding. In MC, the main drawbacks are that dynamical behavior cannot be calculated, and the molecular motions are artificial and give no information on the dynamic motions of real molecules [28]. A further drawback of MC is that it is basically a serial method and does not lend itself readily to parallelization of the code, so that it is harder to take advantage of massively parallel machines. MD has the advantage that dynamical behavior and transport properties are readily calculated. In addition, the molecular motions occur naturally under the influence of the intermolecular forces and any external fields, making it possible to directly observe diffusive, convective, and other modes of motion at the molecular level. Sampling conformations for complex molecules is generally simpler in MD than in MC. As a result, there are many more MD codes that work for all molecular classes than is the case for MC. 2.5.5 Limitations of Monte Carlo and Molecular Dynamics Simulation Techniques MC and MD simulations are limited by the accuracy of the force field used and by the speed (and to some extent the memory) of current computers (Figure 2.5). Computer speed limits the real time that can be studied in MD, making it difficult to study phenomena that require more than a microsecond to evolve. It also serves as a serious barrier in studying large molecules such as polymers, proteins, or slow processes such as micelle formation or protein folding. For some systems and phenomena long-range correlations are present, requiring very large systems to be modeled on the computer, providing a further limitation [29]. One such example is provided by systems with long-range Coulombic forces that fall off as the inverse separation between particles; examples include electrolytes, fused salts, and plasmas. Systems of molecules with short-range forces can also exhibit long-range
43
Computational Modeling of Nanoparticles
N ~ 500 –1,000,000 Periodic boundaries Prescribed intermolecular potential Monte Carlo
Molecular dynamics
Specify N, V, T
Specify N, V, E
Generate random moves
Solve Newton’s equations Fi = miai
Sample with Pacc exp (–u/kT )
Calculate ri(t), vi(t)
Obtain equilibrium properties Take averages
Take averages Obtain equilibrium and nonequilibrium properties
FIGURE 2.5 Comparison of MC and MD simulation techniques.
correlations between molecules, for example, fluids near to a critical point, where correlations between the large density fluctuations can occur over large distances, or diffusion in very narrow pores (particularly single-file diffusion), where long-range dynamical correlations occur due to molecular collisions under confinement [30].
2.6 Modeling of Electronic Transport of Nanoparticles When electrons move in a material they scatter from impurities, other lattice defects, and phonons. Usually, the scattering from other electrons is less important [31]. The scattering causes electrical resistance. In normal-size electronic components this resistance follows Ohm’s law for the current I and voltage V, so that the resistance R is proportional to the length L of the device: V = RI = ρR
L I A
(2.23)
where ρR is the resistivity of the material A is the perpendicular area of the device Ohm’s law is understandable, because typically the scatterers are uniformly distributed in the material. A longer device has also more scatterers to destroy the collective electron drift movement. If we make the device smaller, the number of scatterers diminishes. In this regime it is not surprising that the statistical Ohm’s law is invalid and other theories have to be used. The electron transport properties of devices can be characterized by the de Broglie wavelength, the mean free path, and the phase-relaxation length of electrons. Ohm’s law is valid for devices that have dimensions much longer than these lengths. The de Broglie wavelength is here the electron wavelength at the Fermi level. If the size of
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Computational Nanotechnology: Modeling and Applications with MATLAB®
the device is of the same order of magnitude as the de Broglie wavelength, quantummechanical phenomena appear. For example, electrons have a discrete energy spectrum. The mean free path of the electrons tells how long a distance an electron moves in average before it has lost its original momentum by scattering. Because in a typical collision an electron loses only a part of the momentum the mean free path is longer than distances between individual collisions. The phase-relaxation length is the average length, which an electron can move before collisions destroy its original phase. As in the case of the mean free path a single collision destroys only a part of the phase. More specifically, in order to affect the electron phase, collisions have to be inelastic, so that typically the phaserelaxation length is longer than the mean free path. If the size of the device is smaller than the phase-relaxation length, waves of electrons interfere and quantum-mechanical phenomena appear. These phenomena have large effects on the transport properties of devices. This is the case in a typical nanostructure at least as a first approximation. In this work we concentrate only on the coherent transport of electrons and ignore the inelastic effects. In two-dimensional (2D) nanostructures, the phase-relaxation length is typically large and the de Broglie wavelength of an electron is comparable to the size of the device. Consider next a 2D wire of length L between two large electrodes or electron reservoirs. In the wire electron states in the perpendicular direction of the wire are quantized. The electron density per unit length (L) corresponding to a given perpendicular state in the momentum ranges between k and k + dk and spin σ is given by nσ (k )dk =
1 L 1 f σ (k )dk = f σ (k )dk L 2π 2π
(2.24)
where nσ(k) is the electron density per unit length (L) and spin σ in the momentum ranges between k and k + dk fσ(k) is the Fermi distribution function k is the projection of the wave vector along the wire direction For small bias voltages the electrodes are approximately in equilibrium and have local quasi Fermi levels for the right and left leads, respectively. In contrast, the wire region does not have a well-defined Fermi level. The electron current carried by one perpendicular state, the so-called conducting mode, is ∞
I=
∞
k ⎛ f Rσ (k ) ⎞ ⎛ f Lσ (k ) ⎞ − dk 2π ⎟⎠ ⎜⎝ 2π ⎟⎠ e
∑ ∫ eυ (k)n (k)dk = ∑ ∫ e m* ⎜⎝ σ
σ
0
σ
σ
0
(2.25)
where e is the elementary charge υσ(k) is the electron velocity along the wire having electron density σ k is the projection of the wave vector along the wire direction nσ(k) is the electron density per unit length (L) me* is the electron effective mass ћ is the Planck constant f Rσ(k) is the Fermi distribution function with σ as electron density of the wave vector along the right direction of the wire f Lσ(k) is the Fermi distribution function with σ as electron density of the wave vector along the left direction of the wire
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Computational Modeling of Nanoparticles
In the zero temperature limit the Fermi distributions are step functions so that Equation 2.25 has the form
I=
∑∫ σ
2me*μ R / 2me*μ L/
e
k 1 dk = me* 2π
e2 μR − μL = e
∑h σ
e2
∑ hV
B
(2.26)
σ
where I is the electronic current me* is the electron effective mass σ is the electron density μR/L are the chemical potentials of the leads and their relative values depend on the bias voltage VB so that μR − μL = eVB From Equation 2.26 we see that the conductance of one conducting mode, the so-called ⎛ μR − μL ⎞ conductance quantum, can be obtained as I = GV, where V is the applied potential ⎜ ⎟⎠ ⎝ e and where G is the conductance given by G=
2e h
2
(2.27)
It is the maximum conductance of a single conducting mode with two spin states. In practice, a mode is not necessarily fully conducting, because electrons can scatter in the nanodevice and at its connections to the electrodes. The probability for an electron to pass the device in the conducting mode i is marked by Tσi. Now the total conductance of the wire has the form G=
∑ σ ,i
e2 Tσi h
(2.28)
where Tσ i is the probability for an electron to pass the device in the conducting mode i G is the conductance This equation is called the Landauer formula for the conductance [32]. It implies that the conductance of a thin wire stays infinite even for small values of the length L. The conductance increases to infinity, meaning zero resistance, if we have a large device for which the number of conducting modes is infinite. However, a very large number of conducting modes means a large device that is no longer considered to be a nanodevice. This kind of quantization of the electron conductance is seen in many systems, for example, in quantum wires and in atomic chains. Transport properties of nanostructures are modeled using different quantum-mechanical methods approximating different properties of the structure. In some simulations electron interactions are included carefully so that many-particle effects are included. These models include, for example, the Kondo model and the Anderson impurity model [33]. Nanostructures in these models do not have any
47
Computational Modeling of Nanoparticles
The calculations start from the retarded Green’s function Gr defined by (ω − Hˆ (r ))G r (r , r ʹ; ω ) = δ(r − r ʹ)
(2.29)
where ω is the electron energy ˆ is the DFT Hamiltonian of the system H r and r′ are the coordinates of the electrons Gr is Green’s function The equation also gives the advanced Green’s function Ga as the other solution. In practice, the separation of these two solutions is done using boundary conditions. When Gr is known, the so-called lesser Green’s function G< can be calculated. When there is no bias voltage, the system is in equilibrium and G< is calculated as G <(r , r ʹ ; ω) = 2 f L/R (ω)G r (r , r ʹ ; ω)
(2.30)
where f L/R are the Fermi functions of the leads in the left and right directions, respectively (f L = f R in equilibrium). For a finite bias voltage a more complicated form has to be used. The system is divided into two parts, ωL and ωR, as explained: ⎛ ˆ ⎜ ω − H0 − ⎝
r
∑
r
(ω ) −
L
⎞
∑ (ω)⎟⎠ G (r , rʹ; ω) = δ(r − rʹ) r
(2.31)
R
where ˆ is the Hamiltonian of the isolated region Ω H 0 r L are the so-called self-energies of the leads R
•
This name comes from an analogy with the many-body Green’s function theories, where self-energies are calculated for different interactions, for example, the electron–electron r L or electron–phonon interactions. is the interaction term between the electrons in R Ω and the electrons in ΩL/R. In Green’s function method many-body interactions can be in principle fully included. Also, by using Green’s function method, it is possible to calculate the electron tunneling probability and the current through the system for a finite bias voltage over the structure. An unfavorable feature of the Green’s function method is that the numerical calculations are heavier than those in typical wave function methods. 2D nanostructures can be fabricated at GaAs/AlGaAs interfaces (Figure 2.7). One of the simplest structures is a quantum wire. It is a thin nanoscale constriction between two bulk electrodes. If the wire is short compared to its width it is called as quantum point contact. We have modeled quantum wires using the 2D version of our Green’s function solver. 2D nanostructures have interesting properties. The impurities responsible for the doping regions are located far away from the interface, reducing electron scattering from impurities and increasing the mean free path. The Fermi wavelength is also large as compared to
•
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Computational Nanotechnology: Modeling and Applications with MATLAB®
FIGURE 2.10 Comparison between 2D simulation and quasi-3D simulation.
Rectangle
Any arbitrary shape
(2D simulation)
(Quasi-3D simulation)
Elastic properties: Plane strain or plane stress, only in-plane direction
Elastic properties: No strain, in-plane and out-of-plane direction
(Nuclei per second) * Time interval (s). The time interval is the time when grains grow by one pixel, which is calculated as pixel length (nm) divided by the growth rate (nm/s) [41]. The pixel length is the same as the diameter of a stable nucleus, which is assumed to be 5 nm based on measurement. Making use of the experimental findings [42], the nucleation rate (N) is approximated as a function of temperature (T) as below: N = −0.0007444 * (T ) + 5.058 (Nuclei/μm 2/s)
(2.32)
Moreover, by optimizing the average grain radii (nm) versus time (s) at three different temperatures, the growth rate and average grain size are determined as a function of temperature. The theoretic equations for the nucleation rate (N) and growth rate (G) of the deposition process are as follows [43]: N≈
PN A (2πMRT )1/2
G=
k s hg C g ( k s + h g )N 0
where k s ∝ e −( E/RT )
(2.33)
where P is the pressure T is the temperature M is the molecular weight NA is the Avogadro number E is the characteristic activation energy involved ks is the rate constant for surface reaction hg is the gas phase mass transfer coefficient Cg is the gas concentration N0 is the atomic density R is the gas constant From the above equations, the nucleation rate is a function of pressure and temperature while growth rate is mainly a function of temperature. Moreover, the nucleation rate is
51
Computational Modeling of Nanoparticles
linearly proportional to the pressure. Ohring [43] expressed the grain size (lg) as a function of nucleation rate and growth rate as below: ⎛ G⎞ lg = 0.9 ⎜ ⎟ ⎝ N⎠
1/3
(2.34)
The grain size is defined as the radius of a circle that has the same area as the grain. The grain size is proportional directly to the nucleation rate (N) and inversely to the growth rate (G). Since nucleation rate is linearly proportional to the pressure, the dependence of grain size on the pressure can be described as below: lg 1 ⎛ P2 ⎞ = lg 2 ⎜⎝ P1 ⎟⎠
1/3
(2.35)
where lg1 is the grain size of first particle lg2 is the grain size of second particle P2 is the pressure on second particle P1 is the pressure on first particle The elastic modulus is an intrinsic property, which is mainly determined by chemical composition. Since the elastic modulus of bulk materials is hardly influenced by microstructure change, alterations such as plastic deformation, cold work, and heat treatment do not change elastic modulus. However, producing a preferred orientation can change the elastic modulus. For the MEMS thin films, which have a small number of grains, it is not confirmed whether the microstructure change has influences on the elastic properties. The concept of elastic modulus of polycrystalline material can be better understood in continuum mechanics in which the general relationship between stress and strain is described by σij = Cijklε kl
i, j, k , l = 1, 2, 3, …
(2.36)
where σ and ε are the second order tensor of the Cauchy stress and infinitesimal strain, respectively C is the fourth order tensor of stiffness matrix that has 81 components The number of independent components in C is reduced to 21 due to the symmetry of stress, strain, and C itself. The C can be reduced to three independent components depending on the crystallographic structure for anisotropic materials and two independent components for isotropic materials [44]. The relation between the crystallographic structure and the independent number of components in C is shown in Table 2.1. In certain simulations, the coordinate transformation of this C is performed depending on the plane normal texture as well as in-plane grain orientation (Figure 2.11). This transformation of C can be expressed as Cpqrs = Cijkllipl jqlkr lls where C indicates the number of components in the direction p, q, r, s lip, ljq, lkr, and lls are direction cosines
(2.37)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
TABLE 2.1 Relation between the Crystallographic Structure and the Independent Number of Components in C Independent Components Number in C
Crystal Structure Monoclinic Cubic Tetragonal Orthorhombic Hexagonal Trigonal Isotropic
13 [C11, C12, C13, C16, C22, C23, C26, C33, C36, C44, C45, C55, C66] 3 [C11, C12, C44] 6 [C11, C12, C13, C33, C44, C66] 9 [C11, C12, C13, C22, C23, C33, C44, C55, C66] 5 [C11, C12, C13, C33, C55] 6 [C11, C12, C13, C14, C33, C44] 2 (E, v)
Original matrix (C)
Transformation due to plane normal texture
Transformation due to in-plane grain orientation FIGURE 2.11 Transformation of stiffness matrix in thin films simulation.
Final matrix (C)
For the silicon for [001] plane normal texture, the values are given as below: ⎡165.7 ⎢ 63.9 ⎢ ⎢ 63.9 C=⎢ ⎢ 0 ⎢ 0 ⎢ ⎢⎣ 0
63.9 165.7 63.9 0 0 0
63.9 63.9 165.7 0 0 0
0 0 0 79.6 0 0
0 0 0 0 79.6 0
0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ ( GPa ) 0 ⎥ 0 ⎥ ⎥ 79.6 ⎥⎦
By means of electron microscopy, it is observed that thin films have a column structure. Therefore, a quasi-3D microstructure thin film is generated by extrusion of a 2D microstructure, and each grain is assigned with a random orientation in addition to a plane normal texture [45]. Figure 2.12 illustrates the extrusion of the plane normal texture assignment to quasi-3D structure and Figure 2.13 presents the mesh generation of a quasi-3D microstructure thin film. Having generated the mesh of quasi-3D microstructure of thin film, boundary conditions are applied as shown in Figure 2.14. To determine the effective (homogenized) elastic properties of a thin film, the total reaction force (Fx) is obtained from analysis, stress along the x direction can be calculated as σx = Fx/A
53
Computational Modeling of Nanoparticles
[100] Plane orientation
FIGURE 2.12 Plane normal texture assignments to the quasi-3D microstructure.
[110] Plane orientation
FIGURE 2.13 Mesh generation of quasi-3D microstructure thin film for polysilicon (24.5 × 9.8 × 5 μm).
Y Z
X
ΔX
FIGURE 2.14 Boundary condition of elastic properties simulation.
where A is cross-sectional area; in addition, strain is found by εx ΔX/L where L is the original length of x direction. Based on obtained stress and strain, Young’s modulus can be obtained as Ex =
σx εx
(2.38)
where Ex is Young’s modulus σx is the stress along the x direction εx is the strain along the x direction Poisson’s ratio (v) also can be calculated by vxy = −
εy εx
(2.39)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
and vxz = −
εz εx
(2.40)
where vxy is Poisson’s ratio that characterizes the decrease in the y direction during the tension applied in the x direction εx, εy, and εz are the strain of surface nodes (average) vxz is Poisson’s ratio that characterizes the decrease in the z direction during the tension applied in the x direction Elastic properties are not deterministic in the thin film that contains a small number of grains because the orientation of each grain is stochastic. In this case, multiple simulations should be performed to obtain the elastic properties. However, the number of simulations should be determined before the simulations. Without assuming normality, we need more than 30 simulations (samples) to construct an approximate 95% confidence interval (CI) for the mean elastic properties of a specific size polysilicon thin film (population). The CI is a formula that tells us how to use sample data to determine an interval that estimates a population parameter [44]. The confidence level (i.e., 95%) is the probability that an interval estimator encloses the population parameter. To determine the actual number of simulations, the following equation is used: 2 ⎡⎛ s ⎞ ⎤ n = ⎢⎜ Zα/2 0 ⎟ ⎥ de ⎠ ⎥⎦ ⎢⎣⎝
(2.41)
where s0 is a prior estimate of the standard deviation (usually from previous simulations or pilot experiment) de is the margin of error representing the maximum tolerable error in the estimation α is related to the confidence Zα/2 is the number such that the area under the standard normal density function to the right of Zα/2 is α/2 [46] The margin of error (de) is fixed beforehand, and Table 2.2 shows the set values for de and prior estimates of standard deviation (s0) obtained from previous simulations. The number of simulations, n, is calculated using 95% confidence level. On the basis of Table 2.2, TABLE 2.2 Set Values for de and Prior Estimates of Standard Deviation (s0) Young’s modulus (Ex) Poisson’s ratio (vxy) Poisson’s ratio (vxz)
de
s0
n
2.5 GPa 0.0115 0.0050
6.64 GPa 0.0374 0.0142
27 40 31
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Computational Modeling of Nanoparticles
n = 40 is reasonable. Those 40 simulations can be taken as random samples from the same – distribution (population). Equations of sample mean (X) and standard deviation (s) are described: X=
1 n
s=
∑X
(2.42)
i
S n −1
(2.43)
n
S=
∑ (X − X ) i
2
(2.44)
i −1
where – X is the sample mean s is the standard deviation n is the number of simulations
2.8 Modeling of Optical Properties of Nanoparticles The optical properties of nanoparticles are important for both traditional and emerging technologies [47]. Nanoparticles have long been used as coloring agents in glass and paints. Research on nanoparticle optics soared in the 1970s, due to the increased societal interest in solar-energy applications. Today, metallic nanoparticles are used in commercial coatings that absorb at particular solar wavelengths. Recently, the optical absorption of noble metal nanoparticles has been employed as the basis for novel sensors: the enhanced local fields close to particle surfaces make it possible to detect single molecules by surfaceenhanced spectroscopy [47,48]. When designing optical coatings, it is vital to have accurate models of their optical properties. Nanoparticle-based coatings can be optimized for specific applications by modifying their particle size, shape, volume fraction, microstructure, etc. Because the optics of nanoparticles is a complex subject, different classes of models are needed in different situations [49]. A comprehensive set of working models for submicron particles embedded in supporting media was developed; and experimental studies on metal–insulator composite solar absorbing coatings, nanostructured transparent conductors, pigmented polymers, and paint coatings were carried out [50]. The experimental results were closely related to the predictions of the appropriate theoretical models. Figure 2.15 gives a qualitative picture of regions of validity for three different nanoparticle-optics models: the effective-medium region, the independent scattering region, and the dependent scattering region. Coatings that contain particles much smaller than the wavelength of light can be modeled by effective-medium theories (EMT) [50,51]. In this case, we can conveniently use electrostatics to model the optical properties of the composite. The coating is modeled as a homogeneous “effective medium” with effective optical properties. For larger particles, both scattering and absorption are important, and multiple scattering must be taken into account. In the independent scattering approximation, the
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Computational Nanotechnology: Modeling and Applications with MATLAB®
1000
FIGURE 2.15 The optics of nanoparticle coatings can be predicted using three different models, depending on the size of the particles and their density. We assume that the particles are placed in a simple cubic structure. The size parameter is defined as 2πR/λ, where R is the radius and λ is the wavelength of light. (From Niklasson, G.A., Modelling of optical properties of nanomaterials, SPI news room 10.1117/2.1200603.0182. With permission.)
Size parameter
100
Independant scattering
10
1
0.1 0.01
Dependant scattering
Effective medium region 0.1
0.2
0.3
0.4
0.5
0.6
Volume fraction of particles
coating material is characterized by effective scattering and absorption coefficients that are obtained by summing up the contributions from each particle [52]. This approximation is restricted to dilute media, so each particle scatters light independently of the others. The third region lies between the two extremes. In dense media, dependent scattering effects become important. In this case, a full solution of Maxwell’s equations for the appropriate structure is necessary. EMT can be used for visible light when the particle size is roughly 10–20 nm or smaller. More rigorous limits of validity can be obtained for specific material combinations and wavelength ranges by computations. The EMTs are sensitively dependent on the detailed arrangement of the nanoparticles. EMTs developed for simple model microstructures provide good approximations in many cases. The two most common are the Maxwell–Garnett model for particles randomly embedded in a homogeneous matrix material and the Bruggeman model for a random mixture of two kinds of particles. These cases often provide good working models for applied work. Fractal particle aggregates are special cases that can be treated by electrical-impedance-network analogues [48,49]. The 2D problem of particles on a substrate surface is more difficult because interactions with image charges in the substrate cannot be ignored. However, the problem has been treated in an excellent work by the Leyden group, which is regrettably little used by the optics community. Our multiple-scattering approach for composite coatings on a substrate is based on the four-flux approximation, which considers the scattering and absorption of direct and diffuse light beams propagating through the coating. The scattering and absorption of single particles are computed exactly and used as inputs to the model. The theory is phenomenological, however, and contains several adjustable parameters. Systematic employment of order-of-scattering expansions to develop methods to calculate these parameters has led to more rigorous versions of the four-flux theory. Dependent scattering occurs when the distance between adjacent particles is less than about 0.3 times the wavelength. This occurs for homogeneously distributed particles at high-volume fractions, as shown in Figure 2.15. Dependent scattering models are also needed when nanoparticles stick to one another and form large aggregates. In this case, we must find rigorous solutions for the electromagnetic field that takes into account interference between fields scattered by different particles. Many numerical techniques have been developed and calculations for model structures containing several hundred particles are common today. A number of pieces of computer
Computational Modeling of Nanoparticles
57
Reflectance (R) and transmittance (T)
1.0 Experiment Theory
T
0.8 0.6 0.4 0.2 0
R 0
0.5
1.0
1.5
2.0
2.5
Wavelength (μm)
FIGURE 2.16 The reflectance and transmittance of a porous coating consisting of 16 nm diameter nanoparticles of tin-doped indium oxide (ITO) are shown as a function of wavelength [51]. Excellent agreement is found between experiments and the effective-medium theory for a mixture of ITO and air. (From Niklasson, G.A., Modelling of optical properties of nanomaterials, SPI news room 10.1117/2.1200603.0183. With permission.)
code are available. Our understanding of the optical properties of nanoparticles and nanocomposites has increased considerably in recent years. This has been very useful for our research in nanostructured optical coatings. Selective solar-absorber coatings such as Ni–Al2O3 and Ni–NiO employ nanometer size metal particles embedded in a dielectric. Their optical properties can be accurately modeled by EMT, which can also be used as a design tool. Figure 2.16 shows the excellent agreement between experimental optical data and EMT modeling, which has been obtained for porous transparent conducting coatings consisting of tin-doped indium-oxide nanoparticles [53]. We can also use four-flux theory to optimize solar-absorbing paints with respect to pigment material and particle size as well as film thickness. These are only a few examples of the use of state-of the-art models for the optical properties of nanostructured materials in optical modeling and coating design [54].
2.9 Modeling of Bionanoparticles Biological molecules and systems have a number of attributes that make them highly suitable for nanotechnology applications (Figure 2.17). For example, proteins fold into precisely defined 3D shapes, and nucleic acids assemble according to well-understood rules. Antibodies are highly specific in recognizing and binding their ligands, and biological assemblies such as molecular motors can perform transport operations. Nanoparticles and nanospheres have considerable utility as controlled drug delivery systems [55]. When suitably encapsulated, nanoparticles can be delivered to the appropriate site, their concentration can be maintained at proper levels for long periods of time, and they can be prevented from undergoing premature degradation. Nanoparticles (as opposed to micron-sized particles) have the advantage that they are small enough that they can be injected into the circulatory system [56,57]. Highly porous materials are ideal candidates for controlled drug delivery and tissue engineering. Phospholipid bilayers can self-assemble into long cylindrical tubes with diameters usually below a micron and lengths up to hundreds of microns. Tubules prepared from phospholipid bilayers are ideal for such applications because they are biocompatible [58]. Dendrimers can be prepared so that they are of discrete size and contain specific functional groups. Examples include gene transfer agents
59
Computational Modeling of Nanoparticles
is obviously too lengthy and too costly to be useful in personal medicine. As research in nanotechnology extends the tools for fabrication of integrated circuits to nanometer dimensions, alternative technologies for faster and cheaper DNA sequencing are likely to emerge [63,64]. For example, a device for high-throughput DNA sequencing may be built around a 2 nm diameter pore in a thin (2–5 nm thick) synthetic membrane. The sequence of a DNA molecule can be read by such a device, in principle, through a semiconductor detector, integrated within the pore, which records electric signals induced by DNA passing through the pore. Sequencing DNA strands with a nanopore device has not been achieved yet. As A, C, G, and T DNA nucleotides differ from each other by only a few atoms, in order to relate the sequence of DNA to the measured electrical signals, it is essential to characterize the conformations of DNA inside the pore in atomic detail [65]. At present, MD is the only methodology that can provide realistic images of microscopic events taking place inside the nanosensor. Atom by atom, we build microscopic models of the experimental systems and simulate, adapting the methodology developed by us for membrane proteins, translocation of DNA through pores in synthetic membranes [66,67]. To construct a microscopic model of a nanopore, we can first build an Si3N4 membrane by replicating a unit cell of a β-Si3N4 crystal. By removing atoms from the Si3N4 membrane, we produce pores of symmetric double-conical (hourglass) shapes with radii that correspond to the experiment [68,69]. After placing a DNA molecule in front of the pore, we add water and ions on both sides of the membrane. The final systems measure from 10 to 30 nm in the direction normal to the membrane and include up to 2,00,000 atoms [67]. When a uniform electrical field is applied to all atoms in our system, it induces, at the beginning of the simulation, a rearrangement of the ions and water that focuses the electrical field to the vicinity of the membrane, neutralizing the field in the bulk [70]. The resulting voltage bias V across the simulated system depends both on the magnitude of the applied field E and the dimension Lz of the system in the direction of the field, that is, V = ELz; the electrostatic potential near the pore is not uniform. Applying a 1.4 V bias over a 5.2 nm thick membrane we can observe, within the timescale of an MD simulation, a full permeation of a short (20–60 base pairs) DNA fragment through the nanopore. A typical translocation event is shown in Figure 2.18. At the beginning of the simulation, a 20 base pair fragment of double
0 ns
(a)
4.5 ns
(b)
(c)
24.3 ns
39.9 ns
(d)
48.9 ns
(e)
FIGURE 2.18 MD simulation of DNA translocation through a nanopore in an Si3N4 membrane. (a) Beginning of the simulation. (b) The moment when the terminal Watson–Crick base pair is split at the narrowest part of the pore. (c) A moment during the time interval of 8 ns that DNA spends in the conformation shown without moving. (d) The moment when DNA exits the pore while one base at the DNA end remains firmly attached to the surface of the nanopore. (e) End of the simulation, when most of the DNA have left the pore and the ionic current has returned to the open pore level. (From Lu, D., Aksimentiev, A., Shih, A.Y., Cruz-Chu, E., Freddolino, P.L., Arkhipov, A., and Schulten, K., Phy. Bio., 3:S40–S53, 2006. With permission.)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
stranded (DS) DNA is placed in front of a 2.2 × 2.6 nm2 pore as shown in Figure 2.18a. Within the first 2 ns, the end of DNA nearest to the pore is pulled into the pore by its charged backbone. The rest of the molecule follows that end, which, after 4 ns, encounters the narrowest part of the pore [71]. At this point, the terminal Watson–Crick base pair is split, and the freed nucleotides adhere with their bases to the surface of the pore, as shown in Figure 2.18b. The translocation proceeds further until it halts after 20 ns in the conformation similar to the one shown in Figure 2.18c. At this point, the three Watson–Crick base pairs that were first to enter the pore are split; one of the terminal bases continues to adhere strongly to the pore surface [70,71]. The system is driven out of this metastable conformation by increasing the voltage bias from 1.4 to 4.4 V for 0.3 ns after approximately 30 ns from the beginning of the simulation. Subsequently, the simulation was continued at a 1.4 V bias. DNA slowly exits the pore, while one of the bases holds firmly to the surface of the pore (Figure 2.18d). After about 50 ns, most of the DNAs have left the pore, but a few bases remain attached to the pore surface (Figure 2.18e); 9 of 20 base pairs are split. In addition to driving DNA through the nanopore, the electric field induces displacements of ions, forcing them to move through the pore in opposite directions. When DNA enters the pore, it reduces the ionic current; however, after the DNA translocation is completed, the ionic current recovers to the open pore level. Using this simple principle, the ionic current blockades produced by RNA and DNA strands can be electrophoretically driven through the transmembrane pore of the α-hemolysin channel suspended in a lipid bilayer. Using this approach, one can discriminate different sequences of RNA, DNA homopolymers, as well as the segments of purine or pyrimidine nucleotides within a single RNA strand. Single nucleotide resolution has been demonstrated for DNA hairpins, raising the prospect of creating a nanopore sensor capable of reading the nucleotide sequence directly from a DNA or RNA strand. Using MD simulation in case of, synthetic nanopores, current blockades do not necessarily reflect translocation of DNA through these nanopore. The ionic current can be computed from an MD trajectory by summing up local displacements of all ions over a time interval: I (t) =
1 ΔtLz
N
∑ q [z (t + Δt) − z (t) i
i
i
(2.45)
i 1
where zi and qi are the z coordinate and the charge of atom i, respectively Lz is the length of the simulated system in the direction of the applied electric field (the z-axis in our case) The sum runs over all ions. Nanodiscs are nanometer-sized lipid bilayers, which are stabilized by amphipathic proteins. They provide a platform in which to embed, solubilize, and study single membrane proteins. Membrane proteins can be difficult to study experimentally and are often studied under non-native conditions such as in high concentrations of detergents and in micelles. It is therefore desirable to be able to study membrane proteins, especially integral membrane proteins, in a more native lipid bilayer environment. This has motivated scientists to engineer nanometer-sized discoidal lipid bilayers stabilized by amphipathic helical proteins, termed nanodiscs, which furnish the desired nanoenvironment for membrane proteins. Nanodiscs are homogeneous protein–lipid particles of discrete size and composition [72]. The proteins forming nanodiscs are uniquely engineered to solubilize a small lipid bilayer. These proteins, termed membrane scaffold proteins, are long helical amphipathic proteins, having a hydrophobic face and a hydrophilic face.
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Computational Nanotechnology: Modeling and Applications with MATLAB®
found in nanodiscs. In the case of synthetic nanopores the issue is single molecule electrical recording with the eventual aim of cheap, fast, and accurate DNA sequencing. In this case modeling needs to adapt to new materials, Si3N4 as well as SiO2. Furthermore, modeling needs to provide images of the translocation processes and electrostatics of the permeant-pore systems, which are essential for guiding experiments and technological development. The images provided by the modelers, tested through predictions like blockage currents or ion conductivities verified by observation, are invaluable to the engineers. Not presented, but already accomplished, are predictions of electronic signals measured along the nanopores translocating DNA; such predictions combine MD simulations with the theory of electronic devices and yield clear estimates regarding the sensitivity of measured signals to DNA properties like base pair sequence. In the case of nanodiscs the issue is to make available a nanoscale lipid bilayer membrane for study of related biomedical processes like receptor binding or cholesterol uptake. Proteins native to human cells need to be redesigned for the technical purpose; additionally, the actual equilibrium structures and dynamical properties of proteins and cholesterol in nanodiscs need to be visualized. Biomolecular modeling is involved in both aspects: testing the properties of redesigned scaffold proteins maintaining nanodisc shapes and illustrating the most likely assembly of nanodiscs. Both roles are essential for guiding the technical use of nanodiscs [80].
2.10 Modeling of Polymer Nanocomposites The desirable properties attained by polymers alone are inadequate for new technologies and uses. Therefore, a prominent area of interest lies in the enhancement of material properties in polymer systems. The primary method of property enhancement is the addition of a secondary phase, usually on the nanoscale such as layered silicate clays or carbon nanotubes. These systems are reported to improve properties such as storage modulus [81], gas permeability [82], thermal stability [83], and flame retardance [83]. Several factors influence the extent of improvement, including the chemistry of the polymer matrix, the structure of the clay or nanotube, the dispersion of nanoclay or nanotube within the polymer matrix, and the method of preparation. Currently, these systems are normally only characterized by volume fraction, morphology, and an average description of orientation through orientation factors. In order to develop more robust structure–property relationships by more fully understanding the inner workings of a material system, advanced characterization techniques must be developed. It is believed that texture as defined by the orientation distribution of clay plate normals in a polymer–clay nanocomposite is an important microstructural feature that needs characterization. A technique x-ray diffraction (XRD) pole figure analysis characterizes polymer–clay nanocomposites by observing texture as defined by nanoclay plate normal distributions [84–89]. The technique for constructing pole figures to describe texture of intercalated nanocomposites is devised by calculating the full width at half maximum (FWHM) of the clay basal spacing peak after λ integration for each XRD image. The FWHM and the center of the peak are computed using Lorentzian curve-fitting in the Fityk peak-fitting software to create a template that defines the region of each XRD image to be included in calculations. Radial integration of the edited images provided intensity values that correspond to the angles α and β, the two angles in spherical coordinates that describe the direction of the normal to a clay platelet on the unit sphere. The distribution of α and β can be plotted as a stereographic projection to form a pole. Because the clay trilayers are treated as transversely isotropic discs, only
65
Computational Modeling of Nanoparticles
×10–3
(a)
3
3
2.5
2.5
2
TD
×10–3
1.5
2
TD
1.5
1
1
0.5
0.5 (b)
ND
ND ×10–3
×10–3 9 8
8 7
7
6
6 5
TD
4
5
TD
4 3
3
(c)
ND
2
2
1
1 (d)
ND
0
FIGURE 2.22 The effect of morphology on clay plate normal distribution with different clay loadings: (a) 2 wt% 30B intercalated, (b) 5 wt% 30B intercalated, (c) 2 wt% 30B phase separated, and (d) 5 wt% 30B phase separated (generated using MATLAB®).
two angles can be used to fully describe their orientation. Each group of trilayers has two normals. All normals residing in the southern hemisphere are converted to the equivalent normal in the northern hemisphere prior to pole figure construction. Image analysis can be performed using MATLAB® codes. Images can be converted from the Rigaku OSC format to a format usable by MATLAB with the MOSFLM program, a supported part of the CCP4 software suite. Figure 2.22 displays how pole figures change depending on when the clay loading is increased in the polymer. The intercalated structures are on the top of Figure 2.22a and b while the phase separated structures are at the bottom (Figure 2.22c and d). The two pole figures on the left of Figure 2.22a and c represent 2 wt% 30B while those on the right (Figure 2.22b and d) represent 5 wt% 30B. The 2 wt% and 5 wt% samples exhibit significantly different textures. It is clear from the figure that pole figures describing the texture of the composite system show a greater dispersion of clay plate normals than do the phase separated peaks. This again would support the theory that the plates are more difficult to shift from their original position when they have been intercalated by polymer. This technique provides a more in-depth look into microstructure, which can subsequently be used to more fully understand and develop efficient structure–property relationships [90].
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Computational Nanotechnology: Modeling and Applications with MATLAB®
MATLAB codes for Figure 2.22 clear; directory = 'Z:\Masters\XRD_Data\Results\30B0.02--T1--1'; base = '30B0.02--06.19.07--1'; tic cd(directory); start_image = 1; num_images = 72; iteration = 1; % Override original_base = base(4:7); new_base = original_base; %#ok new_base = '0.05'; base(4:7) = new_base; if strcmp(base(1:3),'30B') if strcmp(base(4:7),'0.02') theta = 0.04133.* ones(180,1); image_dim = 399; BG = imread('F:\Masters\Images\Background--... small_inverted_edited_intercalated.jpg'); else theta = 0.08343.* ones(180,1); image_dim = 699; BG = imread('F:\Masters\Images\Background--... small_inverted_edited.jpg'); end elseif strcmp(base(1:3),'25A') if strcmp(base(4:7),'0.02') theta = 0.04133.* ones(180,1); image_dim = 399; BG = imread('F:\Masters\Images\Background--... small_inverted_edited_intercalated.jpg'); else theta = 0.08308.* ones(180,1); image_dim = 699; BG = imread('F:\Masters\Images\Background--... small_inverted_edited.jpg'); end elseif strcmp(base(1:3),'Na+') if strcmp(base(4:7),'0.02') theta = 0.04133.* ones(180,1); image_dim = 399; BG = imread('F:\Masters\Images\Background--... small_inverted_edited_intercalated.jpg'); else theta = 0.13195.* ones(180,1); image_dim = 699; BG = imread('F:\Masters\Images\Background--... small_inverted_edited.jpg'); end elseif strcmp(base(1:2),'PS') theta = 0.16923.* ones(180,1);
Computational Modeling of Nanoparticles
image_dim = 1699; BG = imread('F:\Masters\Images\Background--... PS_Vol_Correction.jpg'); end num_images_str = int2str(num_images); center = [1495.4 1506.0]; polefig_dim = 250; firstrun = 1; polefig = zeros(180*num_images,8); h = waitbar(0,['0/', num_images_str, 'images processed']); load 'Z:\Masters\XRD_Data\Extra_Data\correction_factor.mat';... %volume correction chi = (0:179)'.* (pi/180) + (pi/2); %radians %% for eta_int = start_image:iteration: (start_image+num_images−1) if eta_int < 10; tempstr = '_00'; else tempstr = '_0'; end eta_str = int2str(eta_int); if strcmp(base(4:7),'0.02') base(4:7) = original_base; filename = [directory, '\', base, tempstr, eta_str,... '_small_inverted_edited_intercalated.jpg']; base(4:7) = new_base; else base(4:7) = original_base; filename = [directory, '\', base, tempstr, eta_str,... '_small_inverted_edited.jpg']; base(4:7) = new_base; end E = imread(filename)−BG; volume_correction = correction_factor(eta_int)*ones(size(chi)); [R,xp] = radon(E,0:179,3); Intensity = (R(2,:))'.* volume_correction; eta = ((((eta_int*5)−5)+2.5) * (pi/180)).*... ones(size(Intensity)); %radians alpha = acos(cos(theta).*sin(chi)); %radians beta = atan2((−sin(eta).*sin(theta)+cos(eta).*cos(theta).*... cos(chi)),(−cos(eta).*sin(theta)−sin(eta).*... cos(theta).*cos(chi))); %radians operate_these = find(alpha > (pi/2)); alpha(operate_these) = pi−alpha(operate_these); for ii = 1:size(beta(operate_these)) if beta(operate_these(ii)) > 0 beta(operate_these(ii)) = beta(operate_these(ii))−pi; else beta(operate_these(ii)) = beta(operate_these(ii))+pi; end end %% % Stereographic Projection c = round((sin(alpha)./(1+cos(alpha))).*cos(beta).*... (polefig_dim./2))+(polefig_dim./2)+1; d = round((sin(alpha)./(1+cos(alpha))).*sin(beta).*... (polefig_dim./2))+(polefig_dim./2)+1;
67
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Computational Nanotechnology: Modeling and Applications with MATLAB®
if firstrun == 1 polefig=[c d Intensity alpha beta eta chi theta]; firstrun = 0; polefig_length = size(polefig,1); else polefig(polefig_length+1: polefig_length+size(c,1),:) =... [c d Intensity alpha beta eta chi theta]; polefig_length = size(polefig,1); end eta_str2 = int2str(eta_int−start_image+1); waitbar((eta_int−start_image+1)/num_images, h,... [eta_str2,'/',num_images_str, 'images processed']); end save(['polefig_', directory(29:size(directory,2)),'.mat'],'polefig'); close(h); %% Normalize pole figure unnormalized_intensity = polefig(:,3); sin_alpha = sin(polefig(:,4)); integrand = unnormalized_intensity.* sin_alpha; total_intensity = sum(integrand).* ones(size(integrand)); normalized_intensity = ((2*pi).* unnormalized_intensity)./... total_intensity; polefig(:,3) = normalized_intensity; %% Pcolor plot figure X1 = 1:polefig_dim; Y1 = (1:polefig_dim)'; [X1,Y1,Z1] = griddata(polefig(:,1),polefig(:,2),polefig(:,3),... X1,Y1,'cubic'); pcolor(Z1); shading interp; colormap(jet(20)); axis equal; xlim([0 polefig_dim]); ylim([0 polefig_dim]); colorbar; set(gca, 'XTick',[],'YTick',[]); text(0,125,'TD','horizontalAlignment','right','verticalAlignment'... ,'middle','FontSize',22) text(125,0,'ND','horizontalAlignment','center','verticalAlignment',... 'top','FontSize',22) t=toc %#ok
2.11 Opportunities and Challenges in Computer Modeling of Nanoparticles Modeling is a rather new research and development methodology, unfamiliar to many bench scientists and still considered not trustworthy. However, in entirely new technological areas like bionanotechnology, qualitative concepts, pictures, and suggestions are sorely needed, and we hope that our examples have shown that biomolecular modeling can serve a critical role in this respect. Till date, nanotechnology has been developed mostly from the basis in physics, chemistry, material science, and biology. As nanotechnology is a truly multidisciplinary field, the cooperation between researchers in all related areas is crucial to the success of nanotechnology. Computer science has taken a role mostly in research tools, for example, a virtual-reality system coupled to scanning probe devices
Computational Modeling of Nanoparticles
69
in nanomanipulator project [91]. However, the third and fourth generations of nanotechnology rely heavily on research in computer science. Computer science is today a broad field with many aspects that may affect nanotechnology. Earlier sections have outlined the use of graphics and imaging with nanomanipulators. Other current uses of computer science for nanotechnology include developing software systems for design and simulation. A research group at NASA has been developing a software system, called NanoDesign, for investigating fullerene nanotechnology and designing molecular machines. The software architecture of NanoDesign is designed to support and enable their group to develop complex simulated molecular machines. However, here we focus on intelligent systems. Research in intelligent systems involves the understanding and development of intelligent computing techniques as well as the application of these techniques for realworld tasks, often including problems in other research areas. The techniques in intelligent systems comprise methods or algorithms in artificial intelligence (AI) including knowledge representation/reasoning, machine learning, and natural computing or soft computing. An exciting new development at the time of writing is a project called PACE (programmable artificial cell evolution) [92]. This large interdisciplinary project aims to create a “nanoscale artificial protocell able to self replicate and evolve under controlled conditions.” The protocells in this work are intended to be the “simplest technically feasible elementary living units (artificial cells much simpler than current cells).” These are intended to act as nanorobots, comprising an outer membrane, a metabolism, and peptideDNA to encode information. Evolutionary modeling is being used extensively in PACE, to analyze real and simulated protocell dynamics, their possible evolution, and the evolution of (potentially noisy) protocellular networks. In addition to this work, computer modeling of embryogenesis and developmental systems is becoming increasingly popular in computer science. Such models will provide a method for their genes to be programmed in order to enable the growth of larger, multicellular forms. Apart from genetic algorithms and other evolutionary algorithms (EAs) that have promising potential for a variety of problems (including automatic system design for molecular nanotechnology), another emerging technique is swarm intelligence, which is inspired by the collective intelligence in social animals such as birds, ants, fish, and termites. Typical uses of swarm intelligence are to assist the study of human social behavior by observing other social animals and to solve various optimization problems. There are three main types of swarm intelligence techniques: models of bird flocking, the ant colony optimization (ACO) algorithm, and the particle swarm optimization (PSO) algorithm. Different techniques are suitable for different problems [93]. Although the field of computer science is still young, swarm intelligence is becoming established as a significant method for parallel processing and simultaneous control of many simple agents or particles in order to produce a desired outcome. Likewise, BT’s Future Technologies Group developed a software platform known as EOS, for EAs and ecosystem simulations [94]. Systems such as these will become increasingly important for modeling molecular machine systems. They are also being investigated as a solution to provide self-healing, adaptive, and autonomous telecommunications networks. Another potential benefit of such techniques for complex adaptive systems in this area would be to control intelligently the manufacture of nanometer-scale devices, where no exact mathematical model of the system exists. Many intelligent systems’ techniques have been successfully applied in control system of various complex applications. Although at nanometer scale the principles and properties of materials are altered, researchers have attempted to solve other dynamic problems using soft computing techniques and have been developing new techniques to cope with such problems. Also inspired by emergent collaborating behaviors of social insects, the Autonomous Nanotechnology Swarm (ANTS)
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architecture for space exploration by NASA Goddard Space Flight Center is claimed to be revolutionary mission architecture [93]. Researchers at the center have been developing a framework to realize the autonomous intelligent system by using an evolvable neural interface (ENI). As a result, the interface allows cooperation between higher level neural system (HLNS) for elementary purpose actions and lower-level neural system (LLNS) for problem solving as required in real-world situations. In the plan, each autonomous unit will be capable of adapting itself for its mission, and the ANTS structures will be based on carbon nanotube components. Recently, a new swarm algorithm, called the perceptive particle swarm optimization (PPSO) algorithm has been developed [93]. The PPSO algorithm is an extension of the conventional PSO algorithm for applications in the physical world. By taking into account both the social interaction among particles and environmental interaction, the PPSO algorithm simulates the emerging collective intelligence of social insects more closely than the conventional PSO algorithm; hence, the PPSO algorithm would be more appropriate for real-world physical control problems. This is the first particle swarm algorithm to be explicitly designed with nanotechnology in mind. Because each particle in the PPSO algorithm is highly simplified (each able to detect, influence, or impact local neighbors in limited ways) and the algorithm is designed for working with a large number of particles, this algorithm would be truly suitable for programming or controlling the agents of nanotechnology (whether nanorobots, nanocomputers, or DNA computers), whose abilities are limited, to perform effectively their tasks as envisioned [95]. This is seen as a crucial “missing link” in bottom-up nanotechnology: the control of the nanosized agents. A billion (or trillion) tiny particles, whether complex molecules or miniature machines, must all cooperate and collaborate in order to produce the desired end result. None will have, individually, sufficient computing power to enable complex programming. Like the growth of crystals, the development of embryos, or the intelligent behavior of ants, bottom-up nanotechnology must be achieved through collective, emergent behaviors, arising through simple interactions amongst itself and its environment. Computer science, and especially fields of research such as swarm intelligence, will be critical for the future of bottom-up nanotech.
2.12 Conclusion As the development of nanotechnology progresses in several disciplines including physics, chemistry, biology, and material science, computer scientists must be aware of their roles and brace themselves for the greater advancement of nanotechnology in the future. This chapter has outlined the development of nanotechnology. It is hoped that this gentle review will benefit computer scientists who are keen to contribute their works to the field of nanotechnology. The possible opportunities that computer science can offer have also been discussed, which can benefit other nanotechnologists from other fields by helping them be aware of the opportunities from computer science. As computer scientists who are interested in the field of nanotechnology, one of the future works is to build a system that consists of a large number of particles automatically forming into a designed structure. By using the computer simulation techniques to control the swarm of particles, each particle can be utilized to perform lightweight computations and hold only a few values. It is anticipated that such models will lead to successful bottom-up nanotechnology systems in the future.
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Acknowledgment The corresponding author wishes to thank DST-SERC for granting research project under “Fast Track Scheme For Young Scientists” vide sanction no SR/FT/CS-012/2008.
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3 Micromagnetics: Finite Element Analysis of Nano- Sized Magnetic Materials Using MATLAB® Shin-Liang Chin and Timothy Flack CONTENTS 3.1 Introduction.......................................................................................................................... 75 3.2 Theory of Micromagnetics ................................................................................................. 76 3.3 Landau–Lifshitz Gilbert Equation..................................................................................... 79 3.4 Finite Element Approach ....................................................................................................80 3.4.1 Mesh Generation ......................................................................................................80 3.4.2 Interpolation Function and Coordinate Transformation ................................... 81 3.4.3 Numerical Integration.............................................................................................83 3.4.4 Scalar Potential Calculation ...................................................................................84 3.4.5 Galerkin Weighted Residual Calculation for Landau–Lifshitz Gilbert Equation....................................................................................................... 87 3.4.6 MATLAB® Implementation .................................................................................... 89 3.4.7 Post-Processing......................................................................................................... 91 3.5 Applications.......................................................................................................................... 92 3.6 Conclusions........................................................................................................................... 94 References........................................................................................................................................ 95
3.1 Introduction The origin of magnetism is electron spin. In the macroscale (typically >1 μm), quantum mechanical effects are negligible, and the behavior of magnetic materials is typically simulated using just Maxwell’s equations. In recent decades, advancement in e-beam lithography and fabrication methods have made nano-scale magnets possible. At such dimensions, quantum mechanical effects become significant and Maxwell’s equations alone will not be sufficient to describe the behavior of magnetic materials. Obviously, simulating these nano-magnetic materials from a quantum physics point of view would give the most “accurate” description. In practice, however, such a simulation would be difficult, if not impossible, because the calculation will involve millions of atoms or molecules. Therefore, it is useful to find a compromise between Maxwell’s theory (field theory approach) and the quantum mechanical approach. The formalism of micromagnetics by Brown in the 1960s [1,2] is a semiclassical approach to model magnetic materials at the nano scale. In essence, the theory of micromagnetics incorporates quantum mechanical effects by replacing the electron spin distribution with a continuous vector field and approximates the exchange energy based on this vector field. The main advantage of this micromagnetics theory, from a simulation point of view, is that it allows us to simulate magnetic materials 75
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applied field. The energy due to any externally applied field will be considered below in the Zeeman energy. This magnetostatic energy, also commonly known as the demagnetizing energy, is governed by Maxwell’s equations, that is, D = 0, ⋅B ∇
(3.3)
D = 0, ×H ∇
(3.4)
D = μ( H D + M ), B
(3.5)
where μ is the permeability of free space H˜ D and B˜D are the magnetic induction and the magnetic field vectors for the demagnetizing field, respectively Combining Equations 3.3 through 3.5 gives D = ∇ D + M )⎤ = 0 , ⋅B ⋅ ⎡μ( H ∇ ⎣ ⎦
(3.6)
and since the curl of H˜ D is zero, H˜ D can be expressed as a gradient of a scalar potential: D = −∇ϕ. H
(3.7)
When this is substituted into Equation 3.6, we obtain ⋅ M. ∇ 2ϕ = ∇
(3.8)
Solving this for the potential, subject to the following boundary conditions, ϕEXTERNAL = ϕINTERNAL ,
(3.9)
ϕ EXTERNAL ) ⋅ n ⋅ ϕ INTERNAL ) ⋅ n ⋅n − (∇ =M , (∇
(3.10)
where ñ is a unit vector pointing outward from the surface of the magnetic material, we can obtain the expression of the potential scalar that consists of a magnetic surface charge density component and a magnetic volume charge density component: ϕ (r ) = −
1 4π
∫ V
⋅M (r ʹ ) 1 ∇ dV ʹ + 4π r − rʹ
∫ S
(r ʹ ) ⋅ n M dSʹ , r − rʹ
(3.11)
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where r and r′ are the position vector of the source and destination, respectively, and the del operator acts only on r′. The energy in relation to the demagnetizing energy is then given by EMAG = −
μ ⋅H D )dV . (M 2
∫
(3.12)
V
Equations 3.11 and 3.12 show that the magnetostatic energy calculation requires a sixfold integration, which is obviously computationally expensive. The Zeeman energy arises from an externally applied magnetic field. In the presence of such a magnetic field, the magnetic moments will align themselves such that they are parallel to it. The Zeeman energy is thus given by ⋅H A dV , EZEEMAN = −μ M
∫
(3.13)
V
where H˜ A is the magnetic induction vector of the externally applied field. In addition to the above energy contributions, there is also the anisotropy energy, which is intrinsic to the material. The most common form of anisotropy is the stress anisotropy (also known as magnetostriction) and the crystalline anisotropy. Here, we will briefly discuss only crystalline anisotropy. Crystalline anisotropy occurs due to the spin–orbit interaction of the electrons. Depending on their lattice structure, the magnetic materials have a “preferred” direction of magnetization. This is called “the easy axis.” For example, a hexagonal crystal lattice structure will have uniaxial anisotropy and the anisotropy energy can be described by [3] E = KV sin 2 θ + other higher terms where θ is the angle between the easy axis and the magnetization K is the anisotropy constant V is the sample volume Combining the contributions from the exchange energy, magnetostatic energy, and the Zeeman energy (the anisotropy energy is neglected here), the total Gibbs free energy is ETOTAL = EEXCHANGE + EMAG + EZEEMAN =
C m y )2 + ( ∇ mz )2 dV − 1 ( H D ⋅ M ) dV − μ M ⋅H A dV . ( ∇ m x )2 + ( ∇ 2 2
∫ V
∫ V
∫
(3.14)
V
We can deduce from Equation 3.12 that the magnetostatic energy will reduce if the contributions due to each small sub-domain do not superimpose. In turn this occurs if the magnetic material breaks down into many small domains, with the magnetization in each domain pointing in different directions (refer to Figure 3.2). The overall demagnetizing field will then be smaller as adjacent magnetizations cancel each other out. However, breaking down into small domains means that the magnetization direction would need to rotate rapidly over a small distance and this, therefore, increases the exchange energy. The Zeeman energy, on the other hand, is at a minimum when the magnetization is aligned
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Micromagnetics: Finite Element Analysis of Nano-Sized Magnetic Materials
++++
– – – –
++
––
––
+ +
+
–
–
+
+ –
– +
FIGURE 3.2 Ferromagnetic materials form domain walls to reduce the demagnetizing energy.
with the externally applied field. Thus, the energy ground state deduced from the micromagnetics theory is the result of the competing energy terms. The equilibrium or the energy ground state of the magnetization distribution can be found by using common minimization methods such as the conjugate gradient method or the Newton–Raphson method. However, to find the dynamic response of the magnetization in response to an externally applied field, the Landau–Lifshitz Gilbert (LLG) equation has to be solved.
3.3 Landau–Lifshitz Gilbert Equation The magnetization reversal and movement of domain walls are some examples of the dynamic process, which cannot be determined using the energy minimization method based on Equation 3.14. In the presence of an external field, the magnetic moment precesses about the applied field until a new energy ground state is achieved. This dynamic process is governed by the LLG equation ∂M γ ×H eff − γ ⋅ α [ M × (M ×H eff )], M =− 2 ∂t 1+ α 1 + α2 M
(3.15)
where γ is the gyromagnetic ratio α is the damping constant H˜ eff is the effective magnetic field The effective magnetic field is found by applying variational principle to Equation 3.14 eff = − ∂ETOTAL = C ∇ 2m A + μH D, + μH H MS ∂M
(3.16)
where ∇2m ˜ = ∇2mxi˜ + ∇2myj˜ + ∇2mzk˜. We would expect a ½ factor for the demagnetizing field 1 term in H˜ eff since EMAG = EZEEMAN , but the ½ factor has “disappeared” because H˜ D depends 2 on magnetization of the material whereas H˜ A does not. By substituting Equation 3.1 into Equation 3.15, the LLG equation can be further reduced to ∂m γ eff − γ ⋅ α [m eff )]. ×H × (m ×H =− m 2 ∂t 1+ α 1 + α2
(3.17)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
The first term describes the precession motion of the magnetic moment and the second term is the damping term. Without the damping term, the magnetic moment will precess forever without reaching an energy ground state. The damping factor must be determined experimentally and if set close to unity will be equivalent to finding the ground state of the magnetization using the steepest descent approach. A very good complete derivation of the LLG equation can be found in Ref. [4].
3.4 Finite Element Approach The finite element method is a numerical method for solving boundary-valued partial different equations (PDEs). It is widely used in structural analysis and electromagnetic field computation. This method is particularly useful when trying to model an arbitrary geometry by discretizing the geometry into smaller and identical shapes such as prisms and tetrahedrons. However, this comes at the cost of higher complexity in implementation and higher computation resources. In general, there are five major steps in the finite element method. First, the domain is discretized into identical shapes (but with different sizes) called finite elements. Then, the solution of the PDE in each of these elements is approximated using interpolation functions. A global matrix system of equations is formed by assembling the equations for each element and the boundary condition is imposed. This will form a linear system of equations, which is solved to find the coefficients of the interpolation functions such that the distance from the actual solution is minimum. 3.4.1 Mesh Generation There are a lot of free mesh generators available online. For the purpose of this chapter, it is sufficient to have a mesh generator that generates only two-dimensional (2D) triangle elements. The triangles can then be easily transformed into a triangular prism. After the mesh is generated, the following information should be made available: X(i) is the x coordinate for the ith node Y(i) is the y coordinate for the ith node Z(i) is the z coordinate for the ith node Node (e, i) is the element information, where the index e denotes the element number and i = 1, 2, …, 6 denotes global nodal numbering for that element When generating a mesh for the finite element method, it is important to identify if a surface is shared between two elements. These surfaces have their normal vectors pointing in opposite directions and therefore the surface integral for these shared surfaces cancel each other out. Identifying these shared surfaces and avoiding calculating the surface integrals for these surfaces can save a significant amount of computational resources. The accuracy of a micromagnetics simulation depends on the mesh size. For magnetic soft materials, it is important that the mesh size is comparable to or smaller than the exchange length, lex, which is given by the following expression: lex =
C 2 . μM
(3.18)
Micromagnetics: Finite Element Analysis of Nano-Sized Magnetic Materials
81
It should be noted that using a mesh size comparable to the exchange length does not guarantee reliable results. One should always run the simulation a few times with incrementally smaller mesh size to ensure convergence. 3.4.2 Interpolation Function and Coordinate Transformation It is assumed that the basics of the finite element method are known to the reader. For those readers who are unfamiliar with this method, [6–8] are very good references to begin with. In general, the very first step in the finite element method is to choose a proper interpolation function for the discretized domain. Tetrahedral elements are commonly used for three-dimensional (3D) finite element structures. Such elements will be useful, for example, if we are simulating a sphere. However, in this chapter, a triangular prism is used for two reasons. Firstly, a lot of magnetic materials currently under investigation are magnetic thin films. Thin films are really just a 2D structure where the thickness of the films is usually uniform. Thus, a triangular prism is usually sufficient. Secondly, the implementation of a triangular prism finite element is an easier example to follow. A coordinate transformation is performed on each of the discretized triangular prisms such that their coordinates in the solution domain (x, y, z) are mapped onto the “natural coordinates” (ε, η, ξ) with their nodes positioned as shown in Figure 3.3. The linear interpolation function for each node on the triangular prism element is then N1e =
ε(1 − ξ) , 2
(3.19)
N 2e =
η (1 − ξ ) , 2
(3.20)
N 3e =
(1 − ε − η)(1 − ξ) , 2
(3.21)
N 4e =
ε(1 + ξ) , 2
(3.22)
N 5e =
η (1 + ξ ) , 2
(3.23)
η Node 5 (0, 1, 1)
Node 2 (0, 1, –1)
Node 3 (0, 0, –1)
Node 6 (0, 0, 1)
Node 4 (1, 0, 1)
ε
Node 1 (1, 0, –1)
FIGURE 3.3 A triangular prism in natural coordinates (ε, η, ξ).
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Computational Nanotechnology: Modeling and Applications with MATLAB®
N 6e =
(1 − ε − η)(1 + ξ) . 2
(3.24)
Note that N ie = 1 only at the ith nodes. The reason for this coordinate transformation is that the integration limits are the same when considering different unit elements. This makes the integration over a large area with large numbers of elements substantially easier. Any variable φ, including the x, y, and z coordinates, can then be interpolated within the triangular prism by 6
ϕ (ε , η, ξ) =
∑ N (ε, η, ξ) ϕ , e i
e i
(3.25)
i 1
where φie is the variable’s value at the ith node. These values have to be known at the start of the numerical simulation. Furthermore, the volume and surface integral of the finite element method would necessitate the evaluation of the derivative of the interpolation function with respect to the natural coordinates. The relationship between the derivative in the transformed natural coordinate and the derivative in the Cartesian x, y, z coordinates can be obtained using the chain rule of differentiation: ⎡ ∂N i ⎤ ⎡ ∂x ⎢ ⎥ ⎢ ⎢ ∂ε ⎥ ⎢ ∂ε ⎢ ∂N i ⎥ ⎢ ∂x ⎢ ∂η ⎥ = ⎢ ∂η ⎢ ⎥ ⎢ ⎢ ∂N i ⎥ ⎢ ∂x ⎢ ∂ξ ⎥ ⎢ ∂ξ ⎣ ⎦ ⎣
⎡ ∂N i ⎤ ∂z ⎤ ⎡ ∂N i ⎤ ⎥⎢ ⎥ ⎢ ⎥ ∂ε ⎥ ⎢ ∂x ⎥ ⎢ ∂x ⎥ ∂z ⎥ ⎢ ∂N i ⎥ ∂N = J ⎢⎢ i ⎥⎥ ∂η ⎥ ⎢ ∂y ⎥ ∂y ⎥⎢ ⎢ ⎥ ⎥ ∂z ⎥ ⎢ ∂N i ⎥ ⎢ ∂N i ⎥ ⎢⎣ ∂z ⎥⎦ ∂ξ ⎥⎦ ⎢⎣ ∂z ⎦⎥
∂y ∂ε ∂y ∂η ∂y ∂ξ
⎡ ∂N i ⎤ ⎡ ∂N i ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ε ⎥ ⎢ ∂x ⎥ ∂N ⎢ ∂N ⎥ ⇒ ⎢⎢ i ⎥⎥ = J −1 ⎢ i ⎥ , ∂η ∂y ⎢ ⎥ ⎢ ⎥ ∂ ⎢ Ni ⎥ ⎢ ∂N i ⎥ ⎢ ∂ξ ⎥ ⎢⎣ ∂z ⎥⎦ ⎣ ⎦
(3.26)
where J is known as the Jacobian matrix, and by using Equation 3.25, it can be expressed in terms of the interpolation function and the nodal values of the Cartesian coordinates: ⎡ ⎢ ⎢ ⎢ ⎢ J=⎢ ⎢ ⎢ ⎢ ⎢⎣
6
6
6
∂N i xi ∂ε
∑
∂N i yi ∂ε
∑
∂N i xi ∂η
6
∑
∂N i yi ∂η
∑
6
∂N i xi ∂ξ
6
∂N i yi ∂ξ
6
∑ i 1 6
i 1
∑ i 1
i 1
i 1
∑ i 1
∂N i
⎤
∑ ∂ε z ⎥⎥ i 1 6
i 1
∑ i 1
i
⎥ ∂N i ⎥ zi . ∂η ⎥ ⎥ ∂N i ⎥ zi ⎥ ∂ξ ⎥⎦
(3.27)
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83
Having performed the coordinate transformation, the surface integral and volume integral are then expressed as
∫ dS(x, y, z) = ∫ det J dS(ε, η, ξ), S
(3.28)
S
1 1 1− η
∫
dxdydz =
∫ ∫ ∫ det J dεdηdξ . −1 0
V
(3.29)
0
3.4.3 Numerical Integration The integration can be performed numerically using Gaussian quadrature, and the classical book Numerical Recipes in C [9] gives a good account of implementing it in C as well as some fundamental theory behind this method. In order to carry out this integration using Gaussian quadrature, weighting coefficients at specific points (abscissas) of the function needs to be known. (The weighting coefficients can be generated using the formulae described in the book or by simply using function files available on file exchange at MATLAB® central.) Given N known weighting coefficients and their corresponding abscissas, an integration can be approximated by b
N
∫ f (x)dx ≈ ∑ ω f (x ). i
(3.30)
i
i 1
a
Then for any variable interpolated within the triangular prism the volume integral is (refer to Equation 3.29) given by
1 1 1− η
N
N
N
∫ ∫ ∫ φ det J dεdηdξ = ∑ ∑ ∑ ω ω ω φ det J i
−1 0
⇒
i 1
0
k
6
N
N
i 1
j 1 k 1 l 1
N
j
j 1 k 1
∑ ∑ ∑ ∑ ω ω ω N (ε i
j
k
l
ijk
, ηijk , ξijk )ϕl det J (εijk , ηijk , ξijk ) ,
where ω is the weighting coefficient J is the Jacobian (ε, η, ξ) are the natural coordinates as defined before
(3.31)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
Since the value of N, its derivative, and the determinant of the Jacobian changes for each abscissa, it would be useful to create a MATLAB function that returns these values when the abscissa is given. (Note that the following function actually outputs the Jacobian, not the determinant of the Jacobian. However, the determinant can be easily found using the MATLAB function det( J)). %this function returns the value for the Jacobian, %interpolation function (N) and its derivative. function [J,N,dnde]=Jac(x,y,z,e1,e2,e3) %e1, e2 and e3 are the abscissas for Gaussian quadrature integration %e1 and e2 are interchangeable, but e3 must refer to the specific %gaussian points generated with the range from −1 to 1. %x(i),y(i), z(i) are the i-th node coordinates for each element %i=1,2,…,6 %interpolation function N(1)=e1*(1−e3)/2; N(2)=e2*(1−e3)/2; N(3)=(1−e1−e2)*(1−e3)/2; N(4)=e1*(1+e3)/2; N(5)=e2*(1+e3)/2; N(6)=(1−e1−e2)*(1+e3)/2; %i-th row is Ni, j-th column is the natural coordinate dnde(1,:)=[(1−e3)/2,0,−e1/2]; dnde(2,:)=[0,(1−e3)/2,−e2/2]; dnde(3,:)=[(e3−1)/2,(e3−1)/2,(e2+e1−1)/2]; dnde(4,:)=[(1+e3)/2,0,e1/2]; dnde(5,:)=[0,(1+e3)/2,e2/2]; dnde(6,:)=[−(1+e3)/2,−(1+e3)/2,(1−e1−e2)/2]; %Jacobian matrix xyz=[x', y', z']; J=dnde'*xyz; end
3.4.4 Scalar Potential Calculation Substituting Equation 3.25 into Equation 3.1 gives (for each element) ⎛ e = Ms ⋅ ⎜ M ⎜ ⎝
6
∑ i 1
N i mxi i +
6
∑ i 1
N i myi j +
6
⎞
∑ N m k ⎟⎟⎠ , i
zi
i 1
where mxi, myi, and mzi are the magnetization components at the ith node of each element Ni is the interpolation function at the ith node of each element
(3.32)
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Micromagnetics: Finite Element Analysis of Nano-Sized Magnetic Materials
Substituting this into the volume charge density term in Equation 3.11, we obtain ϕ( x0 , y0 , z0 )volume ⎧ ⎪ 6 Ms ⎪ =− ⎨ 4π ⎪ i 1 ⎪ ⎩
∑
⎛ ⎜⎝
⎛ dNi ⎞ dV ⎟ mxi + ⎜ ⎠ V dx ⎝
∫
⎛ ⎜⎝ x0 −
∑
6 j
2
⎞ dN i ⎛ dV ⎟ myi + ⎜ ⎝ V dy ⎠
∫
⎞ ⎛ N j x j ⎟ + ⎜ y0 − ⎠ ⎝ 1
∑
6 j
2
⎫ ⎪ ⎪ ⎬. 2 ⎞ ⎪ N jzj ⎟ ⎪ ⎠ ⎭ 1
dN i ⎞ dV ⎟ mzi ⎠ V dz
∫
⎞ ⎛ N j y j ⎟ + ⎜ z0 − ⎠ ⎝ 1
∑
6 j
(3.33)
From this equation, we can see that the coefficients for mxi, myi, and mzi are only dependent on the geometry of the magnetic sample, that is, the x, y, z coordinates of each nodes in each element. Unless adaptive meshing is used, these values usually remain constant throughout the integration time step. Therefore, we can speed up the numerical integration, at the expense of requiring more memory, by computing these coefficients only once during the initialization phase (before the time stepping). The scalar potential can be obtained by multiplying the same coefficients with the updated values of mxi, myi, and mzi at the beginning of each time step iteration. %this computes the coefficients for the volume integrals %as described in Equation 3.33 %xx,yy,zz corresponds to x0, y0, z0 %pnt1, pnt2, pnt3 are the abscissas %w1, w2, w3 are the corresponding weighting coefficients %volx, voly, volz are the coefficients for volume integrals volx=zeros(6,1); voly=zeros(6,1); volz=zeros(6,1); for u=1:size(pnt3,1) for v=1:size(pnt2,1) for w=1:size(pnt1,1) %function Jac generates the Jacobian, N and its derivative [J,N,dnde]=Jac(x,y,z,pnt1(u,v,w),pnt2(u,v),pnt3(u)); dndx=(J\dnde')'; x_temp=0; y_temp=0; z_temp=0; %generate the denominator first. for i=1:6 x_temp=x_temp+N(i)*x(i); y_temp=y_temp+N(i)*y(i); z_temp=z_temp+N(i)*z(i); end denum=sqrt( (xx−x_temp)∧2+(yy−y_temp)∧2+(zz−z_temp)∧2); for i=1:6 volx(i)=volx(i)+w1(u,v,w)*w2(u,v)*w3(u)*det(J)… *dndx(i,1)/denum;
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voly(i)=voly(i)+w1(u,v,w)*w2(u,v)*w3(u)*det(J)… *dndx(i,2)/denum; volz(i)=volz(i)+w1(u,v,w)*w2(u,v)*w3(u)*det(J)… *dndx(i,3)/denum; end end end end
Similarly, for the surface charge density term in Equation 3.11,
ϕ( x0 , y0 , z0 )surface ⎧ ⎡ ⎛ ⎪ ⎢ Nodes ⎜⎝ Ms ⎪ ⎢ = ⎨ 4π ⎪ surface ⎢ i 1 ⎛ ⎢ ⎜⎝ x0 − ⎪ ⎢⎣ ⎩
∑ ∑
⎞ ⎛ N i nx dS⎟ mxi + ⎜ ⎠ ⎝ S
∫
∑
6 j
⎞ ⎛ Ni ny dS⎟ myi + ⎜ ⎠ ⎝ S
∫
2
⎞ ⎛ N j x j ⎟ + ⎜ y0 − ⎠ ⎝ 1
∑
6 j
⎤⎫ ⎥⎪ ⎥⎪ ⎬, 2 ⎥ ⎞ ⎥⎪ N j z j ⎟ ⎥⎪ ⎠ ⎦⎭ 1
⎞ N i nz dS⎟ mzi ⎠ S
∫
2
⎞ ⎛ N j y j ⎟ + ⎜ z0 − ⎠ ⎝ 1
∑
6 j
(3.34) where nx, ny, and nz are the unit normal vectors pointing in the x, y, and z directions, respectively. The integration for the surface charge density term is not as straightforward for two reasons. Firstly, the surface of the triangular prism consists of two different shapes, the triangle and the rectangle. The surface integral for these two different shapes have to be evaluated separately and then summed. The summation index, i, is summed from 1 to the number of nodes for the surface, which can be 3 or 4 depending on the surface the integration is taking place. Secondly, it is necessary to calculate the unit normal vector for each of the surfaces. The normal vector, Ñ (not to be confused with the interpolation function), can be obtained with the help of the Jacobian matrix [10] (ε, η) = ∂( y , z) i + ∂( z , x) j + ∂( x , y ) k N ∂(ε, η) ∂(ε, η) ∂(ε, η)
∫
⇒ N i nx dS( x , y ) = S
∂f ∂( f , g ) ∂u = det where ∂g ∂( u , v ) ∂u
∂f ∂v = det J . ∂g ∂v
∂( y , z)
∫ N ∂(ε, η) dS(ε, η), i
S
(3.35)
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The volume and surface charge terms are then summed for each element and then assembled into a global matrix just like any other finite element method: ϕ element = ϕ ( x0 , y0 , z0 )surface + ϕ ( x0 , y0 , z0 )volume .
(3.36)
3.4.5 Galerkin Weighted Residual Calculation for Landau–Lifshitz Gilbert Equation The weak formulation and Galerkin’s method of weighted residual are applied to the LLG equation (refer to Equation 3.17). To do this, we use the interpolation functions Ni as the weighting functions for the Galerkin method. We multiply these weighting functions with Equation 3.17 and integrate over the entire volume. We then obtain the weak formulation of the LLG equation by using Green’s theorem, which states that − ∇ ∫ ϕ∇ φ dV = ∫ (ϕ∇ φ) ⋅ ndS ∫ ϕ ⋅ ∇ φdV. 2
V
S
(3.37)
V
Applying Equation 3.25 to the variables and performing numerical integration we obtain the following expression for each element [11]: 6
∑ j 1
Aij
dmxj γμ = dt 1 + α2 6
−
⎧ C ⎪ ⎨ Ms μ ⎩⎪ 6
∑∑ j 1 k 1
6
6
∑∑
6
Q jik (myj mzk − mzj myk ) +
j 1 k 1
6
∑ ∑ ϕ (m T
z yj ijk
k
− mzjTijky )
j 1 k 1
⎧ ⎫ αγμ ⎪ C ⎡⎢ ⎪ Aijk (myj hzk − mzj hyk )⎬ + ⎨ 1 + α 2 ⎪ Ms μ ⎢ ⎭⎪ ⎣ ⎩
6
6
6
∑∑∑R
ijkl
(myj mxk myl
j 1 k 1 l 1
+ mxj mzk mzl + mxj myk myl + mzj mxk mzl + 2mxj mxk mxl ) 6
+
6
6
∑∑∑R
jkil
j 1 k 1 l 1
6
+
6
⎤ (mxj mzk mzl + mxj mxk mxl + mxj myk myl )⎥ ⎥ ⎦
6
∑∑∑m m yj
xk
(Sijykl ϕ l − Aijkl hyl )
j 1 k 1 l 1 6
+
6
6
∑∑∑
6
z mxj mzk (Sijkl ϕ l − Aijkl hzl ) +
j 1 k 1 l 1
∑∑∑m m xj
⎫
∑ (a ϕ − A h )⎪⎬⎪ , x ij
j 1
j
6
xk
x (Sijkl ϕ l − Aijkl hxl )
j 1 k 1 l 1
6
−
6
(3.38)
ij xj
⎭
where Aij =
∫ N N dV i
v
j
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Computational Nanotechnology: Modeling and Applications with MATLAB®
Aijk =
∫ N N N dV i
j
k
v
Aijkl =
∫ N N N N dV i
j
k
l
v
Qijk =
∫ N ∇ N ⋅ ∇ N dV i
j
k
v
Rijkl =
∫ N N ∇ N ⋅ ∇ N dV i
j
k
l
v
N j ) dV = aijx i + a y j + aijz k aij = ( N i∇ ij
∫ v
N k ) dV = Tijkx i + T y j + Tijkz k T ijk = ( N i N j∇ ijk
∫ v
y x z N l ) dV = Sijkl S ijkl = v ( N i N j N k ∇ i + Aijkl j + Aijkl k
∫
Again, we can see that the coefficients remain the same throughout the time-stepped integration procedure. Therefore, we can pre-generate these coefficients to speed up the calculation, again at the expense of requiring more memory. Equation 3.38 can be expressed in terms of matrix Aij Fj (mx ) = Gi ⇒ F(mx ) = A −1G,
(3.39)
where F(mx) is the derivative of mx with respect to time Gi is the combined terms on the left hand side of Equation 3.38 Aij is the coefficient for F(mx) and it is a symmetrical matrix The derivative of my and mz with respect to time is just the cyclic permutation of all the vector terms in G, that is, y x z Aij Fj (mx ) = Gi (mx , my , mz , hx , hy , hz , aijx , aijy , aijz , Tijkx , Tijky , Tijkz , Sijkl , Sijkl , Sijkl , …)
(3.40)
y z x Aij Fj (my ) = Gi (my , mz , mx , hy , hz , hx , aijy , aijz , aijx , Tijky , Tijkz , Tijkx , Sijkl , Sijkl , Sijkl , …)
(3.41)
y z x Aij Fj (mz ) = Gi (mz , mx , my , hz , hx , hy , aijz , aijx , aijy , Tijkz , Tijkx , Tijky , Sijkl , Sijkl , Sijkl , …).
(3.42)
Micromagnetics: Finite Element Analysis of Nano-Sized Magnetic Materials
89
The matrix A is an N2-sized symmetrical matrix, and MATLAB can deal with this kind of large matrix efficiently using the function sparse. Using the function sparse will not only save significant memory but also speed up the matrix inversion calculation in Equation 3.39. The global matrix for the finite element method is assembled from these elemental matrices in the usual way. We can now advance the solution from m ˜ n to m ˜ n+1 over a time step h using the fourth-order Runge–Kutta formula [9] n +1 = m n + m
k1 k 2 k 3 k 4 + + + , 6 3 3 6
(3.43)
where n) k1 = hF(m ⎛ n k1 ⎞ k 2 = hF ⎜ m + ⎟ ⎝ 2⎠ ⎛ n k2 ⎞ k 3 = hF ⎜ m + ⎟ ⎝ 2⎠ n + k3 ) k 4 = hF(m A time step on the order of 10 fs is usually required for good accuracy. However, there are times during the simulation process when not much is happening. For example, during a magnetization reversal process, the magnetic moment does not move a lot at the beginning or at the end of the hysteresis curve. The time step during these periods can be increased to the order of 0.1 ns without any loss of accuracy. Therefore, it is often very useful to implement the adaptive stepsize control for Runge–Kutta [9]. The time step integration does not guarantee that the following constrain is met: mx 2 + my 2 + mz 2 = 1.
(3.44)
Therefore, it is necessary to normalize the magnetization component at the end of every time step iteration. 3.4.6 MATLAB ® Implementation With all the steps described above, the only thing left to do is to determine when to stop the time step iteration. The easiest way to do this is to terminate when the rate of change of the magnetization component is less than 1° per nanosecond. If the time step is in nanoseconds, then this rate of change is given by ⎡m n ⋅m n+1 ⎤ dθ 180 = cos −1 ⎢ n n + 1 ⎥ × osecond. < 1 per nano m dt π × timestep ⎢⎣ m ⎥⎦
(3.45)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
Generate mesh Initialize material parameters Initialize magnetization and the external field
Generate the coefficients for (i) Scalar potential (ii) Galerkin formulation of LLG equation (iii) Fourth-order Runge–Kutta
Start time step integration Update magnetization and time step
Calculate scalar potential Perform fourth-order Runge–Kutta
Determine error and the next time step Normalize the magnetization component Terminating criteria met? FIGURE 3.4 Micromagnetics finite element program flowchart.
No
Yes Stop
The flowchart for the finite element implementation for micromagnetics is shown in Figure 3.4. The step that consumes a large amount of time and memory is obviously going to be the scalar potential calculation. This is because the scalar potential calculation requires 3 × N2 memory storage space for the coefficients, one for each coordinate. Suppose we have a 100 nm × 100 nm × 4 nm magnetic bar made out of permalloy. The exchange length of a permalloy is about 5 nm, which means that the mesh size also has to be at least approximately 5 nm. Then there will be approximately 20 × 20 × 2 elements and 21 × 21 × 2 nodes. If the floating precision is double (8 bytes), then the required memory storage space would be 3 × (21 × 21 × 2)2 × 8/106 ≈ 19 MB. This memory requirement would increase very rapidly with increasing number of nodes due to the square dependence.
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Micromagnetics: Finite Element Analysis of Nano-Sized Magnetic Materials
3.4.7 Post-Processing The variables such as mx, my, mz, x, y, z and the scalar potential φ can be interpolated from the nodes of the element to any point within the element using Equation 3.25. The exchange, magnetostatic, and Zeeman energies can be calculated using Equations 3.2, 3.12, and 3.13, respectively:
EEXCHANGE =
C 2 C N jm ) dV = i ) ⋅ (∇ j ) dV ( ∇m ( ∇N i m 2 2
∫ V
=
EMAG = −
∫ V
⎤ C⎡ C N j ) dV ⎥ ⋅ (m ⎢ ( ∇N i ⋅ ∇ im j ) = Bij (m i ⋅m j) 2⎢ 2 ⎥ ⎦ ⎣V
∫
μ ⋅H D ) dV = μ Ms (M 2 2
∫ V
=
∫ (N m ⋅ ∇ N φ ) dV i
i
j
j
V
⎤ μ Ms μ Ms ⎡ N j ) dV ⎥ ⋅ (m ⎢ (Ni ⋅ ∇ iφ j ) = aij ⋅ (m iφ j ) 2 2 ⎢ ⎥ ⎦ ⎣V
∫
⋅H A dV = −μ Ms EZEEMAN = −μ M
∫ V
(3.46)
(3.47)
∫ (N m ⋅ N H ) dV i
i
j
j
V
⎤ ⎡ j ) = −μ Ms Aij (m j ). iH i ⋅ H = −μ Ms ⎢ ( N i ⋅ N j ) dV ⎥ ⋅ (m ⎥ ⎢ ⎦ ⎣V
∫
Aij and ãij are the same as in Equation 3.38 and Bij = mentation of this is shown below:
(3.48)
∫ ∇ N ⋅ ∇ N dV. The MATLAB impleV
i
j
%energy for each elements energy_z=0;energy_d=0;energy_e=0; for e=1:TNELMS %the matrix 'node' contains element information as %described in earlier section mx_elem=mx(node(e,1:6)); my_elem=my(node(e,1:6)); mz_elem=mz(node(e,1:6)); pot_elem=pot(node(e,1:6)); for i=1:6 for j=1:6 %Aij, Bij and axij are pre-generated coefficients energy_d=energy_d−axij(e,i,j)*mx_elem(i)*pot_elem(j)− ayij(e,i,j)*my_elem(i)*pot_elem(j)−azij(e,i,j)*mz_elem(i)*pot_elem(j);
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energy_z=energy_z+Aij(e,i,j)*(mx_elem(i)*hx(j)+my_elem(i)*hy(j)+mz_ elem(i)*hz(j)); energy_e=energy_e+Bij(e,i,j)*(mx_elem(i)*mx_elem(j)+my_elem(i)*my_ elem(j)+mz_elem(i)*mz_elem(j)); end end end %total energy for the entire geometry Zeeman(ite)=-mu*Ms*energy_z; Demag(ite)=-mu*Ms/2*energy_d; Exchange(ite)=C/2*energy_e;
In MATLAB, we can easily display a 3D vector plot for the magnetization components by using the function quiver3. If this function is invoked together with getframe in each time step iteration, MATLAB will display the change of magnetization components continuously as they relax toward the energy minimum. %this plots the vector for magnetization component on the bottom surface %depth=total z distance %cellsize=cell size in the z-direction no_vert=depth/cellsize; surf_layer=length(X)/(no_vert+1); %plots only the bottom surface quiver3(X(1:surf_layer),Y(1:surf_layer),Z(1:surf_layer),… mx(1:surf_layer),my(1:surf_layer),mz(1:surf_layer),.5); %adjust the axis and viewing angle Xlen=max(X)−min(X); Ylen=max(Y)−min(Y); axis ([min(X)−Xlen/2 max(X)+Xlen/2 … min(Y)−Ylen/2 max(Y)+Ylen/2]); view(0,90) getframe;
Sometimes it is very useful to see how only a particular component of the magnetization, for example, mx, changes across the x−y plane. We want a gradient of color as the magnetization changes (Figure 3.5). The function scatter3 will do the trick: scatter(X(1:surf_layer),Y(1:surf_layer),… 150,mx(1:surf_layer),'filled');
3.5 Applications Micromagnetics simulation is an important tool for research in many areas of physics and engineering. In physics, it has revealed interesting magnetization configurations and magnetization reversal processes in nano-magnetic materials [12].
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30 20 10 0 –10
0
20
40
60
80
100
120
140
160
(a)
(b) FIGURE 3.5 Magnetic domain wall formed in a magnetic nanowire. (a) The magnetization component of bottom layer of a nanowire. The x and y axes are in the units of nanometers. (b) Dark color indicates x component of the magnetization while light color indicates y component. A domain wall is clearly seen forming at the center of the nanowire as the magnetization changes direction from left to right.
25 20 15 10 5 0 –5 –10 –15 –20 –25
50 40
nm
nm
30 20 10 0 0 (a)
10
20
nm
30
40
50 (b)
0
5 10 15 20 25 30 35 40 45 50 nm
FIGURE 3.6 Magnetization configuration: (a) leaf state and (b) vortex state.
Figure 3.6 shows two configurations known as the leaf and vortex state forming on permalloy, which has the following parameters: Exchange stiffness, A = C/2 = 1.3−11 J/m Magnetic saturation, MS = 8 × 105 A/m Gyromagnetic ratio, γ = 2.211 × 105 The vortex state is obtained using the finite element micromagnetics simulation by initializing the magnetization in the following manner and allowing it to relax under zero magnetic field:
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%this initializes magnetization for a vortex configuration %initialize magnetization to rotate %about the center of the disk (x0,y0) angle=atan((X−x0)./(Y−y0)); mx=cos(angle)'; my=−sin(angle)'; mz=zeros(TNNODES,1); %normalize anorm=sqrt(mx.∧2+my.∧2+mz.∧2); mx=mx./anorm; my=my./anorm; mz=mz./anorm;
This vortex configuration has been verified experimentally in Ref. [13]. Micromagnetics simulations have also been used to characterize the magnetic properties of ferromagnetic rings [14] and triangles [15]. It was also shown that if several magnetic dots are coupled closely together, new ground states of magnetization configuration can be formed [16]. Most of these magnetic materials under investigation are planar structures, on which the finite element method can be easily applied using the triangular prism element described here. For certain magnetic material structures, such as the sphere, it would be more useful to use tetrahedrons as the basic element. Micromagnetics simulation is also an important tool for many areas of research in digital memory storage devices. As the storage devices shrink to the nanometer regime, it becomes increasingly important to investigate the effect of the stray field from the readand-write head on the magnetic bits. Bit patterned media [17] have been proposed to be the next generation of digital memory storage devices. Parkin [18] has recently proposed the use of racetrack memory as the next generation digital memory storage device. In all of this research, it will be interesting to use micromagnetics theory to simulate the operation and reliability of such devices. Cowburn [19] and Allwood et al. [20] showed that nano-magnetic materials can be used to construct magnetic logic gates. Just like the conventional electronic logic gates, these magnetic logic gates perform basic logic operations such NAND, NOT, and NOR but instead of using electronic charge they use magnetization. One such way to achieve this is by using nano-magnetic dots [21] and the cellular automata architecture [22]. There have also been proposals to use domain wall motions in carefully designed magnetic nanowires as magnetic logic gates. In many of these researches, a popular finite difference micromagnetics simulation software called OOMMF [23] is used to verify the experimental results.
3.6 Conclusions Basic micromagnetics theory has been introduced. The micromagnetics theory is best used to describe the behavior of nano-sized magnetic materials, for which Maxwell’s equations alone can no longer be adequate. These nano-magnets exhibit significantly different magnetic properties compared to the bulk materials, for example, nano-sized magnets have single magnetic domain with interesting configurations such as the vortex and the leaf states.
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Such nano-magnets are gaining popularity among researchers for their applications in digital data storage technology as well as for fundamental studies in physics. The finite element approach has been applied to the governing equation of micromagnetics, and it was shown how this can be implemented using MATLAB. In this chapter, only simple and common techniques for finite element micromagnetics are introduced. The algorithm described here is by no means the most efficient in terms of computation speed or memory resources. For example, multipole approximation [24], hierarchical algorithm [25], or a boundary and finite element hybrid method [4] can speed up the calculation of the magnetic scalar potential, which is usually the limiting factor in the computational performance of finite element micromagnetics simulation. Other than the exchange, Zeeman, magnetostatic, and crystalline anisotropy energies considered here, it is also possible to include thermal effects [26] and apply spin torques to the magnetic materials [27]. Two popular micromagnetics simulation packages, OOMMF [23] and Nmag [28], are available free online at the following Web sites: OOMMF: http://math.nist.gov/oommf/ Nmag: http://nmag.soton.ac.uk/nmag/ OOMMF, developed by NIST, uses the finite difference method whereas Nmag, developed at the University of Southampton, uses the finite element method.
References 1. W.F. Brown, Micromagnetics, Interscience Publishers, New York, 1963. 2. W. Brown Jr., Domains, micromagnetics, and beyond: Reminiscences and assessments, J. Appl. Phys. 49, 1937–1942, 1978. 3. J. Fidler, R.W. Chantrell, T. Schrefl, and M. Wongsam, Micromagnetics I: Basic principles, in Encyclopedia of Materials: Science and Technology, K.H.J. Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, and S. Mahajan, Eds., Elsevier Science Ltd, Oxford, England, 2001, pp. 5642–5651. 4. T. Schrefl, D. Suess, W. Scholz, H. Forster, V. Tsiantos, and J. Fidler, Finite element micromagnetics, in Lecture Notes in Computational Science and Engineering, Springer, New York, 2002. 5. J. Fidler and T. Schrefl, Micromagnetic modelling—The current state of the art, J. Phys. D Appl. Phys. 33, R135, 2000. 6. A. Polycarpou, Introduction to the Finite Element Method in Electromagnetics, Morgan & Claypool Publishers, San Rafael, CA, 2006. 7. A. Bondeson, Computational Electromagnetics, Springer, New York, 2010. 8. P.P. Silvester and R.L. Ferrari, Finite Elements for Electrical Engineers, Cambridge University Press, New York, 1996. 9. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge University Press, 1992. 10. P.V. O’Neil, Advanced Engineering Mathematics, CL-Engineering, Punjab, 2006. 11. M. Perry, Numerical micromagnetic modelling of the static and dynamic properties of integrated nanoscale ferromagnetic systems, PhD, University of Cambridge, 2003. 12. M. Kläui and C.A.F. Vaz, Magnetization configurations and reversal in small magnetic elements, Handbook of Magnetism and Advanced Magnetic Materials, H. Kronmüller and S. Parkin, Eds., John Wiley & Sons, Chichester, U.K., 2007.
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13. R. Cowburn, D. Koltsov, A. Adeyeye, M. Welland, and D. Tricker, Single-domain circular nanomagnets, Phys. Rev. Lett. 83, 1042–1045, 1999. 14. J. Lee, T. Hayward, S. Holmes, B. Hong, J. Llandro, K. Cooper, D. Anderson, G. Jones, and C. Barnes, Influence of thermal excitation on magnetization states and switching routes of magnetic multilayer rings, J. Appl. Phys. 105, 07C107–07C107-3, 2009. 15. L. Thevenard, H. Zeng, D. Petit, and R. Cowburn, Six-fold configurational anisotropy and magnetic reversal in nanoscale Permalloy triangles, J. Appl. Phys. 106, 063902, 2009. 16. M. Perry, T. Flack, D. Koltsov, and M. Welland, Simulation of the micromagnetic properties of sub-micron permalloy dot arrays, J. Magn. Magn. Mater. 314, 75–79, 2007. 17. S. Chou, Patterned magnetic nanostructures and quantized magnetic disks, Proc. IEEE. 85, 652–671, 1997. 18. S. Parkin, Racetrack memory: A storage class memory based on current controlled magnetic domain wall motion, Dev. Res. Conf.—Conf. D. DRC, University Park, PA, 2009, pp. 3–6. 19. R. Cowburn, Where have all the transistors gone?, Science 311, 183–184, 2006. 20. D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson, D. Petit, and R.P. Cowburn, Magnetic domain-wall logic, Science 309, 1688–1692, September 2005. 21. R. Cowburn, Magnetic nanodots for device applications, J. Magn. Magn. Mater. 242–245, 505–511, 2002. 22. G. Snider, A. Orlov, I. Amlani, X. Zuo, G. Bernstein, C. Lent, J. Merz, and W. Porod, Quantumdot cellular automata: Review and recent experiments (invited), J. Appl. Phys. 85, 4283–4285, 1999. 23. M.J. Donahue and D.G. Porter, OOMMF User’s Guide, Version 1.0, National Institute of Standards and Technology, Gaithersburg, MD, 1999. 24. S.W. Yuan and H. Bertram, Fast adaptive algorithms for micromagnetics, IEEE Trans. Magn. 28, 2031–2036, 1992. 25. X. Tan, J.S. Baras, and P. Krishnaprasad, Fast evaluation of demagnetizing field in three dimensional micromagnetics using multipole approximation, Proc. SPIE Int. Soc. Opt. Eng. 3984, 195–201, 2000. 26. T. Schrefl, J. Fidler, D. Suess, W. Scholz, and V. Tsiantos, Micromagnetic simulation of dynamic and thermal effects, in Handbook of Advanced Magnetic Materials, Springer, Germany, 2006, pp. 128–146. 27. Z. Li and S. Zhang, Domain-wall dynamics and spin-wave excitations with spin-transfer torques, Phys. Rev. Lett. 92, 207203, 2004. 28. T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag, Magn. IEEE Trans. 43, 2896–2898, 2007.
4 System-Level Modeling of N/MEMS Jason Vaughn Clark CONTENTS 4.1 Introduction.......................................................................................................................... 98 4.2 Formulation of System-Level Equation of Motion for Complex- Engineered N/MEMS ................................................................................ 101 4.2.1 Unified Energy Variables...................................................................................... 102 4.2.2 Constitutive Laws of State .................................................................................... 103 4.2.2.1 Kinetic Energy ......................................................................................... 103 4.2.2.2 Potential Energy ...................................................................................... 107 4.2.2.3 Path-Independent Dissipators............................................................... 110 4.2.3 Constitutive Laws of Constraint.......................................................................... 113 4.2.3.1 Path-Dependent Dissipation.................................................................. 113 4.2.3.2 Sources...................................................................................................... 115 4.2.3.3 Transformers and Transducers ............................................................. 116 4.2.3.4 Transactors ............................................................................................... 116 4.2.3.5 Displacement Constraints...................................................................... 117 4.2.3.6 Flow Constraints ..................................................................................... 118 4.2.3.7 Tangible Variables ................................................................................... 119 4.2.3.8 Implicit Effort Constraints ..................................................................... 120 4.2.3.9 Dynamic Constraints ............................................................................. 122 4.2.4 Work......................................................................................................................... 124 4.2.4.1 Actual, Possible, and Virtual Displacements ...................................... 124 4.2.4.2 Virtual Work ............................................................................................ 125 4.2.4.3 Virtual Work of Constraint Efforts....................................................... 127 4.2.4.4 Efforts........................................................................................................ 127 4.2.5 Differential Algebraic Equation of Motion ........................................................ 129 4.2.5.1 Work–Energy Equivalence..................................................................... 129 4.2.5.2 Lagrangian DAE...................................................................................... 130 4.3 N/MEMS Models............................................................................................................... 138 4.3.1 Free Particles........................................................................................................... 139 4.3.2 Interacting Particles............................................................................................... 140 4.3.3 Mechanical Flexure ............................................................................................... 143 4.3.4 Electrical–Mechanical Flexure............................................................................. 147 4.3.5 Electrical Capacitor................................................................................................ 148 4.3.6 Electrostatic Gap Actuator.................................................................................... 149 4.3.7 Electrical Transformer........................................................................................... 150 4.3.8 Effort Source ........................................................................................................... 150
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4.3.9 Fluid Differential Transformer ............................................................................ 151 4.3.10 Microgear Pair........................................................................................................ 152 4.3.11 Microhinged Beam ................................................................................................ 152 4.3.12 Inductor ................................................................................................................... 153 4.3.13 Microlever ............................................................................................................... 154 4.3.14 Transducer .............................................................................................................. 155 4.3.15 Positive Displacement Pump................................................................................. 155 4.3.16 Rack and Pinion ..................................................................................................... 156 4.3.17 Ram Pump .............................................................................................................. 156 4.3.18 Resistor .................................................................................................................... 157 4.3.19 Slider ........................................................................................................................ 158 4.3.20 Transactors.............................................................................................................. 159 4.3.21 Thermal Expansion ............................................................................................... 159 4.3.22 Residual Stress......................................................................................................... 160 4.3.23 Residual Strain Gradient....................................................................................... 160 4.3.24 Noninertial Reference Frame ............................................................................... 160 4.4 System-Level Design of N/MEMS .................................................................................. 161 4.4.1 Configuring N/MEMS with a Netlist................................................................. 161 4.4.2 Configuring N/MEMS with a Mouse- or Pen-Driven GUI ............................. 177 4.5 System-Level Model Verification ..................................................................................... 178 4.6 Conclusion .......................................................................................................................... 185 References...................................................................................................................................... 185
4.1 Introduction In this chapter, we discuss the design and modeling of complex nano- or microelectromechanical systems (N/MEMSs) at the systems level. Such complex systems include the types of MEMS shown in Figure 4.1. We describe such systems with coupled differential equations and algebraic constraints. The methods presented here have been implemented in an open source MATLAB® tool called PSugar [1]. There are many benefits to using system-level, lumped analysis for the design, modeling, and simulation of N/MEMSs. In general, simulation greatly reduces the time and cost associated with the usual fabricate-and-test iterative process. However, with increasing complexity in N/MEMS design, fabricate-and-test methods become impractical. The need for simulation tools has led to the development and widespread use of computer aided engineering (CAE) tools based on distributed or finite element analysis (FEA) [2–5]. Although these tools have been successful in simulating the behavior of new or simplefunction components, they have not been as successful in simulating the behavior of more complex systems on a personal computer due to memory requirements, nor within a practical amount of time, often requiring hours. The needs and challenges of CAE for MEMS can be found in [6,7]. In essence, depending on how well CAE software facilitates the design process, reduces the time of computation, and agrees with reality, the software can be an invaluable aid for technological advancements in N/MEMS. With the ultimate goal of quickly and accurately simulating complex systems, in this chapter, we present efficient methods to configure, model, and simulate N/MEMS that are composed of a large number of lumped components using MATLAB.
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99
(a)
(b)
(c)
FIGURE 4.1 Complex-engineered MEMS. These scanning electron microscope images of MEMS designed by Sandia National Labs include some of the most complex MEMS to date. They may comprise path-dependent constraints, inequality constraints, and rack and pinion mechanisms as shown in (a); microscale hinges, sliders, and flexible plates as shown in (b); or complicated gear trains with contact friction as shown in (c). Common components such as electrostatic actuators, flexures, and on-chip electrical, circuits that integrate with these devices are not shown.
Three reasons why there is a need for system-level tools that use lumped models are as follows: First, there are a significant number of designers who use (or would like to use) a common set of N/MEMS components in their designs. For instance, a set of commonly used components might comprise carbon nanotubes (CNTs), buckyballs, electromechanical flexures, comb drives, microhinges, microgears, resistors, operational amplifiers, voltage sources, etc. Second, there are a significant number of simple, linear components for which the relative error between their lumped analysis and FEA is less than the relative error between their experimental analysis and FEA. This is due to the uncertainties in structural geometries and material properties from variations in the fabrication process. Third, as the number of MEMS components or time steps increase, FEA may need more memory than accuracy allows or may consume an impractical amount of time. In this chapter, we present several common parameterized models for system-level analysis. We also discuss how new models, such as those extracted from theory, experiment, lumped or FEA, can be developed. For the models presented, their relative errors are less than 3% of their FEA counterparts, within nominal operating range. And we show that the performance of a complete system composed of a multitude of these models fairly agrees with the true performance. Such results represent a good cost/accuracy trade-off. By using a SPICE-like netlist in MATLAB, we show that the time to configure a system can be significantly reduced, and by using computationally efficient models, the time for computation can be significantly reduced. Such capabilities facilitate the simulation of more complex systems and may give designers insights that they would not otherwise have. The organization of this chapter is as follows. In Section 4.2, we discuss an unconventional formulation of the equations of motion that we use to represent N/MEMS for lumped, system-level analyses. This formulation facilitates the modeling and simulation of electromechanical elements as well as molecular dynamics. Using this formulation, in Section 4.3, we discuss how we model disparate components of N/MEMS. In Section 4.4, we
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TABLE 4.1 Nomenclature (in Standard International Units) W E T T* V ⋅ V D S ⋅ S U L H
Work in Joules Stored energy Kinetic energy, temperature Kinetic coenergy Potential energy, voltage, volume Volume rate, fluid volume flow Dissipation of power, content Entropy Entropy rate Work function Lagrangian Hamiltonian
ψ Ψ ϕ Φ λ Λ γ Γ Ω ∇ f q σ (·)T (·)f (·)q d(·) δ(·) (·)CL H P s F k kθ b bθ μ N C Cf R Rf L
Flow constraint Vector of flow constraints Displacement constraint Vector of displacement constraints Dynamic variable constraint, flux linkage Vector of dynamic variable constraints Effort constraint Vector of effort constraints, torque Sum of efforts Differential operator, gradient Generalized flow Generalized displacement Surface charge density, stress Matrix transpose Jacobian with respect to flow Jacobian with respect to displacement Differential operator Variational operator Constitutive law Angular momentum Power, pressure Dynamic variable Force Stiffness Torsional stiffness Mechanical damping Torsional damping Coefficient of friction Normal force, number of dimensions Electrical capacitance Fluid capacitance Electrical resistance Fluid resistance Electrical inductance
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TABLE 4.1 (continued) Nomenclature (in Standard International Units) If I M x θ ω p i v pP Q e eγ eV eT eD es eϕ
Fluid inertance Moment of inertia, second moment of area Mass Translational displacement Angular displacement Angular rate Momentum Electrical current Velocity Pressure momentum Vector of efforts Generalized effort Implicit efforts Potential efforts Kinetic efforts Content efforts Dynamic variable efforts Displacement constraint efforts
discuss how we configure these models into complex N/MEMS using an efficient graphical user interface (GUI) and powerful netlist language. Using this system configuration method, in Section 4.5, we verify our methodology against distributed element analysis and validate our methodology against experiment. In Section 4.6, we summarize our methodologies for modeling complex-engineered N/MEMS. The nomenclature used in the sections of this chapter is given in Table 4.1.
4.2 Formulation of System-Level Equation of Motion for Complex-Engineered N/MEMS We describe the behavior of N/MEMS as being governed by the manipulation of energy between the system’s components, within its components, and between the system and its surroundings. And we describe such a system as an entity that operates on a pair of power variables. Power variables are quantities whose product is power. Given that the system may be decomposed into subsystem components, these components manipulate the power in a characteristic fashion, which is observed as the dynamic response of the N/MEMS. N/ MEMS equation of motion given in this chapter is rooted in the general analytical system dynamics efforts by Layton and Fabien in [8–12]. There are many opensource solvers for differential algebraic equations (DAEs) available. An in-depth discussion of the numerical solution of DAEs is beyond the scope of this chapter, we direct the interested reader to [13–15]. This section is organized as follows. In Section 4.2.1, we define the power variables of our unified system. These variables are used to describe the power of each element. It is the power that is exchanged between components that describe the system’s behavior. We also discuss unifying variables and constitutive laws (CLs) of state, which give rise to a system’s energy
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state functions. These state functions include a system’s kinetic energy, potential energy, and path-independent dissipation. In Section 4.2.2, we discuss CLs of state and of constraint, which are laws that affect the system’s behavior by imposing constraints or boundary conditions upon the system. This discussion includes path-dependent dissipation, transformers, and transducers. In Section 4.2.3, we culminate the previous discussions into a system’s DAE of motion and provide examples of symbolic computation using MATLAB. 4.2.1 Unified Energy Variables N/MEMSs are in general multidisciplinary systems. System components might include the physical domains of electrical, mechanical, fluid, thermal, and possibly others. To consolidate the many types of variables that might be involved in the modeling of N/MEMS, it is advantageous to make use of a common set of unifying variables such as power or energy. To systematically describe the storage, transmission, and transformation of power or energy, we define a unified set of variables—effort e, flow f, momentum p, and displacement q, which are related to power. That is, the derivatives of momentum and displacement are effort and flow, and the product of effort and flow is power. Power variables are pairs of physical quantities whose product is power. That is, the power Pj of the jth component of a micro- or nanosystem is the product of two variables associated with that component, effort ej and flow fj, such that the total power of the assembled system is N
P≡
N
∑ ∑e f . Pj =
j 1
(4.1)
j j
j 1
The physical quantities represented by the effort and flow power variables in terms of mechanical, electrical, fluid, and thermal domains are listed in Table 4.2. In terms of power variables, momentum and displacement are defined in integral form as t
∫ e dt,
p≡
(4.2)
−∞ t
q≡
∫ f dt
(4.3)
−∞
TABLE 4.2 Physical Quantities of Power Variables Domain Mechanical translational Mechanical rotational Electrical Fluid Thermal
Effort e
Flow f
Force F Torque Γ Voltage V Pressure P Temperature T
Velocity v Frequency ω Current i ⋅ Volume rate V ⋅ Entropy rate S
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TABLE 4.4 Examples of Some Constant Kinetic Stores Domain
fCL
1 •p Kinetic Store 1 p m 1 ω CL = H I 1 iCL = λ L vCL =
Mechanical translational Mechanical rotational Electrical
1 V = pP If
Fluid
Kinetic Store Mass, m Moment of inertia, I Inductance, L Fluid inertance, If
The energy stored in a kinetic store by virtue of its momentum is called kinetic energy T. Energy at time t is related to the power variables e and f by t
∫ P dt
E≡
−∞ t
∫ ef dt.
=
(4.7)
−∞
Substituting the dynamic requirement from (4.4) in (4.7), we have t
E=
∫ ef dt
−∞ t
=
dp
∫ dt f dt
−∞ p
=
∫ f dp,
(4.8)
0
where f (p = 0) = 0. Since the flow is due to the kinetic store f = f(p), the energy is a function of momentum, which we define as kinetic energy T. We therefore rewrite (4.8) as p
T ( p) ≡
∫ f (p)dp.
(4.9)
0
Example: Kinetic Energy of a Nonlinear Electrical Inductor If the CL relating flow (current i) and momentum (flux linkage λ) is i(λ, q) = (β1 + q)λ/β0, then the kinetic energy of the nonlinear inductor is
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λ
∫
T (λ , q) = i(λ , q)dλ 0
λ
=
∫ 0
=
(β 1 + q ) λ dλ β0
(β1 + q) 2 λ . 2β0
(4.10)
◊
For a system of N components, the total kinetic energy of the system is p
N
T ( p) = T0 +
∑ ∫ f (p)dp,
(4.11)
j
j 1 p0
where p0 = p(t0) and T0 = T(p0). Given the total kinetic energy of a system, the constitutive behavior of the jth kinetic store is fj =
∂T , ∂p j
(4.12)
which is a kinetic flow. Flows that are not related to momentum are nonkinetic flows. For instance, current in a circuit that does not contain an inductor component will have a nonkinetic current flow and will not store electrical kinetic energy. A list of some common constant kinetic stores is given in Table 4.5. On the other hand, kinetic coenergy T* expresses the kinetic energy stored by virtue of flow. Its CL of state (4.13) is the inverse of what is used for kinetic energy. That is, −1
p( f ) = ⎡⎣ f ( p)⎤⎦ .
(4.13)
The designation of kinetic energy versus kinetic coenergy is determined by the functional dependence in the CLs f(p) versus p( f ): E( p) =
∫ f (p)dp = T(p)
versus E( f ) =
∫ p( f )df = T *( f ).
TABLE 4.5 Some Common Constant Kinetic Stores Domain
T ( p) =
p2 1 2 Kinetic Store
Kinetic Store
Mechanical translational
T ( p) =
1 p2 2 m
Mass, m
Mechanical rotational
T(H ) =
1 H2 2 I
Moment of inertia, I
Electrical
T (λ ) =
1 λ2 2 L
Inductance, L
Fluid
T ( pP ) =
1 1 2 pP 2 If
Inertance, If
(4.14)
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TABLE 4.6 Kinetic Coenergy of Some Common Constant Kinetic Stores Domain
T *( f )
1 Kinetic Store • f 2 2
Kinetic Store
Mechanical translational
T *(q ) =
1 2 mq 2
Mass, m
Mechanical rotational
T *(ω) =
1 I ω2 2
Moment of inertia, I
Electrical
T *(i) =
1 2 Li 2
Inducance, L
Fluid
T *(v ) =
1 2 Ifv 2
Intertance, If
4.2.2.2 Potential Energy Given effort and displacement, we describe how to obtain the potential energy of an arbitrary component. Dissipation is discussed in the following section. A potential store is a quantity that stores potential energy. A CL of state that relates effort e and displacement q is a potential store. In general, effort and displacement applied to a component may be nonlinear and depend on time: eCL (q, t) or qCL (e , t).
(4.20)
CLs of effort and displacement are conventionally written in terms of the effort (or displacement) being applied to the component. Such efforts (or displacements) are equal and opposite to that being applied to the system. That is, the relationship between an effort or displacement applied to the system and an effort or displacement applied to the component is e = −eCL
or q = −qCL .
(4.21)
That is, an effort eCL applied to a component is equal to −eCL applied to the system. And a displacement qCL applied to a component is equal to −qCL applied to the system. A list of some common constant potential stores is given in Table 4.7. Example: Linear Potential Store of Mechanical Spring If force that is applied to a flexure of spring constant k is related to displacement by FCL = kx, then the spring is a potential store, and potential energy is stored in the stiffness k. The force that the spring applies to the system is F = −FCL = −kx. TABLE 4.7 Some Constant Potential Stores eCL = Potential Store × q
Potential Store
Mechanical translational Mechanical rotational
FCL = kx ΓCL = kθθ
Spring, k Torsional spring, kθ
Electrical
VCL =
1 q C
Capacitor,
Fluid
PCL =
1 v Cf
Fluid capacitor,
Domain
1 C 1 Cf
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Example: Linear Potential Store of an Electrical Capacitor If voltage that is applied to a capacitor C is related to charge by VCL = (1/C)q, then the capacitor is a potential store, and potential energy is stored inside C. Now, the voltage that 1 the spring applies to the system is V = −VCL = − q. C Example: Nonlinear Potential Store of a Torsional Mechanical Spring If a torque that is applied to a nonlinear torsional spring k(θ) = k1 + k2θ2 is related to displacement angle by ΓCL = k(θ)θ, then the torque that the torsional spring applies to the system is Γ = −ΓCL = −k1θ + k2θ3. The energy stored in a potential store by virtue of its displacement is called potential energy P. Energy at time t is related to the power variables e and f by t
∫ P dt
E≡
−∞ t
∫ ef dt.
=
(4.22)
−∞
Substituting the kinematic requirement f = q˙ in (4.22), we have t
E=
∫ ef dt
−∞ t
=
dq
∫ e dt dt
−∞ q
=
∫ e dq,
(4.23)
0
where it is assumed that e(q = 0) = 0 in the last equality. This energy of a potential store is a function of displacement: E = E(e(q)) = E(q) = U (q).
(4.24)
This energy in this potential store is called the work function or potential function, U. This work function represents the amount of work that the potential store is able to do by virtue of its displacement. By convention, the negative of this work function is called the potential energy V: V = −U .
(4.25)
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Therefore, the potential energy may be expressed as follows: q
∫
V (q) ≡ − e(q)dq.
(4.26)
0
Example: Potential Energy of a Mechanical Spring If force is related to displacement by the CL FCL = kx, then potential energy stored in the spring is x
V ( q) = − F(q)dx
∫ 0
x
∫
= − − FCL dx 0
x
∫
= − − kx dx 0
1 2 kx . ◊ 2
=
(4.27)
Example: Potential Energy of an Electrical Capacitor If voltage is related to charge by the CL VCL = (1/C)q, then potential energy stored in the capacitor is q
∫
V (q) = − V (q)dq 0
x
∫
= − −VCL dq 0
x
∫
=− − 0
=
1 q dq C
q2 . ◊ 2C
(4.28)
For a system of N components, the total potential energy of the system is N
V (q) = V0 −
q
∑ ∫ e (q)dq, j
j 1 q0
(4.29)
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TABLE 4.8 Some Common Potential Energy Expressions V (q) =
Domain
1 Store q2 2
Potential Store
Mechanical translational
V ( x) =
1 kx 2 2
Translational spring, k
Mechanical rotational
V (θ) =
1 kθ θ 2 2
Torsional spring, kθ
Electrical
V ( q) =
1 1 2 q 2 C
Capacitor, C
Fluid
V (v ) =
1 1 v 2 Cf
Fluid capacitor, Cf
where q0 = q(t0) and V0 = V(q0). Given the total potential energy of a system, the constitutive behavior of the jth potential store is ej = −
∂V ∂q j
≡ eVj ,
(4.30)
where such efforts are called potential efforts, eVj (q). The potential energies of linear potential stores are given in Table 4.8. 4.2.2.3 Path-Independent Dissipators Given a relationship between effort and flow, we describe how to obtain the content and power dissipation of an arbitrary component. Path-dependent dissipation is discussed in the following section. A path-independent dissipator is characterized by a CL of state that relates effort e and flow f. That is, the dissipation depends on the state and not the path the component took to get to a state. The CL has the form eCL ( f ), or
f CL (e).
(4.31)
In general, effort and flow may be nonlinear and depend on displacement and time, that is eCL ( f, q, t), or
f CL (e , q, t).
(4.32)
Some linear path-independent dissipators, where the dissipators are constant, are given in Table 4.9. Nonlinear dissipators may be functions of q, f, p, e, t. CLs of effort and flow are conventionally written in terms of the effort (or flow) being applied to the component. However, such efforts (or flows) are equal and opposite to that being applied back to the system. So efforts and flows applied to the system are related to (4.32) by e = −eCL
and
f = − f CL .
(4.33)
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TABLE 4.9 Some Common Constant Dissipators Domain Mechanical translational Mechanical rotational Electrical Fluid
eCL = Dissipator × f FCL = bx⋅ ΓCL = bθω VCL = Ri ⋅ PCL = RfV
Dissipator Damping, b Torsional damping, bθ Resistor, R Fluid resistance, Rf
That is, an effort eCL applied to a component is equal to −eCL applied to the system. And a flow fCL applied to a component is equal to −fCL applied to the system. Example: Linear Dissipation of Mechanical Damper If frictional force applied to a damper b is related to its velocity by FCL = bv, then the damper is a dissipator, and power is lost through b. Now, the force that the damper applies to the system is F = −FCL = −bv, which is a force that opposes velocity. ◊ Example: Linear Dissipation of an Electrical Resistor If voltage that is applied to a resistor R is related to current by VCL = Ri, then the resistor is a dissipator, and power is lost through R. Now, the voltage that the resistor applies to the system is V = −VCL = −Ri, which is a voltage drop. ◊ Example: Nonlinear Dissipation of an Electrical Resistor If voltage that is applied to a resistor R is related to current by VCL = Ri|i|, then the resistor is a nonlinear dissipator, and power is lost through R. Now, the voltage that the resistor applies to the system is V = −VCL = −Ri|i|, a voltage drop. ◊ Power is related to the power variables e and f by P = ef .
(4.34)
The power dissipated by a component is equal to the negative of the power generated: Pdissipated = −Pgenerated .
(4.35)
Dissipated power may be expressed as follows: Pdisspated = − Pgenerated P
∫
= − dP 0
P
∫
= − d(ef ) 0 f
∫
= − e df − 0
e
∫ f de. 0
(4.36)
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The first term in the result represents the power dissipation of a path-independent dissipator by virtue of its flow. It is called the content: f
∫
D( f ) ≡ − e( f )df ,
(4.37)
0
where e = −eCL for dissipative components. The second term in the result is called the co-content: e
∫
D *(e) ≡ − f (e)de.
(4.38)
0
Example: Content of a Mechanical Dissipator If force is related to flow by the CL FCL = bv, then the content of the dissipator is x
∫
D(q) = − F(v)dv 0
x
∫
= − − FCL dv 0
x
∫
= − − bv dv 0
=
1 2 bv . 2
◊
(4.39)
Example: Content of an Electrical Resistor If voltage is related to current by the CL VCL = Ri, then the content of the resistor is i
∫
D(i) = − V (i)di 0 i
∫
= − − VCL di 0 i
∫
= − − Ri di 0
=
1 2 Ri . 2
◊
(4.40)
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System-Level Modeling of N/MEMS
TABLE 4.10 Content of Some Common Dissipators Domain
D( f ) =
1 Dissipator × f 2 2
Dissipator
Mechanical translational
1 D( x ) = bx 2 2
Mechanical rotational
D(ω) =
Electrical
D(i) =
1 2 Ri 2
Resistor, R
Fluid
D(v ) =
1 2 Rf v 2
Fluid resistance, Rf
Translational damper, b
1 bθ ω 2 2
Torsional damper, bθ
For a system of N components, the total content of the system is f
N
D(i) = D0 −
∑ ∫ e ( f )df , j
(4.41)
j 1 f0
where f0 = f(t0) and D0 = D( f0). Given the total content of a system, the constitutive behavior of the jth dissipator is ej = −
∂D ∂f j
≡ e Dj ,
(4.42)
where such efforts are called dissipative efforts, e Dj ( f ). The content of some common dissipators are given in Table 4.10. 4.2.3 Constitutive Laws of Constraint 4.2.3.1 Path-Dependent Dissipation Given a relationship between effort and displacement path, we describe how to obtain the content and power dissipation of an arbitrary component. Sources are discussed in the following section. Path-dependent dissipation is not a function of state but depends on history (i.e., time or path), which may be described using a CL of constraint. For instance, the system shown in Figure 4.4 shows an object sliding on a surface with Coulomb friction. The system sees the following force opposing the object’s motion: F = −μN , where μ is the friction coefficient between the object and the surface N is the normal force of the object upon the surface due to gravity
(4.43)
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System-Level Modeling of N/MEMS
which results in a path-dependent function. Relating this to work
W=
∫ F ds C
=
∫e
D
ds,
(4.48)
C
we see that ED in (4.47) represents the work done by Coulomb friction. Since this frictional work is negative, ED < 0, then the dissipated energy due to the dissipative effort eD is transmitted from the system to its surroundings. For nonlinear friction models, for example, variable surface textures, the coefficient of friction may be a function of position. In this case, (4.47) may be rewritten as follows:
∫
ED = − μ(s)N ds C
∫
= − e D (s) ds.
(4.49)
C
In formulating the equation of motion, we will treat these efforts from path-dependent functions as externally applied efforts. This is in sharp contrast to the potential efforts e V, which we account for in the potential energy state function. Efforts from pathdependent functions are accounted for in the equation of motion through the vector of applied efforts. 4.2.3.2 Sources Given an effort, flow, momentum, or displacement source, we can describe how to represent the source applied upon arbitrary system components. Sources of power or energy are system components that have CLs that can be thought of as imposing boundary conditions on a system. Sources may be determined a priori, or they may be undetermined, such as in an optimal control problem. Any of the four unified variable may be a source, where the CL is given by e S(t), f S(t), p S(t), or q S(t). Usually, the source is given in terms of effort e S(t), such as a battery, or flow f S(t), such as a current source. Recall, stores and dissipators prescribed two power or energy variables. Since the CL of a source only prescribes one of the variable pairs, the second variable of the pair may be as small or as large as the system demands, within realistic limitations. The physics of the sources is usually modeled separately from the rest of the system, unless the dynamics of the system affects the source. The CLs of flow sources are modeled as flow constraints ψ( f ) = 0, and the CLs of effort sources contribute to the vector of applied efforts Q.
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System-Level Modeling of N/MEMS
fcfs fin
fout
βfin = fout
fces fin
ein = 0
+ –
βfin = eout
ecfs
+ ein –
fout
βein = fout
eout
ein = 0
fin = 0
eces
+ ein –
+ – βein = eout
eout
fin = 0
FIGURE 4.6 Transactors.
between the output on the right and the input on the left. The second power variable on the left is typically zero because transactors are typically designed to extract very little energy from the system when sensing the value of the input quantity. The output flows of fces and eces, and output voltages of fcfs and ecfs, depend on the components that the transactors interface with. In the equation of motion, the effort sources of fces and eces are treated as applied efforts, which are added to the vector of applied efforts Q. The flow sources of fcfs and ecfs are treated as flow constraints ψ( f ). The CLs of constraint are ψ fcfs ( f in , f out ) ≡ βf in − f out = 0
(4.50)
γ fcfs (ein ) ≡ ein = 0 γ ecfs ( f in , f out ) ≡ βein − f out = 0 ψ ecfs (ein ) ≡ f in = 0 ⎡βf in − eout ⎤ γ fces ( f in , f out , ein ) ≡ ⎢ ⎥=0 ⎦ ⎣ ein γ eces (ein , eout , f in ) ≡ βein − eout = 0 ψ eces ( f in ) ≡ f in = 0.
(4.51)
(4.52)
(4.53)
4.2.3.5 Displacement Constraints Given algebraic constraints on displacement, we describe how to apply such constraints on an arbitrary system component. In general, a constraint places limits on possible dynamic behavior. Such constraints are algebraic conditions that the solution trajectory must satisfy in addition to the differential equation. If a constraint can be expressed as a function of displacement q and time t, then
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we call it a displacement constraint. Now if that displacement constraint is expressed in the following form, then we call it a holonomic constraint: φ(q, t) = 0.
(4.54)
The variable q = [q1 q2 q3 … qn]T is a column vector of the n-dimensional configuration space. We write a set of m1 holonomic constraints as follows: ⎡ φ1 ( q , t ) ⎤ ⎢ φ (q, t) ⎥ ⎥ ⎢ 2 Φ(q, t) ≡ ⎢ φ 3 (q, t) ⎥ = 0. ⎥ ⎢ ⎢ ⎥ ⎢ φ m 1 ( q , t )⎥ ⎦ ⎣
(4.55)
The Jacobian of (4.55) is ⎡ ∂ φ1 ⎢ ∂q1 ⎢ ⎢ ∂φ 2 ⎢ ∂q ⎢ 1 ⎢ ∂φ 3 ∂Φ = Φ q ( q, t) = ⎢ ∂q ∂q ⎢ 1 ⎢ ⎢ ⎢ ⎢ ∂φ m 1 ⎢ ∂q ⎣ 1
∂ φ1 ∂q2 ∂φ 2 ∂q2
∂ φ1 ∂q3 ∂φ 2 ∂q3
∂φ 3 ∂q2
∂φ 3 ∂q3
∂ φ1 ⎤ ∂qn ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = 0, ⎥ ⎥ ⎥ ⎥ ∂φ m 1 ⎥ ∂qn ⎦⎥
(4.56)
where the size of the matrix is n columns by m1 rows. 4.2.3.6 Flow Constraints Given algebraic constraints on flow, we describe how to apply flow constraints on arbitrary system components. There are two kinds of flow constraints: integrable flow constraints and nonintegrable flow constraints. Integrable flow constraints may be obtained by differentiating holonomic flow constraint as follows: dφ(q, t) ∂φ dq1 ∂φ dq2 ∂φ dq3 ∂φ dqn ∂φ dt =0 = + + + + + dt ∂q1 dt ∂q2 dt ∂q3 dt ∂qn dt ∂t dt n
=
∂φ
∑ ∂q j 1
j
fj +
∂φ =0 ∂t
= ψ( f , q, t) = 0,
(4.57)
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4.2.3.9 Dynamic Constraints Given the constraints that have not been accounted for in the above analyses, we describe how some of them may be accounted for. Dynamic constraints are constraints involving such quantities as time integrals of displacement q dt , time derivatives of flow df/dt, or time derivatives of effort de/dt, etc. Such
≡
quantities are not directly accounted for in the above forms of displacement constraints ϕ(q, t) = 0, flow constraints ψ( f, q, t) = 0, or implicit effort constraints γ (eγ, s, f, q, t) = 0. This concept is applied by assigning the above quantities to a dynamic variable s, that is ⎧ ⎪ ⎪ s (t) ≡ ⎨ ⎪ ⎪ ⎩
∫ q dt f e etc.
(4.71)
such that ⎧ q ⎪ df ⎪ ds ⎪ dt ≡⎨ dt ⎪ de ⎪ dt ⎪etc. ⎩
(4.72)
In doing so, the form of the dynamic constraint is sk − λ k (e γ, s, f, q, t) = 0.
(4.73)
It is the time derivative of the variable s that makes it a dynamic variable. A set of such dynamic constraints is given by s − Λ(e γ, s, f, q, t) = 0,
(4.74)
⎡ λ 1(e γ, s, f, q, t) ⎤ ⎥ ⎢ λ 2 (e γ, s, f, q, t) ⎥ . Λ(e γ, s, f , q, t) ≡ ⎢ ⎥ ⎢ ⎥ ⎢ γ ⎢⎣ λ m 4 (e , s, f, q, t)⎥⎦
(4.75)
where
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⎤ ⎡ sc − CcVc ⎥ ⎢ se − CeVe ⎥ ⎢ Γ(s, V ) ≡ ⎢ sce + CcVc + CeVe ⎥ = 0, ⎥ ⎢ α cVc − 1)⎥ ⎢ ibc − I sat,bc (e α eVe ⎢i − I − 1)⎥⎦ ⎣ be sat,be (e
(4.80)
where the implicit effort constraint also accounts for the two diodes. Recall, we model diodes as implicit efforts. 4.2.4 Work 4.2.4.1 Actual, Possible, and Virtual Displacements There are three types of displacements: actual Δq, possible dq, and virtual δq. The variational operator δ is similar to the differential operator d, except time does not vary for δ. For instance, if F is a function of x, y, z, t
dF( x , y , z , t) =
∂F ∂F ∂F ∂F dx + dy + dz + dt , ∂x ∂y ∂z ∂t
(4.81)
∂F ∂F ∂F δx + δy + δz. ∂x ∂y ∂z
(4.82)
δ F( x , y , z , t ) =
Actual displacements Δq = q(t0 + Δt) − q(t0) correspond to the actual solution trajectory of a system q(t)
(4.83)
over a time interval Δt. That is, they satisfy both the differential equation and the algebraic constraints. Possible displacements dq only satisfy the Pfaffian forms of displacement and flow constraints: Φ q dq + Φ t dt = 0
(4.84)
Ψf dq + b dt = 0
(4.85)
over the infinitesimal time interval dt. Equation 4.96 is obtained by differentiating the displacement constraint, d Φ(q, t) = 0. And (4.97) is the Pfaffian form of a flow constraint, dt dq ⎛ dq ⎞ ψ ⎜ , q, t⎟ = B(q, t) + b(q, t) = 0. Equation 4.96 accounts for holonomic and integrable flow dt ⎝ dt ⎠
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System-Level Modeling of N/MEMS
constraints and (4.97) accounts for nonintegrable nonholonomic constraints of Pfaffian form. Virtual displacements δq satisfy the virtual forms of displacement and flow constraints: Φq δq = 0
(4.86)
Ψf δq = 0
(4.87)
at an instant, where the time variable t does not vary. Equations 4.98 and 4.99 are obtained from (4.96) and (4.97), such that dt = 0. 4.2.4.2 Virtual Work The total work over a path C is given by W=
∫ dW .
(4.88)
C
This is a path-dependent integral. In other words, dW is an inexact differential. That is, in general B
∫ dW ≠ W(B) − W(A),
(4.89)
A
because the amount of work depends on what path was taken to get from A to B. In contrast, state functions such as kinetic coenergy and potential energy are path independent. That is B
∫ dT * = T *(B) − T *(A) A
(4.90)
B
∫ dV = V(B) − V(A). A
There are two types of virtual work: the effort-displacement form and the flow-momentum form. The effort-displacement form of a differential of work is given by n
dW =
∑ e dq j
j
j 1
= e T dq,
(4.91)
where efforts e acting through possible displacements dq are 1 × n column vectors. Since the unit of work is Joules, which is a unified unit, (4.103) accounts for all n coordinates of a multidisciplinary system. That is, the differential of work is due to all forces, torques,
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voltages, pressures, and temperatures acting through their corresponding displacements, angles, charges, volumes, and entropies. At a particular instant t0, these efforts may be imagined to act through virtual displacements δq. In doing so, all multidisciplinary efforts e acting through virtual displacements δq is called virtual work δW. Virtual work is given by n
δW =
∑ e δq j
j
j 1
= e T δq,
(4.92)
where δq is a prescribed variation in the configuration δW is the resulting variation in work, called virtual work The second form of work is the flow-momentum form. It is derived from the above definition of work as follows: n
dW =
∑ e dq j
j
j 1 n
=
∑ j 1
dp j dq j = dt
n
∑ dp j 1
j
dq j dt
n
=
∑ dp
j
fj.
(4.93)
j 1
At a particular instant t0, if we sum over the flows acting through a deliberate variation in momentum δp, then a second form of virtual work is n
δW =
∑ f δp j
j
j 1
= f T δp,
(4.94)
where the virtual momenta are defined as the infinitesimal quantities that satisfy e T δq = f T δp.
(4.95)
Both virtual displacements and virtual momenta are independent of time and satisfy displacement and flow constraints. However, since the flow-momentum form of work is derived from the effort-displacement form, variations in momenta are not independent of variations in displacement, in general.
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System-Level Modeling of N/MEMS
4.2.4.3 Virtual Work of Constraint Efforts Constraints generate constraint efforts eϕ, which force the configuration into compliance with the constraints. Note, constraint efforts eϕ are not to be confused with effort constraints γ = 0 or Γ = 0 discussed earlier. From d’Alembert, we infer the following principle to define constraint efforts for multidisciplinary systems: The totality of constraint efforts does not contribute to the motion of a physical system of components. Consequently, constraint efforts constitute a set of efforts in equilibrium.
{e } ∈ {e φ
equilibrium
}.
(4.96)
From Bernoulli, we infer the following principle to define static equilibrium for multidisciplinary systems: Static equilibrium may be characterized through the requirement that the work done by force in equilibrium, during a small displacement from equilibrium, should vanish.
dW =
∑e
equilibrium j
dq j = 0.
(4.97)
And therefore, from Lagrange, we infer the following principle to define virtual work for multidisciplinary systems: At each instant in the motion of a physical system, the virtual work of the constraint efforts in their totality vanishes.
δW φ =
∑ e δq = 0. φ j
j
(4.98)
All efforts satisfying (4.110) are defined as constraint efforts eϕ. That is, each term in the sum may be zero or all terms in their totality may sum to zero. 4.2.4.4 Efforts There are six types of efforts that we are concerned with constraint efforts eϕ, dissipative efforts eD, source efforts eS, implicit efforts eγ, potential efforts eV, and kinetic efforts eT or e T * . The constraint efforts are efforts that satisfy Lagrange’s principle of virtual work: δW φ ≡
∑ e δq = 0. φ j
j
(4.99)
The equation of motion is based on virtual work. Since the contribution due to effort constraints vanishes, we can ignore them from analysis. Dissipative efforts satisfy e Dj =
−∂D( f , q, t) . ∂f j
(4.100)
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Source and implicit efforts are applied efforts, which contribute to the vector of applied efforts: e Sj + e γj = Q j .
(4.101)
Potential efforts satisfy −∂V (q, t) . ∂q j
(4.102)
∂T * ( f, q, t) , ∂q j
(4.103)
∂T ( p , q , t ) , ∂q j
(4.104)
eVj = Kinetic efforts satisfy e Tj * = and
e Tj =
where the two kinetic efforts are related through the Legendre transform T ( p , q, t) + T * ( f , q, t) =
∑e p . j
(4.105)
j
Apply the variational operator, we have δ ⎡T ( p, q, t) + T * ( f , q, t)⎤ = δ ⎡⎢ ⎣ ⎦ ⎣ ∂T
∂T
∂T *
∑ ∂p δp + ∑ ∂q δq + ∑ ∂f j
j
j
j
⎛ ∂T
∑ ⎜⎝ ∂p
j
⎞ − f j ⎟ δp j + ⎠
j
⎛ ∂T *
∑ ⎜⎝ ∂f
δf j +
j
∑ e p ⎤⎥⎦ j
∂T *
∑ ∂q
⎞ − p j ⎟ δf j + ⎠
j
∑ f δp + ∑ p δf
δq j =
j
j
⎛ ∂T
∂T * ⎞ ⎟ δq j = 0 j ⎠
∑ ⎜⎝ ∂q + ∂q j
j
j
j
(4.106)
⇓ ∂T
∂T *
∑ ∂q δq = − ∑ ∂q j
j
δq j
j
∑ e δq = − ∑ e * δq , T
j
T
j
where in the third line, we collected terms and cancelled identities fj =
∂T ∂T * , pj = . ∂p j ∂f j
(4.107)
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System-Level Modeling of N/MEMS
The result, which relates the kinetic efforts due to kinetic energy and kinetic coenergy will be used in the next chapter to derive the equation of motion. 4.2.5 Differential Algebraic Equation of Motion The solution trajectory of many complex-engineered N/MEMSs must satisfy a set of differential equations as well as a set of algebraic constraints. Complex engineered systems are often mathematically represented as implicit, nonlinear DAEs. In this chapter, we formulate DAEs of motion that are obtained from the energy functions, constraint equations, and the virtual work of applied efforts in a systematic manner. Two main forms of DAEs are derived from the first law of thermodynamics: the Lagrangian and Hamiltonian DAEs. Both of which are nonconservative, do not require the minimum Lagrange coordinates, and are amenable to nonholonomic and dynamics constraints. 4.2.5.1 Work–Energy Equivalence Our derivation of the DAE of motion begins with the first law of thermodynamics. In differential form, the first law of thermodynamics may be written as follows: dE = dW + dQ ,
(4.108)
where dW is the increment of work done on the system (the work that the system does on the universe is −dW) dQ is the increment of heat added to the system dE is the resulting incremental change in energy stored in the system Both dW and dQ are path-dependent differentials (inexact differentials), and dE is pathindependent differential (exact differential). That is, b
∫ dE = E(b) − E(a) a
b
∫ dW ≠ W(b) − W(a)
(4.109)
a
b
∫ dQ ≠ Q(b) − Q(a). a
Now, if we consider modeling thermal components with a temperature effort acting through an entropy displacement, then we can consider heat to be a form of work. Moreover, in a multidisciplinary system, work encompasses the energy domains of mechanical translational, mechanical rotational, electrical, fluid, and thermal. That is, dW ≡ F dx + T dθ + V dq + P dV + T dS.
(4.110)
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With such a multidisciplinary classification of work, a multidisciplinary form of the first law of thermodynamics may be written as follows: dE = dW .
(4.111)
And to consider a particular instant in time, the first law of thermodynamics may be written as follows: δE = δW ,
(4.112)
where the virtual work δW is path-dependent variation of work due to all efforts acting through a deliberate variation in displacement δq. And δE is the resulting path-independent variation in stored energy or a contemporaneous perturbation. For that reason, it is a variation in stored energy, not a virtual change in stored energy. Stating this result formally: At each instant in the motion of a physical system, an admissible variation in displacement (or momentum) entails a variation of stored energy equal to the virtual work of efforts (or flows) acting over the admissible variation. This work–energy equivalence is expressed as follows: (4.113)
δE = δW , where
∑ e δq = ∑ f δp .
δW =
j
j
j
j
(4.114)
4.2.5.2 Lagrangian DAE The multidisciplinary form of the first law of thermodynamics is δE = δW ,
(4.115)
where δE is the variation in stored energy. The energy is stored within the kinetic and potential stores; that is, E = T + V. And δW is the virtual work done on the system by efforts that do not store energy, that is, efforts such as constraint efforts eϕ, dissipative efforts eD, source efforts eS, and implicit efforts eγ. We therefore express (4.115) as follows: n
δ(T + V ) =
∑ (e
φ j
+ e Dj + e Sj + e γj )δq j
(4.116)
j 1
or n
∑ j 1
⎛ ∂T ⎞ ∂T ∂V δp j + δq j + δq j ⎟⎟ = ⎜⎜ ∂q j ∂q j ⎝ ∂p j ⎠
n
∑( e j 1
φ j
)
+ e Dj + e Sj + e γj δq j .
(4.117)
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System-Level Modeling of N/MEMS
∑
By Lagrange’s principle,
n j 1
e φ δq j = 0, so constraint efforts do not contribute. Source
and implicit efforts are applied efforts Q j = e Sj + e γj . And dissipative efforts are defined as e Dj = −∂D ∂f j . Rewriting (4.117) and collecting terms n
∑ j 1
⎛ ∂T ⎞ ∂T ∂V δp j + δq j + δq j ⎟⎟ = ⎜⎜ ∂q j ∂q j ⎝ ∂p j ⎠
n
⎛ −∂D
∑ ⎜⎜⎝ ∂f j 1
j
⎞ + Q j ⎟⎟ δq j ⎠
(4.118)
⇓ n
∑ j 1
∂T δp j + ∂p j
n
⎛ ∂T
∂V
∂D
j
j
j
∑ ⎜⎝ ∂q + ∂q + ∂f j 1
⎞ − Q j ⎟ δq j = 0. ⎠
(4.119)
The left-most term is a flow-momentum form for virtual work, which can be transformed into an effort-displacement form as follows: ∂T δp j = f jT δp j ∂p j = e Tj δq j =
dp j δq j dt
=
d ∂T * δq j , dt ∂f j
where we have used the definition of kinetic flow in the first line, the relation
(4.120)
∑ e δq =
∑ f δq in the second line, the dynamic requirement in the third line, and definition of momentum in the result. Substituting this result into (4.119), we have n
⎛ d ∂T *
∑ ⎜⎝ dt ∂f j 1
+
j
⎞ ∂T ∂V ∂D + + − Q j ⎟ δq j = 0 , ∂q j ∂q j ∂f j ⎠
(4.121)
where the first term has kinetic coenergy and the second term has kinetic energy. Since the two forms of kinetic efforts are related by (4.120) ∂T
∂T *
∑ ∂q δq = − ∑ ∂q j
j
δq j ,
(4.122)
j
then we may write (4.121) as follows: n
⎛ d ∂T *
∑ ⎜⎝ dt ∂f j 1
j
−
⎞ ∂T * ∂V ∂D + + − Q j ⎟ δ q j = 0. ∂q j ∂q j ∂f j ⎠
(4.123)
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To achieve independence of virtual displacements, we use Lagrange multipliers as follows. Displacement constraints ϕ(q, t) = 0 and flow constraints ψ(q∙, q, t) = 0. Impose the following conditions on virtual displacements: Φ q δq = 0,
(4.124)
Ψ q δq = 0,
such that the virtual displacements are not independent, which implies that the parenthetic coefficient in (4.123) does not identically vanish. For brevity, let us call this coefficient Ω, that is, n
⎛ d ∂T *
∑ ⎜⎝ dt ∂f j 1
j
−
⎞ ∂T * ∂V ∂D + + − Q j ⎟ δq j = 0 , ∂q j ∂q j ∂f j ⎠
j = 1, …, n
n
∑ Ω δq j
j
= 0,
j = 1, …, n
(4.125)
j 1
δq T
Ω = 0.
Using Lagrange’s method of undetermined multipliers, the constraints (4.124) are added to Ω such that the entire sum vanishes. If the constraints Φ(q, t) = 0 and Ψ(q∙, q, t) = 0 are independent, then their gradients Φq and Ψq∙, which are normal to the surfaces formed by the constraints, form a basis for a normal space. The vectors in this normal space can be expressed as linear combinations of the basis vectors; that is, vectors in this space can be expressed as Φ Tq κ and ΨTq μ, where κ and μ are vectors called Lagrange multipliers. Adding these vectors to Ω, we have
(
)
δqT Ω + Φ Tq κ + ΨTq μ = 0.
(4.126)
Now the Lagrange multipliers κ(t) and μ(t) can be chosen such that the expression within the parentheses vanishes. This being the case, we have Ω + Φ Tq κ + ΨTq μ = 0,
(4.127)
which is independent of variation in displacement δq. The form is called the multiplier form of Lagrange’s equation. The multiplier rule is formally stated as follows: At each instant in the motion of a physical system, if the displacement and flow constraints are satisfied Φ(q, t) = 0 Ψ( f , q, t) = 0,
(4.128)
and if admissible variation of configuration δq satisfy δqT Ω = 0,
(4.129)
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System-Level Modeling of N/MEMS
then undetermined multipliers κ(t) and μ(t) exist such that δqT (Ω + Φ Tq κ + ΨTq μ ) = 0 is true for arbitrary δq.
(4.130)
◊
To determine the Lagrange multipliers κ(t) and μ(t), we couple the displacement and flow constraints, Φ(q, t) = 0 and Ψ(q∙, q, t) = 0, to the multiplier form of Lagrange’s Equation 4.130; to allow for the modeling of implicit efforts, we couple implicit effort constraints, Γ(e γ, s, q∙, q, t) = 0, to the multiplier form of Lagrange’s equation; and to allow for the modeling of dynamic variables, we couple dynamic constraints, s∙ − Λ(eγ, s, q∙, q, t) = 0, to the multiplier form of Lagrange’s equation. Hence, we now have an equation of motion that is represented by a set of differential equation coupled to a set of algebraic constraints: d ∇q T * − ∇qT * + ∇qV + ∇q D + Φ Tq κ(t) + ΨTq μ(t) = Q dt Φ(q, t) = 0 Ψ(q , q, t) = 0
(4.131)
Γ(e γ , s, q , q, t) = 0 s − Λ(e γ , s, q , q, t) = 0, d ∇q T * − ∇qT * + ∇qV + ∇q D − Q . We call (4.131) the Lagrangian DAE. The dt first term in (4.131) makes this DAE second order, d2q/dt2, this is because where Ω(q , q, t) =
∂(∇q T * ) dq ∂(∇q T * ) dq ∂(∇q T * ) dt d (∇q T * ) = + + dt dt dt dt ∂q ∂q ∂q = (∇q T * )q q + (∇q T * )q q + (∇q T * )t .
(4.132)
In general, an nth-order differential equation can be converted into a first-order differential equation by introducing time derivative constraints of the form Xn = dn−1Xn−1/dtn−1. We therefore introduce the flow variable constraint, f = q∙, and replace all instances of the second-order variable, q¨, with the first-order variable, f˙, and replace all instances of q∙ with f. Writing (4.131) in first-order form yields q = f T T (∇ f T * ) f f + (∇ f T * )q f + (∇ f T * )t − ∇qT * + ∇qV + ∇ f D + Φ q κ(t) + Ψ f μ(t) = Q Φ(q, t) = 0 Ψ( f , q, t) = 0 Γ ( e γ , s, f , q , t ) = 0 s − Λ(e γ , s, f , q, t) = 0
(4.133)
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or concisely expressed as follows: q = f Mf + Φ Tq κ + Ψ Tf μ = ϒ Φ=0 (4.134)
Ψ=0 Γ=0 s − Λ = 0,
where
ϒ( f, q, t) ≡ Q + ∇qT * − ∇qV − ∇ f D − (∇ f T * )q f − (∇ f T * )t ,
(4.135)
M( f , q , t ) = ( ∇ f T * ) f = ∇ f ( ∇ f T * ) = ∇2f T * .
(4.136)
Equation 4.134 allows for complex-engineered systems to be mathematically represented as a first-order Lagrangian DAE in a systematic manner. This is the form we will use in modeling our multidisciplinary systems. The solution trajectory has the form (q, f, κ, μ, eγ)T. Examples using the Lagrangian DAE using MATLAB ® In this section, we provide examples of using the Symbolic Toolbox within MATLAB to generate the terms in the Lagrangian DAE. In general, the inertial matrix is expressed as follows:
⎛ ∂T * ⎞ ⎛ ∂ ∂T * ⎜ ⎟ ⎜ ⎜ ∂f 1 ⎟ ⎜ ∂f 1 ∂f 1 ⎜ *⎟ ⎜ * ⎜ ∂T ⎟ ⎜ ∂ ∂T 2 M = ∇ f T * = ∇ f ( ∇ f T * ) = ∇ f ⎜ ∂f 2 ⎟ = ⎜ ∂f 1 ∂f 2 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ∂T * ⎟ ⎜ ∂ ∂T * ⎟ ⎜ ⎜ ⎝ ∂f n ⎠ ⎜⎝ ∂f1 ∂f n
∂ ∂T * ∂f 2 ∂f 1
∂ ∂T * ∂f 2 ∂f 2
∂ ∂T * ⎞ ⎟ ∂f n ∂f 1 ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ∂ ∂T * ⎟⎟ ∂f n ∂f n ⎟⎠
(4.137)
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System-Level Modeling of N/MEMS
Given a three-particle system, the MATLAB commands for generating the initial matrix of the system are as follows:
syms
f1
f2
f3
m1
m2
m3
m=[m1;m2;m3]; f=[f1;f2;f3]; T=0.5 * m.’ *f.^2; J=jacobian (T,f) ; M=jacobian (J,f) ;
⎛ m1 → ⎜0 ⎜ ⎝0
0 m2 0
0⎞ 0⎟. ⎟ m3⎠
In general, the Jacobian of momentum with respect to displacement, times flow, is the effort:
( ∇ f T * )q f = ∇ q ( ∇ f T * ) f = ∇ q ( p ) f ⎛ ∂T * ⎞ ⎜ ⎟ ⎜ ∂f 1 ⎟ ⎜ *⎟ ⎛ ⎜ ∂T ⎟ ⎜ = ∇ q ⎜ ∂f 2 ⎟ ⎜ ⎜ ⎟⎜ ⎜ ⎟ ⎜⎝ ⎜ ∂T * ⎟ ⎜ ⎟ ⎝ ∂f n ⎠
f1 ⎞ f2 ⎟ ⎟ ⎟ f ⎟⎠ n
⎛ ∂ ∂T * ⎜ ⎜ ∂q1 ∂f1 ⎜ * ⎜ ∂ ∂T = ⎜ ∂q1 ∂f 2 ⎜ ⎜ ⎜ ∂ ∂T * ⎜ ⎝ ∂q1 ∂f n
∂ ∂T * ∂q2 ∂f1
∂ ∂T * ∂q2 ∂f 2
∂ ∂T * ⎞ ⎟ ∂qn ∂f1 ⎟ ⎟ ⎛ f1 ⎞ ⎟ ⎜ f2 ⎟ ⎟⎜ ⎟. ⎟⎜ ⎟ ⎟ ⎜⎝ f n ⎟⎠ ∂ ∂T * ⎟ ⎟ ∂qn ∂f n ⎠
The corresponding MATLAB commands for a particular system are as follows:
syms T=0.5
f1
f2
f3
m1
m2
m3
q1
q2
q3
* [m1;1/q2;m3]’ .* [f1;f2;f3].^2;
J1=jacobian (T,[f1;f2;f3]) ;
(
)
J2=jacobian J1, [q1;q2;q3] ;
⎛0 ⎜ → ⎜0 ⎜ ⎜⎝ 0
0 −f2 q2^2 0
0⎞ ⎟ 0⎟ . ⎟ 0⎟⎠
(4.138)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
The transpose of the Jacobian of the displacement constraint with respect to displacement, times a Lagrange multiplier, is the effort:
⎛ ⎛ φ1 ⎞ ⎞ ⎜ ⎜ φ ⎟⎟ 2 ⎟⎟ Φ Tq κ = ⎜ ∇q ⎜ ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟⎟ ⎝ ⎝ φ m1 ⎠ ⎠
⎛ ∂ φ1 ⎜ ∂q1 ⎜ ⎜ ∂ φ1 = ⎜ ∂q2 ⎜ ⎜ ⎜ ∂ φ1 ⎜ ⎝ ∂qn
T
⎛ κ1 ⎞ ⎜ κ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ κ ⎟⎠ m1
∂φ 2 ∂q1 ∂φ 2 ∂q2
⎛ ∂ φ1 ⎜ ∂q1 ⎜ ⎜ ∂φ 2 = ⎜ ∂q1 ⎜ ⎜ ⎜ ∂φ m 1 ⎜ ⎝ ∂q1
∂ φ1 ∂q2 ∂φ 2 ∂q2
∂ φ1 ⎞ ∂qn ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∂φ m 1 ⎟ ⎟ ∂qn ⎠
T
⎛ κ1 ⎞ ⎜ κ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ κ ⎟⎠ m1
∂φ m 1 ⎞ ∂q1 ⎟ ⎟ ⎛ κ1 ⎞ ⎟⎜ κ ⎟ ⎟⎜ 2 ⎟ . ⎟⎜ ⎟ ⎟⎜ ⎟ ⎝ κ m1 ⎠ ∂φ m 1 ⎟ ⎟ ∂qn ⎠
(4.139)
For a particular system, the corresponding MATLAB commands are as follows: syms
q1
q2
q3
q4
phi1=q1^2 + q2^2 − L^2; phi2= (q3-q1) ^2 + (q4-q2) ^2 − L^2; phi=[phi1; phi2];
q=[q1;q2;q3;q4]; ⎛ ⎜ → ⎜ ⎜ ⎜⎝
’ J=jacobian ( phi, q ) .;
2*q1 2*q2 0 0
2*q1-2*q3⎞ 2*q2-2*q4⎟ ⎟. 2*q3-2*q1⎟ 2*q4-2*q2⎟⎠
Example: Electrical Circuit with Nonlinear Resistor γ + e –
L q1 S
e (t)
– –
R1 (f1) q2
C
R2
q3
◊
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System-Level Modeling of N/MEMS
The CL of the nonlinear resistor is e γ = k f1 f1 ,
(4.140)
∫
which we will account for as an implicit effort, γ(eγ, f1), instead of a content, D( f1 ) = − e γ ( f1 )df1 . The implicit effort constraint is γ ( e γ , f 1 ) = e γ − k f 1 f 1 = 0.
(4.141)
There is also a flow constraint that relates the flows ψ ( f1 , f 2 , f 3 ) = f1 − f 2 − f 3 .
(4.142)
However, if the analyst needs to account for the charge q2,0 on the capacitor, then the flow constraint (4.142) must be integrated as follows:
∫ = (f − f ∫
φ ( q1 , q2 , q3 ) = ψ ( f1 , f 2 , f 3 ) dt 1
2
− f 3 ) dt
= q1 − q2 − q3 + q2 , 0 ,
(4.143)
where the sign convention is positive for charge building up, or charge preexisting, at the node. Hence, the quantity q2,0 is added (instead of subtracted) as the initial condition. The kinetic coenergy, potential energy, and content of the system are T* =
1 2 Lf1 2
V=
1 2 Cq2 2
D=
1 R2 f 32 . 2
(4.144)
Last, the virtual work due to the applied efforts is δW = e Sδq1 − e γ δq1 = (e S − e γ ) δq1 ,
(4.145)
where the first term is energy added since charge gains energy by passing through the source component, as seen in going from a lower voltage (−) to a higher voltage (+). The second term is subtracted since charge loses energy by passing through the nonlinear resistor, as seen in going from a higher voltage (+) to a lower voltage (−). From (4.145), we get the vector of applied effort ⎛ eS − e γ ⎞ Q=⎜ 0 ⎟ . ⎜ ⎟ ⎜⎝ 0 ⎟⎠
(4.146)
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Since there are no flow or dynamic constraints, and kinetic coenergy is not a function of q or t, we may simplify the Lagrangian DAE q = f
(
Mf + Φ Tq κ + ΨTf μ = Q + ∇qT * − ∇qV − ∇ f D − ∇ f T *
)
q
(
f − ∇ f T*
Φ=0
)
t
(4.147)
Ψ=0 Γ=0 s − Λ = 0 . From the energy and virtual work expressions, we have Mf + Φ Tq κ = Q − ∇qV − ∇ f D ⎛L ⎜0 ⎜ ⎜⎝ 0
0 0 0
⎛ eS − e γ ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ 0⎞ ⎛ 1⎞ ⎟ ⎜ ⎟ 0 f + −1 κ = ⎜ 0 ⎟ − ⎜ q2 C⎟ − ⎜ 0 ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜⎝ −1⎟⎠ ⎜⎝ 0 ⎟⎠ ⎜⎝ 0 ⎟⎠ ⎜⎝ R2 f 3 ⎟⎠ 0⎟⎠
(4.148)
The Lagrangian DAE for this system is q = f ⎛L ⎜0 ⎜ ⎜⎝ 0
0 0 0
⎛ eS − e γ ⎞ 0⎞ ⎛ 1⎞ 0⎟ f + ⎜ −1⎟ κ = ⎜ − q2 C ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜⎝ −1⎟⎠ ⎜⎝ − R2 f 3 ⎟⎠ 0⎟⎠
(4.149)
q1 − q2 − q3 + q2 , 0 = 0 e γ − k f1 f1 = 0. ◊
4.3 N/MEMS Models One of the benefits in using the Lagrangian DAE is that the work of the analyst is systematic and greatly reduced to specifying the component models of the system in terms of energy functions, virtual work, and algebraic constraints. Upon such specification, the resulting functions are put into the Lagrangian DAE to emulate the complete system. In this section, we present several models for use in modeling N/MEMS.
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System-Level Modeling of N/MEMS
4.3.1 Free Particles Free particles are absent of applied forces and interaction potentials. The particles may have kinetic energy stored in their mass due to their velocities. However, since there are no forces acting, the velocities are constant in magnitude and direction. Many particles may be represented by parameterized lumped mass. The six-dimensional (6D) displacement and flow vectors for a system of N particles are ⎡ qxi ⎤ ⎢q ⎥ ⎢ yi ⎥ ⎢ qzi ⎥ qi = ⎢ ⎥ ⎢ qθxi ⎥ ⎢ qθyi ⎥ ⎢ ⎥ ⎢⎣ qθzi ⎥⎦
and
⎡ f xi ⎤ ⎢f ⎥ ⎢ yi ⎥ ⎢ f zi ⎥ fi = ⎢ ⎥ ⎢ fθxi ⎥ ⎢ fθyi ⎥ ⎢ ⎥ ⎢⎣ fθzi ⎥⎦
(4.150)
where The q's consist of displacement translations and rotations about global axes The f's are the corresponding flows The parameterized mass M matrix for each particle i is ⎡mj ⎢ ⎢ ⎢ Mj = ⎢ ⎢ ⎢ ⎢ ⎢⎣
mj mj Ij Ij
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ I j ⎥⎦
(4.151)
where mj and Ij are the particle mass and particle moment of inertia. For a system of N particles with 6N degrees of freedom, we have ⎡ q1 ⎤ ⎢q ⎥ ⎢ 2⎥ ⎢ q3 ⎥ q=⎢ ⎥ ⎢ q4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ qN ⎥⎦
and
⎡ f1 ⎤ ⎢f ⎥ ⎢ 2⎥ ⎢ f3 ⎥ f =⎢ ⎥ ⎢ f4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ f N ⎥⎦
(4.152)
and ⎡ M1 ⎢ ⎢ ⎢ M=⎢ ⎢ ⎢ ⎢ ⎢⎣
M2 M3 M4
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ MN ⎥⎦
(4.153)
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Therefore, we write the energy functions for the DAE as follows: T* = T * =
1 T f Mf 2
(4.154)
V =0
(4.155)
D = 0.
(4.156)
φ = 0,
(4.157)
ψ = 0,
(4.158)
γ = 0,
(4.159)
λ = 0.
(4.160)
δW = 0.
(4.161)
The constraint functions are
And the virtual work is
The equation of motion is found by systematically substituting these expressions into the Lagrangian differential algebraic equation. 4.3.2 Interacting Particles Interacting particles are often subject to Lennard-Jones type of potentials, which is often described as a mixture of attractive and repulsive Coulomb forces. Such interaction potentials account for much of the mechanical behavior of molecules of substances. The 6D displacement and flow vectors for a system of N particles are ⎡ qxi ⎤ ⎢q ⎥ ⎢ yi ⎥ ⎢ qzi ⎥ qi = ⎢ ⎥ ⎢ qθxi ⎥ ⎢ qθyi ⎥ ⎢ ⎥ ⎢⎣ qθzi ⎥⎦
and
⎡ f xi ⎤ ⎢f ⎥ ⎢ yi ⎥ ⎢ f zi ⎥ fi = ⎢ ⎥ , ⎢ fθxi ⎥ ⎢ fθyi ⎥ ⎢ ⎥ ⎢⎣ fθzi ⎥⎦
where The q's consist of displacement translations and rotations about global axes The f's are the corresponding flows
(4.162)
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System-Level Modeling of N/MEMS
The parameterized mass M matrix for each particle i is ⎡mj ⎢ ⎢ ⎢ Mj = ⎢ ⎢ ⎢ ⎢ ⎢⎣
mj mj Ij Ij
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ I j ⎥⎦
(4.163)
where mj and Ij are the particle mass and particle moment of inertia. For a system of N particles with 6N degrees of freedom, we have ⎡ q1 ⎤ ⎢q ⎥ ⎢ 2⎥ ⎢ q3 ⎥ q=⎢ ⎥ ⎢ q4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ qN ⎥⎦
and
⎡ f1 ⎤ ⎢f ⎥ ⎢ 2⎥ ⎢ f3 ⎥ f =⎢ ⎥ ⎢ f4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ f N ⎥⎦
(4.164)
and ⎡ M1 ⎢ ⎢ ⎢ M=⎢ ⎢ ⎢ ⎢ ⎢⎣
M2 M3 M4
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ MN ⎥⎦
(4.165)
We therefore write the energy functions for the DAE as follows: T* =
1 T f Mf 2
⎛ ⎛ σ6 ⎞ 2 σ6 ⎞ 4ε ⎜ ⎜⎜ ⎟⎟ − ⎟ rij ⎟ ⎜ ⎝ rij ⎠ 1 ⎝ ⎠
(4.166)
3N
V=
∑ i, j i≠ j
D = 0.
(4.167)
(4.168)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
where ε is the depth of the potential energy well σ is the distance at which the interparticle potential is zero r is related to the distance between particles i and j as defined in the displacement constraint below The kinetic energy includes 6N terms and the potential energy includes N2 terms if the interactions between all particles are accounted for. However, good approximations may be obtained by only accounting for nearby particle interactions:
⎡ ⎡(q − q )2 + (q − q )2 + (q − q )2 ⎤ 3 ⎤ y1 y2 z1 z2 ⎦ ⎡ r12 ⎤ ⎢ ⎣ x1 x 2 ⎥ ⎢ ⎥ ⎢ 2 2 2 3 ⎥ ⎡ ⎤ r13 ( q 1 − qx 3 ) + ( q y 1 − q y 3 ) + ( qz 1 − qz 3 ) ⎦ ⎥ φ1 = ⎢ ⎥ = ⎢ ⎣ x , ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ r1N ⎥⎦ ⎢ ⎡ 2 2 2 3⎥ ⎤ ⎣ ⎣(qx1 − qxN ) + (qy 1 − qyN ) + (qz1 − qzN ) ⎦ ⎦
(4.169)
⎡ φ1 ⎤ ⎢φ ⎥ 2 Φ = ⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎣φN ⎦
(4.170)
ψ = 0,
(4.171)
γ = 0,
(4.172)
λ = 0.
(4.173)
And the virtual work is 6N
δW =
∑ e δq , i
i
(4.174)
i 1
where ei accounts for additional efforts that are not accounted for within the potential function. Note that the potential energy function ignores rotational displacements. The equation of motion is found by systematically substituting these expressions into the Lagrangian differential algebraic equation.
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System-Level Modeling of N/MEMS
4.3.3 Mechanical Flexure Many flexures of MEMS may be represented by parameterized lumped models. The threedimensional (3D) displacement and excitation vectors for a system of N nodes (generally 6N degrees of freedom for the mechanical domain) are ⎡ qx ⎤ ⎢q ⎥ ⎢ y⎥ ⎢ qz ⎥ q=⎢ ⎥ ⎢ qθx ⎥ ⎢ qθy ⎥ ⎢ ⎥ ⎢⎣ qθz ⎥⎦
and
⎡ fx ⎤ ⎢f ⎥ ⎢ y⎥ ⎢ fz ⎥ f =⎢ ⎥ ⎢ fθx ⎥ ⎢ fθy ⎥ ⎢ ⎥ ⎢⎣ fθz ⎥⎦
(4.175)
where The q's consist of displacement translations and rotations about global axes The f's are the corresponding flows Any electrical elements (or any other domain) are appended onto these elements, creating vectors of length 6NMechanical + NElectrical + NOther. The parameterized mass M, stiffness K, and damping R matrices for a linear flexure component are as follows. For clarity, only nonzero matrix elements are shown. ⎡1 ⎢3 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ M = ρAL ⎢ 1 ⎢ 6 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
13 35
11L 210 9 70
−13L 420
13 35 −11L 210
9 70 13L 420
Iy + Iz 3A
Iy + Iz 6A
−11L 210 L2 105
−13L 420 − L2 140
11L 210
L2 105 13L 420
− L2 140
1 6
1 3
9 70
13L 420 13 35
−11L 210
9 70 −13L 420
13 35 11L 210
Iy + Iz 6A
Iy + Iz 3A
−13L 420 − L2 140
11L 210 L2 105
⎤ ⎥ −13L ⎥ ⎥ 420 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 −L ⎥ 140 ⎥ ⎥ ⎥ ⎥ −11L ⎥ 210 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ L2 ⎥ ⎥ 105 ⎦
(4.176)
144
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ K=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Computational Nanotechnology: Modeling and Applications with MATLAB®
EA L
EA L
12EI z L3
12EI y L3
GJ L
6EI y L2
6EI z L2 12EI z L3
12EI y L3 6EI y L2
6EI z L2
GJ L
6EI y L2 4EI y L
6EI z L2
4EI z L 6EI z L2
6EI y L2 2EI y L
EA L
EA L
2EI z L
12EI z L3
12EI y L3
GJ L
6EI y L2
6EI z L2 12EI z L3
12EI y L3 6EI y L2
6EI z L2
GJ L
6EI y L2 2EI y L
6EI y L2 4EI y L
⎤ ⎥ 6EI z ⎥⎥ L2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2EI z ⎥ L ⎥ ⎥ ⎥ ⎥ 6EI z ⎥ L2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 4E EI z ⎥ ⎥ L ⎦
(4.177) ⎡1 ⎢3 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ μw ⎢ ⎢ D= ρAΔ ⎢ 1 ⎢6 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
13 35
11L 210 9 70
13L 420
13 35 11L 210
9 70 13L 420
Iy + Iz 3A A
Iy + Iz 6A
11L 210 L2 105
13L 420 L2 140
11L 210
L2 105 13L 420
L2 140
1 6
1 3
9 70
13L 420 13 35
11L 210
9 70 13L 420
13 35 11L 210
Iy + Iz 6A
Iy + Iz 3A
13L 420 L2 140
11L 210 L2 105
⎤ ⎥ 13L ⎥⎥ 420 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ L2 ⎥ ⎥ 140 ⎥ , ⎥ ⎥ 11L ⎥ ⎥ 210 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 L ⎥ 105 ⎥⎦
(4.178)
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System-Level Modeling of N/MEMS
where μ is the viscosity Δ is the fluid layer thickness between the beam and substrate L is the beam length w is the beam width h is the beam thickness Iy is the second moment of area about the y-axis Iz is the second moment of area about the z-axis J is the polar second-area moment E is the Young’s modulus G is the shear modulus ν is Poisson’s ratio A = hw is the cross-sectional area of the beam ρ is the material density Since these matrices are in terms of a local coordinate system (i.e., the x-axis is oriented along the length of each element), they must all be rotated to a common global coordinate system before they can be assembled into system matrices. For clarity, let us label the above matrices with the subscript local. The following relations govern these transformations: qglobal = Ωqlocal ,
(4.179)
f global = ΩT f local ,
(4.180)
Mglobal = ΩT MlocalΩ,
(4.181)
Rglobal = ΩT RlocalΩ,
(4.182)
K global = ΩT K localΩ,
(4.183)
and ⎡Ω ⎢ Ω=⎢ ⎢ ⎢ ⎢⎣
Ω Ω
⎤ ⎥ ⎥, ⎥ ⎥ Ω ⎥⎦
(4.184)
− where Ω is the direction cosine matrix corresponding to the orientation of the ith element. ⎡ cos xX ⎢ Ω = ⎢cos yX ⎢⎣ cos zX
cos xY cos yY cos zY
cos xZ ⎤ ⎥ cos yZ ⎥ , cos zZ ⎥⎦
(4.185)
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where cos xY corresponds to cosine of the angle between the x-axis in the local frame and the y-axis in the global frame, for example. The first expression in (4.185) is equivalent to the following three ordered rotations: ⎡ ⎡1 ⎢⎢ Ω = ⎢⎢ ⎢⎢ ⎣⎣
cos θ x − sin θ x
⎤ ⎡ cos θ z ⎥ ⎢ sin θ x ⎥ × ⎢ − sin θ z cos θ x ⎥⎦ ⎢⎣
sin θ z cos θ z
⎤ ⎡cos θ y ⎥ ⎢ ⎥×⎢ 1⎥⎦ ⎢⎣ sin θ y
T
1
− sin θ y ⎤ ⎤ ⎥⎥ ⎥⎥ . cos θ y ⎥⎦ ⎥⎦ (4.186)
That is, by rotating about the y-axis, the updated z-axis, and then the updated x-axis (in that order), we obtain the direction cosine matrix. Given the above mass, stiffness, and damping matrices MEMS flexures, the energy functions are * T * = Tmechanical =
1 T f Mf 2 12
=
⎛ ⎜ ⎝
12
∑∑ j 1
i 1
⎞ 1 Mij ⎟ f i2 2 ⎠
(4.187)
V = Vmechanical =
1 T q Kq 2 12
=
⎛ ⎜ ⎜ ⎝
12
1
⎞
∑ ∑ 2 K ⎟⎟⎠ q , j 1
ij
2 i
(4.188)
i 1
D = Dmechanical =
1 T f Rf 2 12
=
⎛ ⎜ ⎜ ⎝
12
1
⎞
∑ ∑ 2 R ⎟⎟⎠ f . j 1
ij
i
2
(4.189)
i 1
The constraint functions are φ = 0,
(4.190)
ψ = 0,
(4.191)
γ = 0,
(4.192)
λ = 0.
(4.193)
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System-Level Modeling of N/MEMS
And the virtual work is 12
∑ e (q)δq ,
δW =
i
(4.194)
i
i 1
where ei are the applied efforts and q = [q1 … q12]T correspond to mechanical displacements and rotations at the end nodes. The flow is f = q∙. 4.3.4 Electrical–Mechanical Flexure The electrical–mechanical model of a MEMS flexure has the same mass, stiffness, and damping matrices as the purely mechanical model and adds the conductive properties of the flexure. The energy, constraint, and virtual work relationships are as follows: Energy functions: * * T * = Tmechanical + Telectrical 12
=
⎛ ⎜ ⎝
12
⎞
1
∑ ∑ 2 M ⎟⎠ f j 1
ij
i
2
+
i 1
1 2 Lf13 , 2
(4.195)
V = Vmechanical + Velectrical 12
=
⎛ ⎜ ⎝
12
⎞
1
2 1 q13
∑ ∑ 2 K ⎟⎠ q + 2 C(q ) , j 1
2 i
ij
(4.196)
i
i 1
D = Dmechanical + Delectrical 12
=
⎛ ⎜ ⎝
12
∑∑ j 1
i 1
1 ⎞ 2 1 2 Dij ⎟ f i + Df133 . 2 ⎠ 2
(4.197)
Constraints: φ=
∫ ψ dt = 0, ψ = f13 +
if tangible,
∑f
ext
= 0,
(4.198) (4.199)
γ = 0,
(4.200)
λ = 0.
(4.201)
Virtual work: δW =
∑ e (q)δq . i
i
(4.202)
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In general, the displacement vector may be written as follows: T
q = ⎡⎣ x a y a za θ x , a θ y , a θ z , a xb yb zb θ x , b θ y , b θ z , b i ⎤⎦ , which corresponds to the mechanical translational and rotational displacements at the end nodes a and b and electrical displacement through the element (from a to b). The flow is f = q∙. The constraint on flow applies Kirchhoff current law at the end nodes, where f ext is the sum of external current contributions into the node.
•
4.3.5 Electrical Capacitor This is a model of an electrical capacitor with capacitance C, current flow f, and charge displacement q. Energy functions: T * = 0,
V=
(4.203)
1 q12 , 2 C
(4.204)
D = 0.
(4.205)
Constraint functions: φ=
∑q + q
ψ=
∑f
j
0
(4.206)
j
= 0,
(4.207)
or
γ = 0,
(4.208)
λ = 0.
(4.209)
Virtual work: δW =
∑ e (q)δq . j
j
(4.210)
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System-Level Modeling of N/MEMS
4.3.6 Electrostatic Gap Actuator This model of a gap closing actuator is modeled as two elastic beams, which form the variable capacitor. Energy: * T * = Tmechanical 24
=
⎛ ⎜ ⎝
24
∑∑ j 1
i 1
T = Tmechanical or ⎞ 1 Mij ⎟ f i2 2 ⎠
24
=
⎛ ⎜ ⎝
24
∑∑ j 1
i 1
1 1 ⎞ 2 ⎟ pi 2 Mij ⎠
(4.211)
V = Vmechanical + Velectrical 24
=
⎛ ⎜ ⎝
24
∑∑ j 1
i 1
1 ⎞ 2 1 1 2 q25 K ij ⎟ qi + 2 ⎠ 2 C(qqi )
(4.212)
D = Dmechanical 24
=
⎛ ⎜ ⎝
24
1
⎞
∑ ∑ 2 D ⎟⎠ f j 1
ij
i
2
(4.213)
i 1
Constraints: φ(q) =
∫ ψ( f )dt = 0,
ψ= f+
∑f
ext
= 0,
(4.214)
(4.215)
γ = 0,
(4.216)
λ = 0.
(4.217)
Virtual work: 24
δW =
∑ e ( q) δ q , i
i
(4.218)
i 1
where the electrostatic forces and capacitance are functions of geometry and charge displacement.
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4.3.7 Electrical Transformer Model of an ideal electrical power transformer. The transformer is modeled using flow and effort constraints. Branch 1 of the transformer includes voltage e1, current f1, and number of turns N1. Branch 2 of the transformer includes voltage e2, current f2, and number of turns N2. Energy: T * = 0, T = 0,
(4.219)
V = 0,
(4.220)
D = 0.
(4.221)
Constraints: N1 f 2 = 0, N2
ψ( f1 , f 2 ) = f1 −
φ = 0, or φ(q1 , q2 ) =
∫ ψ( f , f )dt 1
2
if q1 , q 2 are tangible,
N1 e2 = 0, N2
γ(e1 , e2 ) = e1 −
λ = 0.
(4.222)
(4.223)
(4.224) (4.225)
Virtual work: 2
δW =
∑ e ( q) δ q . i
i
(4.226)
i 1
4.3.8 Effort Source Model of an effort (i.e., force, torque, temperature, pressure, voltage, etc.) that acts at a node. Energy: T * = 0 or T = 0,
(4.227)
V = 0,
(4.228)
D = 0.
(4.229)
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System-Level Modeling of N/MEMS
Constraints: φ = 0,
(4.230)
ψ = 0,
(4.231)
γ = 0,
(4.232)
λ = 0.
(4.233)
Virtual work: δW =
∑ e (q)δq . i
i
(4.234)
4.3.9 Fluid Differential Transformer Modeling of an ideal fluid differential power transformer. The power transformer is modeled using flow and effort constraints. Portal 1 of the transformer includes pressure P1, fluid flow Q1, and cross-sectional area A1. Portal 2 of the transformer includes pressure P2 , fluid flow Q2, and cross-sectional area A2. P0 is the ambient pressure and V1, V2 are tangible volumes. Energy: T * = 0, T = 0,
(4.235)
V = 0,
(4.236)
D = 0.
(4.237)
Constraints: ψ ( Q1 , Q2 ) = Q1 − φ = 0 or φ (V1 , V2 ) = γ ( P1 , P2 ) =
A1 Q2 = 0, A2
∫ ψ (Q , Q ) dt 1
2
if V1 ,V2 are tangible,
A1 ( P1 − P0 ) + ( P2 − P0 ) = 0, A2 λ = 0.
(4.238)
(4.239) (4.240) (4.241)
Virtual work: 2
δW =
∑ P (V ) δV . i
i 1
i
(4.242)
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4.3.10 Microgear Pair An ideal rotational displacement power transformer may be modeled using flow and effort constraints. Gear 1 of the transformer includes torque τ1, rotation angle θ1, rotation rate ω1, gear radius r1, and I1, which is the moment of inertia. Gear 2 of the transformer includes torque τ2, rotation angle θ2, rotation rate ω2, gear radius r2, and I2, which is the moment of inertia. Hi are the angular momenta. Energy: * T * = Tmechanical 2
=
∑ j 1
T = Tmechanical or
1 I iω i2 2
2
=
∑ j 1
1 H i2 , 2 Ii
(4.243)
V = 0,
(4.244)
D = 0.
(4.245)
Constraints: φ=
r1 θ1 + θ2 = 0, r2
(4.247)
ψ = 0,
γ ( τ1 , τ2 ) = τ1 −
(4.246)
r1 τ2 = 0 , r2
λ = 0.
(4.248)
(4.249)
Virtual work: 2
δW =
∑ τ (θ)δθ . i
i
(4.250)
i 1
4.3.11 Microhinged Beam Model of a hinge attached to the end of a beam. The translational degrees of freedom (q7, q8, q9) of the hinge may be anchored, coupled to another beam, or attached to a sliding surface, ϕ.
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System-Level Modeling of N/MEMS
Energy: * T * = Tmechanical 12
=
⎛ ⎜ ⎝
12
∑∑ j 1
i 1
T = Tmechanical or
⎞ 1 Mij ⎟ f i2 2 ⎠
12
=
⎛ ⎜ ⎝
12
1 1 ⎞
∑ ∑ 2 M ⎟⎠ p , j 1
i 1
2 i
(4.251)
ij
V = Vmechanical 12
=
⎛ ⎜ ⎝
12
∑∑ j 1
i 1
1 ⎞ 2 K ij ⎟ qi , 2 ⎠
(4.252)
D = Dmechanical 12
=
⎛ ⎜ ⎝
12
1
⎞
∑ ∑ 2 D ⎟⎠ f . j 1
ij
i
2
(4.253)
i 1
Constraints: φ = φ(q7 , q8 , q9 ) = 0,
(4.254)
ψ = 0,
(4.255)
γ = 0,
(4.256)
λ = 0.
(4.257)
δW = 0.
(4.258)
Virtual work:
4.3.12 Inductor Model of an electrical inductor with inductance inertia L, electric current flow f, and flux linkage momentum p = λ. Energy: T* =
1 2 Lf 2
or T =
1 p2 , 2 L
(4.259)
V = 0,
(4.260)
D = 0.
(4.261)
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Constraints: φ(q) =
∫ ψ( f )dt = 0, ψ= f+
if q is tangible,
(4.262)
ext
(4.263)
∑f
= 0,
γ = 0,
(4.264)
λ = 0.
(4.265)
δW = 0.
(4.266)
Virtual work:
4.3.13 Microlever Model of a massless leaver. The mechanical power transformer is modeled using displacement and flow constraints. Moment arm 1 of the transformer includes force effort F1, lateral displacement y1, and arm length L1. Moment arm 2 of the transformer includes force effort F2, lateral displacement y2, and arm length L2. Energy: T * = 0 or T = 0,
(4.267)
V = 0,
(4.268)
D = 0.
(4.269)
Constraints: l2 y1 + y 2 = 0 , l1
(4.270)
ψ( F1 , F2 ) = F1 −
l2 F2 = 0, l1
(4.271)
γ (τ1 , τ 2 ) = τ1 −
r1 τ 2 = 0, r2
(4.272)
φ( y1 , y 2 ) =
λ = 0.
(4.273)
Virtual work: 2
δW =
∑ F ( y ) δy . i
i 1
i
(4.274)
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System-Level Modeling of N/MEMS
4.3.14 Transducer Model of a piezoelectric transducer. This signal transducer is modeled using a displacement constraint, with charge q proportional to displacement x, and voltage e between the sides. Energy: T * = 0 or T = 0,
(4.275)
V = 0,
(4.276)
D = 0.
(4.277)
φ(q, x) = q − K q x = 0,
(4.278)
ψ = 0,
(4.279)
γ = 0,
(4.280)
λ = 0.
(4.281)
Constraints:
Virtual work: δW =
∑ e (q)dq . i
i
(4.282)
i
4.3.15 Positive Displacement Pump Model of a positive-displacement pump. This power transducer is modeled using flow and effort constraints, with pump displacement d, angular velocity ω, fluid flow rate Q, pump torque τ, and pressure P. Energy: T * = 0 or T = 0,
(4.283)
V = 0,
(4.284)
D = 0.
(4.285)
Constraints: φ=
∫ ψdt = 0,
if tangible,
(4.286)
ψ(ω , Q) = dω − Q = 0,
(4.287)
γ (τ , P) = τ + dP = 0,
(4.288)
λ = 0.
(4.289)
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Virtual work:
∑ e (q)dq .
δW =
i
(4.290)
i
i
4.3.16 Rack and Pinion Model of a rack and pinion transducer, which converts rotational motion to linear motion. The transducer is modeled using a pinion gear with radius r, angular velocity ω, angular displacement q1 = θ, moment of inertia I, and angular momentum H. The liner rack is modeled with a velocity v, displacement q2 = x, mass M, or momentum p. Energy: * T * = Tmechanical =
T = Tmechanical or
1 2 1 I ω + Mv 2 2 2
=
1 H 2 1 p2 + , 2 I 2M
(4.291)
V = 0,
(4.292)
D = 0.
(4.293)
φ = rθ + x = 0,
(4.294)
ψ = 0,
(4.295)
γ = 0,
(4.296)
λ = 0.
(4.297)
Constraints:
Virtual work: δW =
∑ e (q)dq . i
i
(4.298)
i
4.3.17 Ram Pump Model of a ram pump. This power transducer is modeled with flow and effort constraints. The ram velocity is v, ram area is A, fluid flow rate is Q, ram force is F, and pressures on either side of A are P and P0.
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System-Level Modeling of N/MEMS
Energy: T * = 0 or T = 0,
(4.299)
V = 0,
(4.300)
D = 0.
(4.301)
Constraints:
∫ ψ dt = 0,
φ(q) =
(4.302)
ψ(v, Q) = Av − Q = 0,
(4.303)
γ( F , P) = F + A(P − P0 ) = 0,
(4.304)
λ = 0.
(4.305)
Virtual work:
∑ e (q)dq .
δW =
i
i
(4.306)
i
4.3.18 Resistor Model of an electrical resistor with dissipative resistance R(·) and current flow f. Energy: T * = 0 or T = 0,
(4.307)
V = 0,
(4.308)
1 R(⋅) f 2 . 2
D=
(4.309)
Constraints:
∫
φ(q) = ψ ( f )dt , if tangible,
(4.310)
∑f
(4.311)
ψ= f+
ext
= 0,
γ = 0,
(4.312)
λ = 0.
(4.313)
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Virtual work: δW =
∑ e (q)dq . i
i
(4.314)
i
4.3.19 Slider Model of a slider coupled to a 3D rigid element that is constrained to move along its axial direction. Pin joints are located at both ends. The slider is subject to damping b. Energy: * T * = Tmechanical =
T = Tmechanical or
1 Mv 2 2
=
1 p2 , 2M
V = 0,
(4.316)
1 2 bv . 2
D=
(4.315)
(4.317)
Constraints: ⎡ q2 , global ⎤ ⎡ q2 , local ⎤ ⎢q ⎥ ⎥ ⎢q 3 , local ⎥ = R ⎢ 3 , global ⎥ = 0, φ=⎢ ⎢ q8 , global ⎥ ⎢ q8 , local ⎥ ⎢ ⎥ ⎥ ⎢ ⎣ q9 , local ⎦ ⎣ q9 , global ⎦
(4.318)
ψ = 0,
(4.319)
γ = 0,
(4.320)
λ = 0.
(4.321)
Virtual work: δW =
∑ e (q)dq , i
i
i
where a rotation matrix R transforms the global coordinates to the local frame.
(4.322)
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System-Level Modeling of N/MEMS
4.3.20 Transactors Model of four transactors that contain a controlled effort e or flow f source. Constraints for an ecfs are as follows: ⎡ f1 ⎤ ψ=⎢ ⎥ = 0, ⎣ f 2 − βe1 ⎦
(4.323)
γ = 0,
(4.324)
λ = 0.
(4.325)
ψ = f 2 − βf1 = 0,
(4.326)
γ = e1 = 0,
(4.327)
λ = 0.
(4.328)
ψ = f1 = 0,
(4.329)
γ = e2 − βe1 = 0,
(4.330)
λ = 0.
(4.331)
ψ = 0,
(4.332)
⎡ e1 ⎤ γ=⎢ ⎥ = 0, ⎣ e 2 − β f1 ⎦
(4.333)
λ = 0.
(4.334)
Constraints for an fcfs are as follows:
Constraints for an eces are as follows:
Constraints for an fces are as follows:
4.3.21 Thermal Expansion Model of thermal expansion along the length of a mechanical flexure. It is assumed that the temperature that varies along the length of a beam may be well approximated by the average temperature applied along the length of the beam. Virtual work: δW =
∑ e dq . i
i
i
(4.335)
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where ei = Aσ is applied at opposing ends of the flexure. The stress is σ = EαΔT, where α is the thermal expansion coefficient, E is the Young’s modulus, and ΔT = Tbeam − Tambient. 4.3.22 Residual Stress Model of residual stress applied along the length of a mechanical flexure. Virtual work: δW =
∑ e dq , i
i
(4.336)
i
where the effort is applied at opposing ends of the flexure. The effort is given as follows: (4.337)
FStress = Aσ, where A is the cross-sectional area of the flexure the residual stress σ > 0 implies tensile stress (i.e., beam shortening) σ < 0 implies compressive stress (i.e., beam lengthening) 4.3.23 Residual Strain Gradient
Model of residual strain gradient applied as torque at the ends of the mechanical flexure. Virtual work:
∑ e dq ,
(4.338)
MStrain = EI y Γ,
(4.339)
δW =
i
i
i
where the effort is given as
MStrain is the moment, Γ > 0 curls the beam up out of plane, and Γ > 0 curls the beam down toward the substrate. 4.3.24 Noninertial Reference Frame For noninertial reference frames such as accelerating and rotating substrates, the applied effort is − Mω × (ω × r ) − 2 Mω × r − Mω × r , FInertial = MR
(4.340)
where R(t) is the position vector of the substrate ω(t) is the angular frequency vector r(t) is the node position vector These vectors may change position, magnitude, and orientation in time. The terms on the right side of Equation 4.340 correspond to the translational force, centrifugal force, Coriolis force, and transverse force, respectively.
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4.4 System-Level Design of N/MEMS In this section, we discuss MATLAB-based graphical and text netlist methods that can be used to efficiently design complex-engineered N/MEMS at the network or system level. The MATLAB-based netlist language is a powerful tool for design flexibility. And the MATLAB-coded GUI allows users to quickly configure complex systems in 3D using a computer mouse or pen, faster than can be draw by hand on paper. In Section 4.3.1, we discuss a MATLAB netlist language for design with an example of a buckyball, nanotube, micromirror, and more complex MEMS comprising gears, hinges, slider, circuits, flexures, etc. In Section 4.3.2, we discuss a MATLAB GUI for efficiently configuring complex N/MEMS components in 3D. And in Section 4.3.3, we discuss a MATLAB GUI, which facilitates the exploration of ready-made N/MEMS for novices. 4.4.1 Configuring N/MEMS with a Netlist Netlists had first gained popularity in the 1970s with electrical circuit designers using a computer aided design tool called SPICE [16] (simulation program with integrated circuit emphasis). The first system-level tools for statically determinant MEMS [17–20] appeared about a quarter century after SPICE. The first netlists made specifically for MEMS appeared in a tool called SUGAR [21–25], which was programmed using MATLAB. The first netlists for complex N/MEMS appeared in a tool called PSugar [1], which is discussed here. The present netlist language is based in MATLAB and is more powerful than its predecessors that did not have ready access to a vast library of useful programming functions. This netlist language allows for sophisticated computations such as those dealing with matrices or those needing access to external functions from within the netlist. The netlist language overcomes previous limiting issues by encompassing a much more comprehensive programming language, which is MATLAB in this case. The netlist text file is a MATLAB m-file with the requirement that its output is a cell structure of the form
[model1 {nodes1 }{ parameters1 } ; :
:
:
modelN {nodesN }{ parametersN }] where model is the name of the element model nodes is the list of node names parameters is the list of modeling parameters For instance, a set of parameters might be expressed as follows:
{‘L’ 100e - 6;‘force’combdirve (V,W, H, L, N ); ‘C’ cross ( A, B)} ;
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where each pair of entries corresponds to a parameter label and its corresponding function or value. Here, the value 100e-6 is assigned to parameter L, the output of a user-defined function combdrive(…) is assigned to parameter force, and the built-in MATLAB function cross performs a cross product of vectors A and B and assigns the output to parameter C. These parameters are subsequently passed into the model. This netlist methodology clearly allows the N/MEMS designer to have unfettered access to all MATLAB functions, mathematical operations, workspace quantities, toolboxes such as Simulink®, and other third party plug-ins such as the COMSOL multiphysics FEA tool [5]. Moreover, a netlist language that encompasses a well-known programming language greatly extends the design capabilities of the user. For instance, a main netlist may call upon another program that runs a simulation of a different netlist to obtain results that are required by the main netlist. Or, a simulation reaches an event that requires the original netlist to be modified before continuing in time, such as the case of many finite state machines. Or, a user may desire to represent the configuration of a microrobot inside a netlist, yet model the computational processing of the microrobot outside the netlist. The possibilities are quite far reaching. Example 4.4.1.1: Buckyball The following netlist and Figure 4.7 demonstrate the use of the MATLAB-based PSugar netlist to configure a couple of interacting buckyballs. Information about the carbon mass and atom–atom potentials are given in the model function called buckyball. The coordinates of a 60-carbon buckyball are prescribed in MATLAB using the bucky keyword. The MATLAB display of the following netlist is shown in Figure 4.7.
FIGURE 4.7 MATLAB® simulation and display of two interacting buckyballs. The buckyballs are parameterized by a netlist and models functions. The equation of motion is given by the Lagrangian DAE discussed earlier.
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Netlist—Two interacting buckyballs p = []; p = [p; {'point' {1} {'x' 1e-6; 'y' 2e-6; 'z' 3e-6}}]; p = [p; {'point' {2} {'x' -1e-6; 'y' 0; 'z' 1e-6}}]; p = [p; {'buckyball' {1} {'mass' 3E-26; 'sigma' 4E-10; 'epsilon' 0.15; 'bondlength' 1.4E-10}}]; p = [p; {'buckyball' {2} {'mass' 3E-26; 'sigma' 4E-10; 'epsilon' 0.15; 'bondlength' 1.4E-10}}];
where sigma σ and epsilon ε are defined in model 2.2 in Section 4.2. The units of all values are given in the standard International System of Units. Example 4.4.1.2: Carbon Nanotube The following netlist and Figure 4.8 demonstrates the use of subnets and looping capability in the MATLAB-based PSugar netlist to configure parts of a CNT. The CNT_link is a subnet that configures the three “Y-shaped” bonds between four carbon atoms. The outer carbon atom sites of the Y-shaped component of the CNT shown in Figure 4.8 represents interconnecting nodes in the netlist. The following netlist chains the Y-shaped CNT links together into a single row (see Figure 4.9). We show three such rows in Figure 4.10, and we show a fulllength CNT in Figure 4.11 that is deflected due to a set of applied forces acting at the nodes. a (i + 1, j) a (i + 1, j + 1)
a (i, j)
FIGURE 4.8 MATLAB display of a Y-shaped element of a CNT. The carbon atom sites are shown as spheres. The outermost sites constitute interconnecting nodes of the nanotube.
FIGURE 4.9 MATLAB display of a single row of a CNT. Using the nanotube components from Figure 4.8, the netlist connects a series of the Y-shaped components into a loop. Although each Y-shaped component is slightly rotated, (x, y, z) coordinates are not necessary due to the relative node identifiers.
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FIGURE 4.10 MATLAB display of three rows of a CNT. Carrying the above node connectivity a bit further, three nanotube loops are configured here.
FIGURE 4.11 MATLAB simulation and display of a full-length CNT. Continuing the loop completes the nanotube. Here, the leftmost nodes are fixed in space while a set of forces are applied to a set of nodes.
Netlist—One row of a carbon nanotube p = []; i = 0; for j = 1 : 9 angle = 2*pi/10 * (j-1); p = [p; {'CNT_link' {a(i+1,j) a(i,j) a(i+1,j+1)} {'bondlength' 1.4E-10 'oy' angle}}]; end j = 10; angle = 2*pi/10 * (j-1); p = [p; {'CNT_link' {a(i+1,j) a(i,j) a(i+1,1)} {'bondlength' 1.4E-10 'oy' angle}}];
Example 4.4.1.3: MEMS Scanning Mirror The following netlist and Figure 4.12 demonstrate the use of efficiency of netlists to configure intricate geometries using a small number of lines of netlist code. This example also demonstrates the use of a variety of predefined system-level components, which facilitate efficient parameterization. The individual components with their node connectivities are defined in Table 4.11, which is followed by the netlist, where geometric values of parameters are not shown for simplicity. The resulting 3D MATLAB image of the MEMS structure is shown in Figure 4.13 subnets and looping capability in the MATLAB-based PSugar netlist to configure parts of a CNT. The CNT_link is a subnet that configures the three
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TABLE 4.11 Building Blocks for a MEMS Scanning Mirror Model Name, [Node List]
Building Block/Component 1 r
b h
w
a 2
b
c
l1
mirror (circular plate with rim) [a b]
r (radius) w (rim width) h (plate thickness) h2 (rim thickness)
moment_lever (moment arm lever) [a c b]
l1 (arm length) l2 (arm length) l3 (arm length) w (width) h (thickness)
perf_beam (perforated beam) [a b]
l (length) w (width) h (thickness) n (number of perforations) w1 (rail width) w2 (rung width)
perf_arm [a b]
l (length) w (width) h, h2 (thickness) n (number of perforations) w (main width) w1 (minor width)
beam [a b], and beam c (with center node) [a c b]
l (length) w (width) h (thickness)
V (voltage source) [a b]
V (voltage) l, w (cosmetic length, width)
h2
w l2
l3
h
a
3
h
w
b l w2
w1
a
4
w l
w1
b
w1
a h
h2
5 w a
h
c
b
l
6 a
Selected Parameters
+
b
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p = [p; {'perf_beam' {c(ndirves) c1} {'nholes''w''h''l''whoriz''wvert'}}]; p = [p; {'shaped_beam' {c0 e1} {'l''w''h''qy2''L1''oz1'}}]; p = [p; {'anchor' {e1} {'l''h''w'}}]; p = [p; {'shaped_beam' {c1 e2} {'l''w''h''qy2''L1''oz1'}}]; p = [p; {'anchor' {e2} {'l''h''w'}}]; p = [p; {'voltage' {b4} {'V'}}];
Example 4.4.1.4: Exploration of Design Spaces of Ready-Made N/MEMS Another benefit of the parameterization aspect of system-level configuration and design is the ability to reduce the complexity of exploring the design space for popular designs. This is especially amenable for novice users that are not familiar with netlist programming or advanced users that do not have time to create a design from scratch. A MATLAB tool called SugarCube exploits this ability. In Figure 4.14, we demonstrate how the parameters of the above MEMS scanning mirror can be explored by users without the need to know how to program or use MATLAB. All parameters are adjustable through the SugarCube GUI. In this example, given a constant voltage, the lengths of the beam flexure and moment arm are swept to find the geometry that would produce the optimal deflection per applied voltage. SugarCube of course requires a library of ready-made N/MEMS created by expert users.
FIGURE 4.14 MATLAB simulation and display of SugarCube. Exploration of ready-made N/MEMS for novices. The load button allows users to select from a hierarchical library of ready-made MEMS designed by expert users. Select parameters from the model are displayed. The user is able to select minimum values, resolution, and maximum values. The deflected image of the structure is shown in one window and simulation data is plotted in another window.
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Courtesy of Sandia National Lab FIGURE 4.15 A complex microsystem by Sandia National Lab. A scanning electron microscope image of a micromirror designed by Sandia National Labs. The image shows its gears, rack and pinion, sliders, hinges, comb drives, and flexures. The intended use of this device is to provide hack-proof protection for nuclear warheads. The one-pass mechanical combination lock is out of view in this image. (From Sandia National Labs, New Mexico, Albuquerque, NM. http://www.mems.sandia.gov; Allen, J., Micro Electro Mechanical System Design, Taylor & Francis, Inc., Boca Raton, FL, 2005.)
Example 4.4.1.5: Design of a Complex MEMS Design by Sandia This example demonstrates the complex interplay of system variables that must both satisfy the dynamics of the differential equations as well as the algebraic constraints. Like all examples, the equation of motion is systematically described by the Lagrangian DAE. A scanning electron microscope image of the Sandia device is shown in Figure 4.15. The device comprises thin flexures and plates, gears, slider, hinges, comb drives, electronics, etc. For instance, components with displacement governed by algebraic constraints are microscale gears, sliders, and hinges as illustrated in Figure 4.16. By inspection, the netlist that represents this complex microdevice follows. Measurements of the geometry, material properties, and contact friction for the gear–substrate interface and the slider–substrate interface are yet to be made available from Sandia. We therefore estimate such parameters in our netlist.
FIGURE 4.16 MATLAB display of components with algebraic displacement constraints. Left: Multilinked gear train. Such constraints are modeled using the gear model described in Section 2.10. Right: The micromirror assembly includes rectangular plates, sliders, beams, and gears. In particular, the hinge and sider component are modeled using the descriptions provided by Sections 2.11 and 2.18.
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It should be mentioned that the netlist representation of the Sandia microdevice described below is not unique. For example, the use of subnets can be used to greatly reduce the size of the netlist, as was demonstrated with the examples above. Although rather involved, the output of the netlist is a MATLAB cell structure of the form p = [modelj{nodesj} {model parametersj};…]. It should be noted that it is difficult to simulate such a system using traditional FEA due to the extensive amount of memory that would be required. It is also difficult to simulate such a complex system using traditional system-level tools that do not accommodate algebraic constraints. That is, system-level analysis is made possible by representing the system as a system of DAEs. In fact, PSugar is the first tool to simulate the complex MEMS designed and fabricated at Sandia National Labs. In Figure 4.17, we show the graphical result of the netlist as displayed in MATLAB. And in Figure 4.18, we plot a family of trajectories of gear displacement angle, where the simulation is parameterized by flexure support width. Netlist for the complex Sandia microdevice function p = Netlist_Sandia(nNumOfFinger, length, gap, overlap, width, thickness, nNumOfBone, LBone, WBone, LFlex, WFlex, V1, V2) p = [ ]; if nargin == 0 nNumOfBone = 1; nNumOfFinger = 23; length = 36e-6; %length of finger thickness = 15e-6; %thickness of combdrive width = 1e-6; %width of finger gap = 1.5e-6; %gap of fingers overlap = 0; %overlap of fingers LBone = 312.5e-6; %length of one bone WBone = 10e-6; %width of bone LFlex = 375e-6; %length of flexture WFlex = 1e-6; %width of flexture V1 = sqrt(1-cos(10*pi/20) )*20; %voltage for combdrive 1 V2 = sqrt(sin(10*pi/20) )*20; %voltage for combdrive 2 h1 = 2e-6; %thickness of slim connection beam h2 = 4e-6; %thickness of gears h3 = 3e-6; %thickness of slider h4 = 5e-6; %thickness of mirrors w1 = 10e-6; %width of rod r1 = 22e-6; %radius of gear 1 & 3 r2 = 68e-6; %radius of gear 2 r3 = 72e-6; %radius of gear 4 r4 = 20e-6; %radius of gear 5 end L1=(2*length-gap)+2*WBone; L2=length+WBone/2; overlap2=length-overlap; L3=(LFlex*1.05-nNumOfFinger*(2*gap+2*width) )/3; nNumOfBone = 1; R = 1e6; p = [ p; { 'P' { 's(1)' } { 'y' LFlex; } } ];% starting node for j = 1 : nNumOfBone %the main bone p = [ p; { 'beam3d' {na('s',j) na('b1',j)} { 'L' L1WBone/2+WBone; 'h' thickness; 'w' WBone; } } ];
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p = [ p; { 'beam3d' {na('b1',j) na('b2',j)} { 'L' (LBone-2*L12*WBone)/12+WBone/2; 'h' thickness; 'w' WBone; } } ]; p = [ p; { 'beam3d' {na('b2',j) na('b3',j)} { 'L' 10*(LBone-2*L12*WBone)/12; 'h' thickness; 'w' WBone; } } ]; p = [ p; { 'beam3d' {na('b3',j) na('b4',j)} { 'L' (LBone-2*L12*WBone)/12+WBone/2; 'h' thickness; 'w' WBone; } } ]; p = [ p; { 'beam3d' {na('b4',j) na('s',j+1)} { 'L' L1WBone/2+WBone; 'h' thickness; 'w' WBone; } } ]; %comb 1 p = [ p; { 'beam3d' {na('b1',j) na('fs',(8*nNumOfFinger+4)*(j1)+1)} { 'L' WBone/2+2*gap+3*width/2; 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+1 : (8*nNumOfFinger+4)*(j- 1)+nNumOfFinger-1 p = [ p; { 'beam3d' {na('fs',i) na('fe',i)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; end p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j- 1)+nNumOfFinger) na('fe',(8*nNumOfFinger+4)*(j-1)+nNumOfFinger)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; p = [ p; { 'P' {na('fe',(8*nNumOfFinger+4)*(j-1)+nNumOfFinger) na('t1',j)} { 'L' (length-overlap)+WBone/2; 'rz' pi; } } ]; p = [ p; { 'beam3d' {na('t1',j) na('fs',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+1)} { 'L' gap+width; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+1) na('fe',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('t1',j) na('fs',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+2)} { 'L' gap+width; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+2) na('fe',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+2)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+nNumOfFinger+2 : (8*nNumOfFinger+4)*(j-1)+2*nNumOfFinger p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',i+1) na('fe',i+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; end %resistor p = [ p; { 'resistor' {na('fs',(8*nNumOfFinger+4)*(j1)+nNumOfFinger+1) na('w1',j)} { 'L' 3*L3; 'rz' pi/2; 'R' R; } } ]; %comb 2 p = [ p; { 'beam3d' {na('b1',j) na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+1)} { 'L' WBone/2+2*gap+3*width/2; 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+2*nNumOfFinger+1+1 : (8*nNumOfFinger+4)*(j-1)+2*nNumOfFinger+1+nNumOfFinger-1
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p = [ p; { 'beam3d' {na('fs',i) na('fe',i)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; end p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger) na('fe',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; p = [ p; { 'P' {na('fe',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger) na('t2',j)} { 'L' (lengthoverlap)+WBone/2; 'rz' pi; } } ]; p = [ p; { 'beam3d' {na('t2',j) na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+1)} { 'L' gap+width; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+1) na('fe',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('t2',j) na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+2)} { 'L' gap+width; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+2) na('fe',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+2)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+2*nNumOfFinger+1+nNumOfFinger+2: (8*nNumOfFinger+4)*(j-1)+2*nNumOfFinger+1+2*nNumOfFinger p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',i+1) na('fe',i+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; end %resistor p = [ p; { 'resistor' {na('fs',(8*nNumOfFinger+4)*(j1)+2*nNumOfFinger+1+nNumOfFinger+1) na('w2',j)} { 'L' 3*L3; 'rz' -pi/2; 'R' R; } } ]; %flexture 1 p = [ p; { 'beam3d' {na('b2',j) na('f1',j)} { 'L' LFlex+WBone/2+WFlex/2; 'h' thickness; 'w' WFlex; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('f1',j) na('f2',j)} { 'L' 10*(LBone-2*L12*WBone)/12*(2/5); 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f2',j) na('f3',j)} { 'L' 9*LFlex/10; 'h' thickness; 'w' WFlex; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('f2',j) na('f4',j)} { 'L' 10*(LBone-2*L12*WBone)/12/5; 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d’ {na('f3',j) na('f5',j)} { 'L' 10*(LBone-2*L12*WBone)/12/5; 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f5',j) na('f4',j)} { 'h' thickness; 'w' WFlex; } } ]; p = [ p; { 'beam3d' {na('f4',j) na('f6',j)} { 'L' 10*(LBone-2*L12*WBone)/12*(2/5); 'h' thickness; 'w' WFlex; 'rz' 0; } } ];
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p = [ p; { 'beam3d' {na('f6',j) na('b3',j)} { 'h' thickness; 'w' WFlex; } } ]; %add constraint for flexture 1 p = [ p; { 'constraint' { na('f3',j) } { 'displacement''all'; } } ]; p = [ p; { 'constraint' { na('f5',j) } { 'displacement''all'; } } ]; %flexture 2 p = [ p; { 'beam3d' {na('b2',j) na('f7',j)} { 'L' LFlex+WBone/2+WFlex/2; 'h' thickness; 'w' WFlex; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('f7',j) na('f8',j)} { 'L' 10*(LBone-2*L12*WBone)/12*(2/5); 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f8',j) na('f9',j)} { 'L’ 9*LFlex/10; 'h' thickness; 'w' WFlex; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('f8',j) na('f10',j)} { 'L' 10*(LBone2*L1-2*WBone)/12/5; 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f9',j) na('f11',j)} { 'L' 10*(LBone2*L1-2*WBone)/12/5; 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f11',j) na('f10',j)} { 'h' thickness; 'w' WFlex; } } ]; p = [ p; { 'beam3d' {na('f10',j) na('f12',j)} { 'L' 10*(LBone2*L1-2*WBone)/12*(2/5); 'h' thickness; 'w' WFlex; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('f12',j) na('b3',j)} { 'h' thickness; 'w' WFlex; } } ]; %add constraint for flexture 2 p = [ p; { 'constraint' { na('f9',j) } { 'displacement''all'; } } ]; p = [ p; { 'constraint' { na('f11',j) } { 'displacement''all'; } } ]; %comb 3 p = [ p; { 'beam3d' {na('b4',j) na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+1)} { 'L' WBone/2+2*gap+3*width/2; 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+4*nNumOfFinger+2+1 : (8*nNumOfFinger+4)*(j-1)+4*nNumOfFinger+2+nNumOfFinger-1 p = [ p; { 'beam3d' {na('fs',i) na('fe',i)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; end p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger) na('fe',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'P' {na('fe',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger) na('t3',j)} { 'L' (length-overlap)+WBone/2; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('t3',j) na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+1)} { 'L' gap+width; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+1) na('fe',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ];
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p = [ p; { 'beam3d' {na('t3',j) na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+2)} { 'L' gap+width; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+2) na('fe',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+2)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; for i = (8*nNumOfFinger+4)*(j-1)+4*nNumOfFinger+2+nNumOfFinger+2: (8*nNumOfFinger+4)*(j-1)+4*nNumOfFinger+2+2*nNumOfFinger p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',i+1) na('fe',i+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; end %resistor p = [ p; { 'resistor' {na('fs',(8*nNumOfFinger+4)*(j1)+4*nNumOfFinger+2+nNumOfFinger+1) na('w3',j)} { 'L' 3*L3; 'rz' pi/2; 'R' R; } } ]; %comb 4 p = [ p; { 'beam3d' {na('b4',j) na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+1)} { 'L' WBone/2+2*gap+3*width/2; 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; for i = (8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+1:(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger-1 p = [ p; { 'beam3d' {na('fs',i) na('fe',i)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' -pi/2; } } ]; end p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger) na('fe',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger)} { 'L' L2; 'h' thickness; 'w' width; 'rz' 0; } } ]; p = [ p; { 'P' {na('fe',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger) na('t4',j)} { 'L' (lengthoverlap)+WBone/2; 'rz' 0; } } ]; p = [ p; { 'beam3d' {na('t4',j) na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+1)} { 'L' gap+width; 'w' WBone; 'rz' -pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+1) na('fe',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; p = [ p; { 'beam3d' {na('t4',j) na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+2)} { 'L' gap+width; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+2) na('fe',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+2)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ];
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for i = (8*nNumOfFinger+4)*(j-1)+6*nNumOfFinger+3+nNumOfFinger+2: (8*nNumOfFinger+4)*(j-1)+6*nNumOfFinger+3+2*nNumOfFinger p = [ p; { 'beam3d' {na('fs',i) na('fs',i+1)} { 'L' 2*(gap+width); 'h' thickness; 'w' WBone; 'rz' pi/2; } } ]; p = [ p; { 'beam3d' {na('fs',i+1) na('fe',i+1)} { 'L' L2; 'h' thickness; 'w' width; 'rz' pi; } } ]; end %resistor p = [ p; { 'resistor' {na('fs',(8*nNumOfFinger+4)*(j1)+6*nNumOfFinger+3+nNumOfFinger+1) na('w4',j)} { 'L' 3*L3; 'rz' -pi/2; 'R' R; } } ]; end Le = 50e-6; L4 = (L1-WBone/2+WBone)/4; L5 = LBone-L4/2; L6 = (LBone*2)/3; L7 = L5 - L6; L8 = LFlex*10/11; L9 = LFlex*12/11+L4; L10 = LBone*5/4; L11 = L10 - L6; L12 = L5 - L10; L13 = L4/3; L14 = LFlex + L4; % circuit 1 p = [ p; { 'wire' {na('w4',1) 'a1'} { 'L' L6; 'rz' pi; } } ]; p = [ p; { 'capacitor' {'a1''C1'} { 'L' Le; 'rz' -pi/2; } } ]; p = [ p; { 'P' {'C1''G1'} { 'L' Le; 'rz' -pi/2; } } ]; p = [ p; { 'GND' {'G1''C1'} { } } ]; p = [ p; { 'wire' {'a1''a2'} { 'L' L7; 'rz' pi; } } ]; p = [ p; { 'wire' {'a2''a3'} { 'L' L8; 'rz' pi/2; } } ]; p = [ p; { 'wire' {'a3''temp1'} { 'L' L4; 'rz' pi; } } ]; p = [ p; { 'P' {'temp1''V1'} { 'L' Le; 'rz' pi; } } ]; p = [ p; { 'voltage' {'V1''temp1'} { 'V' V1; } } ]; p = [ p; { 'P' {'V1''G2'} { 'L' Le; 'rz' pi; } } ]; p = [ p; { 'GND' {'G2''V1'} { } } ]; p = [ p; { 'wire' {na('w3',1) 'a4'} { 'L' L6; 'rz' pi; } } ]; p = [ p; { 'capacitor' {'a4''C3'} { 'L' Le; 'rz' pi/2; } } ]; p = [ p; { 'P' {'C3''G3'} { 'L' Le; 'rz' pi/2; } } ]; p = [ p; { 'GND' {'G3''C3'} { } } ]; p = [ p; { 'wire' {'a4''a5'} { 'L' L7; 'rz' pi; } } ]; p = [ p; { 'wire' {'a5''a3'} { } } ]; % circuit 2 p = [ p; { 'wire' {na('w2',1) 'b2'} { 'L' L4; 'rz' -pi/2; } } ]; p = [ p; { 'capacitor' {'b2''C5'} { 'L' Le; 'rz' -pi/2; } } ]; p = [ p; { 'P' {'C5''G8'} { 'L' Le; 'rz' -pi/2; } } ]; p = [ p; { 'GND' {'G8''C5'} { } } ]; p = [ p; { 'wire' {'b2''b3'} { 'L' 3*L4; 'rz' pi; } } ]; p = [ p; { 'wire' {'b3''b4'} { 'L' L9; 'rz' pi/2; } } ]; p = [ p; { 'P' {'b4''V4'} { 'L' Le; 'rz' pi; } } ]; p = [ p; { 'voltage' {'V4''b4'} { 'V' V2; } } ]; p = [ p; { 'P' {'V4''G5'} { 'L' Le; 'rz' pi; } } ]; p = [ p; { 'GND' {'G5''V4'} { } } ]; p = [ p; { 'wire' {na('w1',1) 'b5'} { 'L' L4; 'rz' pi/2; } } ];
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p = [ p; { 'capacitor' {'b5''C7'} { 'L' Le; 'rz' pi/2; } } ]; p = [ p; { 'P' {'C7''G9'} { 'L' Le; 'rz' pi/2; } } ]; p = [ p; { 'GND' {'G9''C7'} { } } ]; p = [ p; { 'wire' {'b5''b6'} { 'L' 3*L4; 'rz' pi; } } ]; p = [ p; { 'wire' {'b6''b4'} { } } ]; %End of netlist
4.4.2 Configuring N/MEMS with a Mouse- or Pen-Driven GUI To configure complex N/MEMS using a mouse- or pen-driven GUI, it is desirable to use as few button clicks as possible during the positioning and orienting of components in 3D space. This is because cumbersome tasks often break the natural flow of ideas during the configuration of a device. To facilitate the configuration of draw planes of device elements in 3D with minimal button clicks, draw planes may be uniquely defined with a minimum of three coordinate points. These three points may be defined by using the mouse or pen to click on three objects, such as an axis plane or a device component that has already been positioned on the screen. Since the position of a computer mouse or pen cursor is defined by two-dimensional coordinates, we project the cursor onto objects in 3D space. For example, in Figure 4.19a, we show the positioning of the first of three coordinates used to define a draw plane using MATLAB. This first button click projects the cursor onto the xz-plane. In Figure 4.19b, we show the positioning of the second coordinate of the draw plane, and in Figure 4.19c, we show the positioning of the final coordinate that uniquely defines the draw plane. The three button sequence shown in Figure 4.19a through c take about 1 s. In Figure 4.19d, we show the positioning of an element upon the draw plane. This two-node resistor element required two button clicks. Any additional resistors that may be configured end-to-end stemming from this initial element will only require one additional button click each. Draw planes may be repositioned by dragging the node of a plane along its interface with another plane, such as the xz-plane. The nodes of elements may be similarly repositioned or dragged across the planes into position. For example, a series of nanotubes may be positioned onto the planes and then slid into differing orientations by the mouse. To assist with free-hand positioning, snap-to-grid and snap-to-node features are implemented. The significant benefit of our methodology is that it allows systems to be configured faster than can be drawn on paper. This design/configuration methodology is bidirectional between graphical GUI and text netlist. That is, any element that is configured in the Graphical Window (GW) immediately appears in the Netlist Window (NW) as an additional line of text. Conversely, any element that is typed into the NW immediately appears in the GW. The benefit of this feature allows the best of both methods to be used when appropriate. For instance, the GW may be best for quickly configuring a filter device; however, creating a layout comprised of a parameterized array of devices that vary by one property along the x-direction and vary by a different property along the y-direction is best done by using a nested for loop within a netlist. Elements are chosen for the GW by using the Element Menu Window (EMW). An element may be elementary, such as a molecule, resistor, operational amplifier, or flexure, or an element may be an assemblage of several elementary components, such as a CNT, a band-pass filter, or a micro gyroscope. The listing of selected elements in a particular EMW may be defined by a user. The EMW, the GW, and the NW are identified in Figure 4.20.
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FIGURE 4.19 MATLAB GUI for design configuration input. (a) Configuring draw plane, first click. (b) Configuring draw plane, second click. (c) Draw plane finalized, third click. (d) Element configured, two clicks.
4.5 System-Level Model Verification When developing a system-level component model, one must verify the component model against a corresponding model that is either simulated using a widely accepted tool (such as FEA) or a widely accepted analytical model. Most often FEA is used because analytical models usually only apply to a small set of very simple geometries. Due to computer memory limitation, FEA is most often used for verifying a small component of a much larger system, instead of the entire system. It is therefore assumed that the prediction of
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FIGURE 4.20 MATLAB display of the main PSugar windows. Deflected MEMS images (GW), interactive netlist (NW), and element menus (EMW).
the entire system can be ascertained from an assembly of its well-understood and verified elemental components. After computer model verification facilitates predictable directions for the designer to explore, the design is usually fabricated such that experimental validation can take place. In essence, this is the difference between verification and validation. Unlike computer model verification where model geometries and material properties are prescribed by the analyst, experimental validation requires knowledge of fabricated geometry and material properties, which is often subject to fabrication process variation, and is therefore difficult to predict. Full experimental validation of system-level models is beyond the scope of this chapter. In what follows, we demonstrate the lumped component verification of a curved flexure in Example 4.5.1, the perforated flexure in Example 4.5.2, the electrostatic comb drive in Example 4.5.3, and we compare the performance of an assembled lumped system against experimental data in Example 4.5.4. Example 4.5.1: Curved Flexure Verification We verify a lumped curved beam by comparing its deflection to FEA. The type of curve that is easiest to configure is that of a semicircle. We parameterize this shape in the lumped model by a radius of curvature and angle subtended, and we form this shape in FEA by using a series of Boolean geometry. Simulations of curved beams in the shape of a quarter of a circle are shown in Figure 4.21. Due to reduced order modeling, the lumped model created in MATLAB is over two orders of magnitude faster than FEA model created in COMSOL. That is, the lumped model has one element, compared to over 1400 elements for FEA. The relative error in the lumped model is less than 1% of nonlinear FEA. In Figure 4.22, we show that the
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Final 7 μN
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FIGURE 4.21 Curved flexure—lumped versus distributed. Static simulations of circular-shaped beams are shown for a lumped model created in MATLAB (left) and a distributed FEA model created in COMSOL (right). To better perceive the deflections, the final states are superimposed onto the initial states. The lumped model consists of one element and the FEA model consists of 1487 elements (the discretization is shown on the initial FEA state). A transverse force of 7 μN was applied at the far ends of the beams. The relative error in the lumped model deflection is with respect to FEA. Parameters for lumped model and FEA model include width = 5 μm, thickness = 2 μm, Young’s modulus = 170 GPa, Poisson’s ratio = 0.3, radius of curvature = 97.5 μm, angle subtended = π/2.
% Relative error, lumped to distributed
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FIGURE 4.22 Model convergence of lumped versus distributed models of a curved flexure. The eight absolute relative errors (circled) were determined by comparing a single deflection from a lumped model and eight disparate deflections from FEA with increased discretizations. The number of elements in the FEA simulations ranged from 44 to 1487; the corresponding degrees of freedom for FEA ranged from 411 to 9783. One curved beam element was used in the lumped simulation; the element had 7 degrees of freedom (3 translational + 3 rotational + 1 electrical). The relative error between the lumped and FEA models is less than 1% for small deflections.
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FIGURE 4.24 MATLAB and COMSOL displays of perforated flexure simulation, lumped versus distributed models. Displacement values are color mapped onto the surface of the FEA simulation. We produced the deflections by directly applying a moment onto the lumped model through the netlist and a pair of surface pressures upon the FEA model. Solution time in MATLAB for the lumped model is over three orders of magnitude faster than FEA. The parameters include length = 120 μm, width = 18 μm, thickness = 18 μm, perforations = 10 μm × 10 μm, Young’s modulus = 170 GPa, Poisson’s ratio = 0.3.
FIGURE 4.25 COMSOL simulation and display of a 2.5D comb drive. A plot of the equipotentials is shown. We approximate the out-of-plane third dimension by extrusion. The simulated comb drive is modeled as a perfect conductor with perfectly flat surfaces and 90° corners.
capacitance due to a larger ambient space would be small since the additional space would be much further away from the electrodes. Moreover, higher precision can be obtained from a maximally refined, small field mesh, as opposed to a larger field that cannot be as densely meshed. Figure 4.25 shows the 2.5D FEA configuration and field solution of the comb drive model. We determine the force by first computing the FEA capacitance
∫
CFEA , i = 2 We , i h dA V 2 of the ith configuration, where We,i is the energy density. We horizontally displace one of the combs by a small amount xi+1 − xi by using the arbitrary Lagrangian–Eulerian moving boundary technique [28]. Then we compute the (i + 1)th capacitance CFEA,i+1. In Figure 4.26, the FEA-based force FFEA =
1 2 ( CFEA , i +1 − CFEA , i ) V 2 xi + 1 − xi
(4.341)
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FIGURE 4.26 Convergence of comb drive force between lumped and distributed models. We determine the data points by comparing the lumped comb drive force in MATLAB against FEA simulations. The number of elements in the FEA simulations range from 214 to 113,416; the corresponding degrees of freedom for FEA range from 612 to 229,195. We represent this model in MATLAB with five electromechanical beam elements and 42 degrees of freedom. The relative error between the lumped model and FEA is less than 0.14% for the most refined FEA simulation. The inset shows an intermediate mesh.
is compared to the parallel-plate approximation F|| = N ε 0V 2
h , g
(4.342)
where N is the number of fingers on the comb that has the least amount of fingers. This force approximates the net force that acts upon each comb. Each comb of the micromirror consists of a multitude of fingers. For computational efficiency, instead of applying hundreds of force vectors, we treat these forces as a uniform distribution about the backbone of the comb drive and compute the equivalent lumped force and moment. We apply these lumped forces and moments at the far ends of each comb. That is, since the force per unit length of the load is P = F||/L, the equivalent force and moment are
∫
F = P ψ F dx =
N ε0 h 2 V 2g
(4.343)
N ε0 hL 2 V , 12 g
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and
∫
M = P ψ M dx =
where ψF and ψM are Hermitian shape functions [29]. We exemplify the use of these forces in the lumped model shown in Figure 4.27.
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FN FN–2
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FIGURE 4.27 MATLAB simulation and display of an efficient lumped comb drive model. The angled view shows a multitude of forces applied at the ends of the comb fingers. The top view shows the equivalent lumped forces and moments applied to the far ends of the comb drive. Since comb drive forces are relatively small, we apply a large voltage to exaggerate the deflection so that the effect of the forces and moments can be seen.
Example 4.5.4: Validation of Assembled System against Experimental Results Using the netlist description of the MEMS micromirror discussed in Example 4.4.1.3, we compare simulation against experiment. Scanning electron microscope images of the micromirror are given in Figure 4.28. Images of the system-level representation of the MEMS are given in Example 4.4.1.3. Measurements of the tilt angle of the micromirror were performed by deflecting laser light from it onto a distant wall. Thirteen angles were V Cosine-shaped beam M
F Pull force Tether
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Tether is attached to the end of the U-shaped moment arm, here
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FIGURE 4.28 Image of a micromirror during actuation [30,31]. The comb drive array converts an electric potential V into a mechanical force F that pulls the pair of tethers. The moment arm converts this translational force F into a moment M that rotates the circular mirror by θ. The other half of the comb drive array and the second cosineshaped flexible beam are outside the view range of this figure. The difference in pigmentation of the structural material is an artifact of SEM imaging due to differing amounts of electric potential.
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FIGURE 4.29 Simulation versus experiment. This is a plot of micromirror angle versus voltage of the system-level model described in Example 4.4.1.3. Experimental measurements are shown as circles. We sweep the voltage in the netlist from 0 to 150 V (solid line).
measured from 0° at 0 to 22° at 125 V. Using voltage as an input parameter to the netlist, we sweep the voltage from 0 to 150 V. The measured data and MATLAB simulations are shown in Figure 4.29 in terms of mirror tilt angle as a function of voltage [30].
4.6 Conclusion In this chapter, we have discussed some of the latest methods in modeling and designing N/MEMS at the system level. Our design and modeling advancements were in effort toward an efficient system-level framework for systematically configuring complexengineered nano and microsystems. Regarding modeling, we discussed a systematic method for representing complex, multidisciplinary components of N/MEMS by their energy functions, constraints, and efforts. The method presented here allows the disparate length scales of MEMS, NEMS, and molecular dynamics to be modeled using the same tool and systematic methodology. Regarding design, we discussed an interactive MATLAB GUI that allows users to quickly and easily configure complex configurations, as well as a most powerful netlist language, which embraces the full flexibility and functionality of the MATLAB environment.
References 1. J.V. Clark and Z. Yi, Design and Simulation Using PSugar 0.5, 17th Biennial IEEE University Government Industry Micro/Nano Symposium, West Lafayette, IN, 2010. 2. ANSYS, Inc. Southpointe, 275 Technology Drive, Canonsburg, PA. http:www.ansys.com 3. Coventorware, 951 Mariners Island BLVD, San Mateo, CA. http://www.coventor.com 4. IntelliSuite, IntelliSense Corporation, Woburn, MA 2000. http://www.intellisense.com
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5. COMSOL, Inc. Los Angeles, CA. http://www.comsol.com 6. S.D. Senturia, CAD challenges for microsensors, microactuators, and microsystems, Proceedings of the IEEE, 86(8), 1611–1626, August 1998. 7. S. Senturia, N. Aluru, and J. White, Simulating the behavior of MEMS devices: Computational methods and needs, IEEE Computational Science and Engineering, 16(10), 30–43, January–March 1997. 8. B.C. Fabien and R.A. Layton, Modeling and simulation of physical systems I: An introduction to Lagrangian DAEs, in Proceedings IASTED International Conference, Robotics and Manufacturing, Honolulu, HI, 1996. 9. B.C. Fabien and R.A. Layton, Modeling and simulation of physical systems II: An approach to solving Lagrangian DAEs, in Proceedings IASTED International Conference, Robotics and Manufacturing, Honolulu, HI, 1996. 10. B.C. Fabien and R.A. Layton, Modeling and simulation of physical systems III: An approach to modeling dynamics constraints, in Proceedings IASTED International Conference, Applied modeling and simulation, Banff, Canada, 1997. 11. C. Fuhrer and B.J. Leimkuhler, Numerical solution of differential algebraic equations for constrained mechanical motion. Numerische Mathematik, 59, 55–69, 1991. 12. R.A. Layton, Principles of Analytical System Dynamics. Boston, MA: Springer, 1998. 13. B. Fabien, Analytical System Dynamics. New York: Springer, 2009. 14. U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia, PA: SAIM, 1998. 15. K.E. Brenan, S.L. Camplell, and L.R. Patzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (Classics in Applied Mathematics). Philadelphia, PA: SAIM, 1995. 16. L. Nagel, SPICE: A computer program to simulate semiconductor circuits, UCB/ERL M520, University of California, Berkeley, CA, May 1975. 17. G. Lorenz and R. Neul, Network-type modeling of micromachined sensor systems, in Modeling and Simulation Microsystems, Santa Clara, CA, April 1998. 18. G.K. Fedder, Structured design of microelectromechanical systems, in Proceedings of the IEEE International Conference on Microelectromechanical Systems, Orlando, FL, January 17–21, 1999. 19. G.K. Fedder and Q. Jing, NODAS 1.3—Nodal design of actuators and sensors, in Proceedings of IEEE/VIUF International Workshop on Behavioral Modeling and Simulation (BMAS’98), Orlando, FL, October 27–28, 1998. 20. J.V. Clark, N. Zhou, and K.S.J. Pister, MEMS simulation using SUGAR v0.5, in Proceedings of Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, pp. 191–196, 1998. 21. J.V. Clark, N. Zhou, and K.S.J. Pister, MEMS simulation using sugar v0.5, in Transducer’s SolidState Sensor and Actuator Workshop, Hilton Head Island, SC, pp. 191–196, June 8–11, 1998. 22. J.V. Clark, N. Zhou, D. Bindel, L. Schenato, W. Wu, J. Demmel, and K.S.J. Pister, 3D MEMS simulation modeling using modified nodal analysis, in Proceedings of the Microscale Systems: Mechanics and Measurements Symposium, Orlando, FL, pp. 68–75, June 8, 2000. 23. J.V. Clark, D. Bindel, N. Zhou, S. Bhave, Z. Bai, J. Demmel, and K.S.J. Pister, Sugar: Advancements in a 3D multi-domain simulation package for MEMS, in Proceedings of the Microscale Systems: Mechanics and Measurements Symposium, Portland, OR, June 4, 2001. 24. J.V. Clark, D. Bindel, W. Kao, E. Zhu, A. Kuo, N. Zhou, J. Nie et al., Addressing the needs of complex MEMS design, in Proceedings IEEE of the Fifteenth Annual International Conference on Micro Electro Mechanical Systems, Las Vegas, NV, January 20–24, 2002. 25. Z. Bai, D. Bindel, J.V. Clark, J. Demmel, K.S.J. Pister, and N. Zhou, New numerical techniques and tools in sugar for 3D MEMS simulation, in Technical Proceedings of the Fourth International Conference on Modeling and Simulation of Microsystems, Hilton Head Island, SC, pp. 31–34, March 19–21, 2001. 26. Sandia National Labs. New Mexico, Albuquerque, NM. http://www.mems.sandia.gov 27. J. Allen, Micro Electro Mechanical System Design. Boca Raton, FL: Taylor & Francis, Inc., 2005.
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28. I. Harouche and C. Shafai, Simulation of shaped comb drive as a stepped actuator for microtweezers application, in Sensors and Actuators A: Physical, 2005. 29. M. Paz, Structural Dynamics Theory and Computation. New York: Litton Education Publishing, Inc., 1980. 30. M. Last and V. Melanovic, U.C. Berkeley, private communication, 2002. 31. Dust Networks, Hayward, CA. http://www.dust-inc.com
5 Numerical Integrator for Continuum Equations of Surface Growth and Erosion Adrian Keller, Stefan Facsko, and Rodolfo Cuerno CONTENTS 5.1 Introduction........................................................................................................................ 189 5.1.1 Continuum Equations for Surface Growth ........................................................ 190 5.1.2 Continuum Equations for Ion Erosion................................................................ 192 5.2 Numerical Integration....................................................................................................... 195 5.2.1 Numerical Integrator “Ripples and Dots” ......................................................... 195 5.2.2 Solving Partial Differential Equations in MATLAB® ....................................... 197 5.3 Results and Discussions ................................................................................................... 199 5.3.1 Anisotropic Case: Ripple Patterns....................................................................... 199 5.3.2 Isotropic Case: Dot Patterns ................................................................................. 203 5.3.3 Anisotropy and Rotation: Dot Patterns.............................................................. 204 5.3.4 Nonlinearities......................................................................................................... 205 5.3.5 Dynamic Scaling.................................................................................................... 207 5.3.6 Two and Multiple Ion Beam Patterning.............................................................. 209 5.4 Conclusions......................................................................................................................... 211 Acknowledgments ...................................................................................................................... 212 References...................................................................................................................................... 212
5.1 Introduction Thin-film growth by physical vapor deposition (PVD) and surface erosion by ion beam sputtering (IBS) are crucial techniques for numerous process steps in nanotechnology, for example, in microelectronic device fabrication, data storage, or chemical sensing (Waser 2005). With the further reduction of the characteristic dimensions of the fabricated structures to less than 100 nm, the nanoscale morphology of deposited films and ion sputtered surfaces is becoming more and more important due to the growing influence of the interfaces on the physical and chemical properties of the nanostructures. Therefore, the morphological evolution of grown and eroded surfaces has gained considerable attention, which has been further promoted by the emergence of scanning probe microscopy techniques that offered the possibility to investigate the surface topography on the nanoscale up to the microscale with unprecedented accuracy (Zhao et al. 2000). Many of these surfaces exhibit spatial and temporal fluctuations that follow certain scaling relations similar to those observed in equilibrium critical phenomena (Plischke and Bergersen 2006) and can be analyzed by means of fractal geometry (Barabási and Stanley 1995). 189
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In addition, self-organization phenomena have been observed during IBS, leading to the formation of periodic nanoscale patterns on the surface (Chan and Chason 2007). At offnormal angle of incidence ripple patterns are observed, whereas at normal incidence hexagonally ordered dot patterns are formed on the sputtered surface. Using low-energy ions (kinetic energy ε < 1 keV), these patterns exhibit a periodicity of 10–50 nm with mediumrange order. Recently, it has been demonstrated that these self-organized surface patterns can be used as templates for the growth of thin films with modified magnetic properties (Bisio et al. 2006; Liedke et al. 2007; Fassbender et al. 2009; Körner et al. 2009), as well as for aligning metallic nanoparticles, which results in pronounced anisotropic optical properties (Oates et al. 2007, 2008; Camelio et al. 2009). In order to tune and tailor the characteristics of these templates an accurate modeling of the surface morphology would be highly desirable. Complementary to atomistic simulations of growth and erosion based on kinetic MonteCarlo (kMC) methods or molecular dynamics (MD) simulations, which have developed tremendously in the last years, continuum equations are able to cover a much larger spatial and temporal scale accessing the macroscopic scales at which nontrivial features of the morphological dynamics occur, that remain out of reach for atomistic approaches. Thus, the study of surface growth and erosion via continuum equations has continuously received attention throughout the last 30 years (Krug and Spohn 1991; Lapujoulade 1994; Barabási and Stanley 1995). Especially the concepts of fractal surfaces and universality classes in growth/erosion have been addressed extensively, leading to new insight into these phenomena. In the majority of cases the continuum equations contain nonlinear and noise terms and need to be solved numerically. 5.1.1 Continuum Equations for Surface Growth Thin-film growth by PVD is of tremendous technological importance (Smith 1995). Thus, the theoretical modeling of the various growth processes has received lots of attention during the last decades (Krug and Spohn 1991; Meakin 1993; Barabási and Stanley 1995; Halpin-Healy and Zhang 1995). In general, three different basic kinetic growth models have been established, which can be approximated by continuum equations: random deposition with surface relaxation, ballistic deposition, and molecular beam epitaxy (MBE) growth with surface diffusion. In the following, we will neglect external driving forces that cause a constant upward (growth) or downward motion (erosion) of the surface since the resulting contributions to the surface evolution can be easily omitted by introducing a co-moving frame of reference. In the random deposition model with surface relaxation, particles follow straight trajectories toward the surface where they are randomly deposited. After deposition, they diffuse along the surface up to a finite distance until they find the position with lowest height, which they then occupy. It was demonstrated that this model can be described by the Edward–Wilkinson (EW) equation (Barabási and Stanley 1995). The EW equation was the first continuum equation used to study the growth of surfaces by particle deposition (Edwards and Wilkinson 1982). It only consists of a linear and a noise term and has the form ∂h = ν∇ 2 h + η. ∂t
(5.1)
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Here, h(x, t) is the surface height as a function of the lateral coordinates x and the time t. The linear coefficient ν > 0 acts like a form of “surface tension” and tends to smooth the surface. The Gaussian white noise term η accounts for the stochastic arrival of the particles and is defined as η(r , t) η (r ʹ , tʹ ) = 2Dηδ d (r − r ʹ ) δ (t − tʹ )
(5.2)
with the noise strength Dη and the dimensionality of the surface d. For instance, in the context of equilibrium statistical physics (Plischke and Bergersen 2006), the EW equation corresponds to the dynamics of thermal fluctuations of the so-called critical Gaussian model around its equilibrium state, Dη being proportional to temperature. Ballistic deposition was originally introduced as a model of colloidal aggregates (Barabási and Stanley 1995) and later gained importance as a description of surface growth by vapor deposition (Family and Vicsek 1985). In the ballistic deposition model, the particles arrive to the surface and stick to the first particle they meet, which induces the formation of overhangs (Meakin et al. 1986). Thus, in contrast to random deposition with relaxation, ballistic deposition leads to lateral growth, that is, growth occurs in the direction of the local surface normal. A suitable continuum description of the ballistic deposition model is the so-called Kardar–Parisi–Zhang (KPZ) equation (Kardar et al. 1986). Since lateral growth requires the presence of nonlinearity, the KPZ equation appears as a nonlinear generalization of the EW equation ∂h λ 2 = ν∇2 h + (∇h ) + η . ∂t 2
(5.3)
Due to its nonlinear character, the KPZ equation cannot be solved in closed form like the EW equation. Hence, in order to study the spatiotemporal behavior of the KPZ equation, one has to apply either approximation methods or numerical simulations. In MBE growth, deposited atoms chemisorb to the surface to saturate bonds. Therefore, atoms relax to the nearest local energy minima, which in general are different from the local height minima (Wolf and Villain 1990; Das Sarma and Tamborenea 1991). In this case, the surface current is driven by differences in the surface chemical potential, which is proportional to the surface curvature (Wolf and Villain 1990). In the continuum equation describing the MBE growth model, this results in a term proportional to the fourth derivative of the surface height: ∂h = − K ∇ 4 h + η. ∂t
(5.4)
The linear MBE equation (5.4) thus combines Mullins’ surface diffusion equation (Mullins 1957) with random deposition (noise). In parallel with the EW–KPZ connection, an important nonlinear generalization of Equation 5.4 corresponds to the incorporation of an additional term on the right-hand side of Equation 5.4 with the form λ(2)∇2 (∇h)2, usually described in this context as a conserved KPZ term due to the fact that it conserves the total mass at the surface (Barabási and Stanley 1995). The ensuing continuum model is termed as the Lai–Das Sarma (Lai and Das Sarma 1991) or as the Sun–Guo–Grant equation (Sun et al. 1989; Janssen 1997), depending on the statistics of the noise.
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The above continuum equations have in common that they are linearly stable. This means that, for their corresponding linear versions, all the Fourier modes of the height decay exponentially in time (in the absence of noise) due to the fact that ν ≥ 0. In contrast, continuum equations modeling ion erosion of surfaces, which to some extent can be interpreted as “negative surface growth,” in general feature a linear instability since ν < 0 for the corresponding physical conditions. In addition, these equations are in general anisotropic due to the directionality of the ion beam. 5.1.2 Continuum Equations for Ion Erosion A major breakthrough for the continuum description of surface pattern formation and evolution during IBS was achieved by the model of Bradley and Harper (BH) (Bradley and Harper 1988), which is based on Sigmund’s theory of sputtering (Sigmund 1973). Under the assumption of a Gaussian distribution for the energy deposited in the target by the impinging ion, Sigmund showed that the local erosion rate at a given point on the surface is proportional to the total energy deposited at this point by the nuclear collision cascades of the ions impinging at all points on the surface. In the case of a rough surface, the local erosion rate then depends on the local curvature of the surface and is higher in depressions than on elevations (Sigmund 1973). This curvature dependence of the erosion rate induces an instability of the surface against periodic disturbances, which leads to an amplification of the initial surface roughness. The presence of a competing smoothing mechanism like surface diffusion further leads to a wavelength selection that is associated with the dynamic dominance of a single Fourier mode of the surface height (Bradley and Harper 1988). Hence, the formation of periodic structures on the surface results from a competition between roughening due to removal of surface material and diffusional smoothing. Approximating the full integro-differential equation until the second order in the height derivative, i.e., the surface curvature, BH obtained the linear deterministic continuum equation (Bradley and Harper 1988) ∂h ∂h ∂2h ∂2h =γ + νx 2 + ν y 2 − K ∇ 4 h . ∂t ∂x ∂x ∂y
(5.5)
Here, the projected direction of the ion beam is parallel to the x-axis. The first term on the RHS of Equation 5.5 represents the lateral motion of the ripple patterns with constant γ and does not influence topography amplification or decay. The linear terms induce a surface instability in the x- and y-directions for negative νx and νy, respectively, which leads to the formation of a periodic pattern with wavelength lc = 2π
2K . min( νx , ν y )
(5.6)
The amplitude of the induced pattern grows exponentially with growth rate R=
min( νx , ν y ) 4K
.
(5.7)
For νx < νy and νx > νy, the wave vector of the observed pattern is k c = kcex and k c = kcey, respectively. Since the negative linear terms tend to increase the surface area while
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conserving the volume, their coefficients νx,y are sometimes referred to as “negative surface tensions.” In the framework of Sigmund’s Gaussian approximation, the linear coefficients depend on the incidence angle of the ion beam, which results in a ripple pattern with the wave vector parallel to the x-axis at near-normal ion incidence, whereas the pattern is rotated by 90° at grazing incidence (Bradley and Harper 1988). This pattern rotation has been verified experimentally on various surfaces including Si (Mollick and Ghose 2009), SiO2 (Keller et al. 2009b), and metals (Rusponi et al. 1996). The BH equation (5.5) is able to reproduce the formation and early evolution of ripple patterns during ion sputtering. At long times, however, nonlinear terms have to be taken into account. To lowest nonlinear order, this leads to (Cuerno and Barabási 1995) 2
2 λ y ⎛ ∂h ⎞ ∂h ∂h ∂2h ∂ 2 h λ ⎛ ∂h ⎞ =γ + νx 2 + ν y 2 + x ⎜ ⎟ + − K∇4 h + η . ∂t ∂x ∂x ∂y 2 ⎝ ∂x ⎠ 2 ⎜⎝ ∂y ⎟⎠
(5.8)
Equation 5.8 is an anisotropic and stochastic generalization of the well-known Kuramoto– Sivashinsky (KS) equation (Kuramoto and Tsuzuki 1976; Sivashinsky 1979). Obviously, Equation 5.8 incorporates all the terms that also appear in the above continuum equations for surface growth, Equations 5.1, 5.3, and 5.4, and can thus be considered as a generalization of these growth models. For short sputtering times, the anisotropic KS equation behaves like the linear BH equation with an exponential increase of the ripple amplitude. Then, at a certain transition time tc ∝
⎛ νx , y ⎞ K ln ⎜ , 2 νx , y ⎝ λ x , y ⎟⎠
(5.9)
the surface enters a nonlinear regime and a saturation of the ripple amplitude is observed, as in experiments (Park et al. 1999; Erlebacher et al. 2000). However, numerical analyses of the stochastic KS equation show that the saturation of the ripple amplitude is accompanied by a transition to kinetic roughening (Park et al. 1999). In this regime, the ripple pattern decays and the surface reaches an unordered state. Although the anisotropic KS equation (5.8) includes KPZ nonlinearities, other higher order terms are neglected (Cuerno and Barabási 1995). The most general nonlinear equation that has been derived following the approach of BH is given by Makeev et al. (2002): ∂h ∂h ∂2h ∂2h λ =γ + νx 2 + ν y 2 + x ∂t ∂x ∂x ∂y 2
2
2 λ y ⎛ ∂h ⎞ ∂3h ∂3h ⎛ ∂h ⎞ + Ω + Ω 2 1 ⎜⎝ ⎟⎠ + ∂x ∂x 3 ∂x ∂y 2 2 ⎜⎝ ∂y ⎟⎠
2 2 ∂4h ∂4h ∂4h ⎛ ∂h ⎞ ⎛ ∂ h ⎞ ⎛ ∂h ⎞ ⎛ ∂ h ⎞ + ξ x ⎜ ⎟ ⎜ 2 ⎟ + ξ y ⎜ ⎟ ⎜ 2 ⎟ − Dxx 4 − Dyy 4 − Dxy 2 2 − K ∇ 4 h + η. ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ∂x ∂y ∂x ∂y (5.10)
Equation 5.10 now features a new type of nonlinearity, so-called dispersive nonlinearities with the coefficients ξx,y. Although it has been established that these nonlinearities are affecting the shape of the formed ripples (Makeev et al. 2002) and in addition contribute to the ripple motion causing a nonuniform velocity (Muñoz-García et al. 2008),
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their dynamical role has not yet been completely assessed (Keller et al. 2009a). Also the terms proportional to Ω1, 2 contribute to the traveling of the ripples along the surface (Makeev et al. 2002). The terms with the coefficients Dij enter Equation 5.10 in the form of diffusion-like terms proportional to the fourth derivative of the surface height function and thus lead to an additional anisotropic smoothing of the surface. Therefore, this relaxation mechanism is usually called effective or ion-induced surface diffusion (ISD) (Makeev and Barabási 1997). However, ISD results from preferential erosion during the sputtering, which appears as a reorganization of the surface and does not involve any mass transport. Thus, ISD is an erosion mechanism that just mimics surface diffusion. This also evolves from the fact that the coefficient Dxx becomes negative at large incidence angles, leading to an additional instability of the surface (Makeev and Barabási 1997). For off-normal incidence, the general continuum (GC) equation (5.10) has a highly nonlinear character with a rich parameter space, which might lead to rather complex morphologies and dynamic behaviors. In the special case of normal ion incidence, however, Equation 5.10 reduces to the isotropic KS equation with γ = ξx = ξy = Ω1 = Ω2 = 0, νx = νy, λx = λy and Dxx = D yy = Dxy/2 (Makeev et al. 2002). A general problem of the isotropic KS equation is its inability to reproduce the formation of hexagonally ordered dot patterns, as occasionally observed in experiments conducted under normal ion incidence (Facsko et al. 1999; Gago et al. 2001) or oblique incidence with simultaneous sample rotation (Frost et al. 2000). However, stationary hexagonal patterns are a feature of the damped version of the isotropic KS equation (Paniconi and Elder 1997; Facsko et al. 2004). In its generalized anisotropic form, the damped KS (dKS) equation reads 2
λy ∂h ∂h ∂2h ∂ 2 h λ ⎛ ∂h ⎞ = −α h + γ + νx 2 + ν y 2 + x ⎜ ⎟ + ∂t ∂x ∂x ∂y 2 ⎝ ∂x ⎠ 2
2
⎛ ∂h ⎞ 4 ⎜⎝ ∂y ⎟⎠ − K ∇ h + η .
(5.11)
The damping term −αh induces the smoothing of all spatial frequencies and, therefore, prevents kinetic roughening, which leads to stable patterns at high fluences as has been observed in several experiments (Bobek et al. 2003; Ziberi et al. 2005a; Keller et al. 2008). The damping coefficient α also enters the effective growth rate of the ripple amplitude Reff = R − α. The comparison of the numerical solution of these continuum equations with experiments shows a remarkable qualitative agreement reproducing many observations. However, still many open questions remain, for example, the origin of wavelength coarsening that is frequently found in experiments (Gago et al. 2006; Hansen et al. 2009; Keller et al. 2009a) or the observation of stable flat surfaces during sputtering (Carter and Vishnyakov 1996; Davidovitch et al. 2007; Keller et al. 2009b). In addition, the coefficients calculated from the simplified Sigmund energy deposition do not match quantitatively (Bobek et al. 2003). Partial improvements on, for example, the existence of coarsening have been achieved through modified equations in which anisotropic conserved KPZ terms λ ij( 2)∂ i 2 (∂ j h)2 appear on the right-hand side of Equation 5.10 (Muñoz-García et al. 2008, 2009, and references therein; Muñoz-García et al. 2010). Furthermore, stable flat surfaces can be reproduced by implementation of an additional smoothing mechanism, the so-called ballistic diffusion, which results in a positive term proportional to ∇ 2h (Carter and Vishnyakov 1996; Davidovitch et al. 2007). Nevertheless, further progress is required in the theoretical modeling of IBS.
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5.2 Numerical Integration 5.2.1 Numerical Integrator “Ripples and Dots” The software package (“Ripples and Dots”) is fully written in MATLAB® and integrates numerically the continuum equations introduced in the foregoing sections, modeling the evolution of the surface height during erosion by sputtering with low-energy ion beams. However, due to the similarity with continuum equations for interface growth, the package can be used as well for studying continuum equations describing growth, like the EW equation (5.1), the KPZ equation (5.3), and the linear MBE equation (5.4). All of these equations are partial differential equations describing atomistic processes at the surface, in a coarse-grained approach. The graphical user interface (GUI) of MATLAB has been used extensively to make the programs user-friendly and to output the results during the calculation. Figure 5.1 shows the main window, which appears after starting the “Ripples and Dots” program. The right side is dedicated to the input parameters. Here, the continuum equation can be selected from a list. The available equations are BH, KS, dKS, KPZ, and GC. Upon selection the corresponding formula appears and the coefficients needed in the equation are activated, whereas all unneeded coefficients are disabled. Here, one can also choose between different configurations of the “sputtering experiment,” like normal incidence, one ion source, two equal ion sources, and one ion source with simultaneous rotation. The azimuthal angles of the ion sources can be chosen arbitrarily.
FIGURE 5.1 Graphical user interface of the program “Ripples and Dots.”
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In the upper part of the input fields the values for the coefficients, like the negative surface tensions νx, νy, the nonlinear coefficients λx/2, λy/2, and the thermal diffusion coefficient K, have to be given. In the lower part parameters connected to the integration and the program can be changed, such as the grid size and the discretization steps Δx, Δy, and Δt, respectively. The program saves the surface morphology every “saving time” step and stops after the given integration time. When “Autosave” is checked the files with the surface morphology are saved as MATLAB data files in the folder “Results/mm_dd/” (mm= month and dd=day) with the name given automatically by the program as “h_gridsize_hhmm_tttt” (hh=hour, mm=minutes of the starting time, and tttt=integration time step). The initial surface for starting can be random rough with white noise with an amplitude of “h0.” Otherwise an arbitrary MATLAB data file consisting of a surface morphology with the same grid size can be chosen. In order to start the integration the input parameters have to be saved. The parameter file is also a MATLAB data file (“*.mat”) with the same filename as the surfaces without the integration time. The lower right box contains important values, like the expected periodicity lc, the calculated effective growth rate Reff, and the estimated saturation time tc. The progress of the calculation can be followed by the “integration time” ruler. The integration is done by a separate program called “RDCalculate.” During the run time of “RDCalculate” the surfaces are saved every “saving time” step and the main program updates regularly the output images in the left part of the window. The biggest figure shows the latest surface height as a color plot. The three figures below show the two-dimensional Fourier transform (2d-FFT), the one-dimensional (1d) power spectral density (PSD) function in the x- and y-directions, and the evolution of the root mean square roughness or global interface width, respectively. The coefficients for the continuum equations describing erosion can be calculated in the framework of the Bradley–Harper model using Sigmund’s theory of sputtering (Bradley and Harper 1988). Herein, the energy deposition of the incoming ions into the surface is assumed to have an asymmetric Gaussian form in an ensemble average E(r ) =
ε ( 2π )
3/2
⎛ ( z + a )2 x 2 + y 2 ⎞ exp ⎜ − − , σμ 2σ 2 2μ 2 ⎟⎠ ⎝ 2
(5.12)
where ε is the ion energy a is the mean penetration depth σ and μ are, respectively, the longitudinal and the lateral widths of the distribution The “Ripples and Dots” package also includes a program for calculating the coefficients in the way following the formulas given by Makeev et al. (2002). The program can be called from the main “Ripples and Dots” program via the menu and the calculated coefficients can be transferred back to the main window. In “CalculateParameters” (see Figure 5.2), the input values on the left side are the microscopic parameters of the energy distribution a, σ, and μ, which can be estimated by simulations of the collision cascade, that is, by TRIM (Ziegler et al. 1984; Bolse 1994) and the ion flux F. In order to calculate the normally unknown proportionality factor p between the deposited energy and the sputtering yield the program gives the opportunity to input the sputtering yield Y. The value for Y can also be simulated by TRIM or can be determined experimentally. Then p is calculated from the given value of Y and used for the calculation
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FIGURE 5.2 Graphical user interface of the program “CalculateParameters.”
of the coefficients. On the right side, the calculated coefficients for the GC equation are listed for the given incidence angle. As an addition, the constant erosion rate, the lateral drift velocity γ, the negative surface tensions νx, νy, the nonlinear terms λx, λy, the diffusion coefficients Dxx, Dyy, Dxy, as well as the growth rate R and the critical time tc can be plotted as a function of the incidence angle and of the microscopic parameters a, σ, and μ, respectively. 5.2.2 Solving Partial Differential Equations in MATLAB ® For calculating the temporal evolution of surfaces during erosion/growth described by the continuum equations the “RDCalculate” program uses a simple finite difference method in an Euler scheme with cyclic boundary conditions (Press et al. 2007). After converting the continuum equation into a difference equation using equidistant spatial (in x- and y-directions; z is the direction perpendicular to the surface into the bulk) and temporal discretizations, the equation can be integrated effectively by matrix multiplications. By using the “sparse” matrices in MATLAB the matrix multiplications are performed very fast and no further sophisticated algorithms are needed. In the Euler scheme the temporal and spatial differentials, ∂h/∂t and ∂nh/∂rn, are substituted by differences Δh/Δt and (Δh)n/(Δr)n. In this way, the partial differential equation turns into a difference equation, which can easily be integrated by n
hi + 1 = hi + Δt ⋅
∑ k 1
⎛ ( Δh)k ( Δh)k ⎞ a + b + k k ⎜⎝ ( Δx)k ( Δy )k ⎟⎠
∑ l,m
l
m
⎛ Δh ⎞ ⎛ Δh ⎞ clm ⎜ ⎟ ⎜ ⎟ . ⎝ Δx ⎠ ⎝ Δx ⎠
(5.13)
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In MATLAB the sum on the right-hand side can be calculated very effectively in a matrix formalism, where the differentiations are matrix multiplications. In addition, the cyclic boundary conditions are incorporated by including the corresponding matrix elements. As an example the first spatial derivatives, ∂h/∂x and ∂h/∂y, are replaced by (hi + 1 − hi )/Δx and implemented by multiplications with ⎡ −1 ⎢0 ⎢ grad = ⎢ 0 ⎢ ⎢0 ⎢⎣ 1
1 −1 0 0 0
0 1 −1 0 0
0 0 1 −1 0
0⎤ 0⎥ ⎥ 0 ⎥. ⎥ 1⎥ −1⎥⎦
(5.14)
By using the “sparse” matrices in MATLAB the integration becomes very fast, because most of the matrix elements are zero. Using this method the spatial derivatives in the x- and y-directions are implemented as multiplications with the grad matrix from left, grad · h, and from right, h · grad, respectively. For the second spatial derivative a symmetric matrix is used, which is equivalent to grad · grad shifted with one index to the left: ⎡ −2 ⎢1 ⎢ delta = ⎢ 0 ⎢ ⎢0 ⎢⎣ 1
1 −2 1 0 0
0 1 −2 1 0
0 0 1 −2 1
1⎤ 0⎥ ⎥ 0 ⎥. ⎥ 1⎥ −2 ⎥⎦
(5.15)
All the other derivatives are composed by the grad and delta matrices, for example, the diffusion term represented by the fourth derivative is calculated by delta· delta h. The first nonlinear term λ/2 (∂h/∂r)2 is calculated following the discretization proposed by Lam and Shin who showed that the simple discretization is not suited for this kind of nonlinearity (Lam and Shin 1998). If the azimuthal angle ϕ is not zero, the ion beam has to be rotated. This is done by multiplying the differential vector with the corresponding rotation matrix R: ⎡cos φ R=⎢ ⎣ sin φ
− sin φ ⎤ . cos φ ⎥⎦
(5.16)
By doing this the continuum equations remain of the same form, however, with mixed terms and new “rotated coefficients.” The rotated dKS equation then reads as follows: 2 ∂h ∂h ∂h ∂2h ∂2h ʹ ∂h = −αh + γ ʹx + γ ʹy + νʹx 2 + ν yʹ 2 + νxy ∂y ∂x ∂y ∂x ∂y ∂t ∂x
λʹ + x 2
2
2 λ ʹy ⎛ ∂h ⎞ λ ʹxy ⎛ ∂h ∂h ⎞ ⎛ ∂h ⎞ − K∇4 h + η + ⋅ ⎜⎝ ⎟⎠ + ⎟ ⎜ ∂x 2 ⎝ ∂y ⎠ 2 ⎜⎝ ∂x ∂y ⎟⎠
(5.17)
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with the new coefficients γ x = γ cos(φ); γ yʹ = γ sin(φ) ν ʹx = λ ʹx =
νx − ν y νx + ν y νx − ν y νx + ν y νx − ν y cos( 2φ); νʹxy = 2 ⋅ sin( 2φ) + cos( 2φ); νyʹ = − 2 2 2 2 2 λx + λy λx − λy λx + λy λx − λy λx − λy + cos( 2φ); λ ʹy = − cos( 2φ); λ ʹxy = sin( 2φ). 2 2 2 2 2 (5.18)
The global damping α and the diffusion coefficient K remain unchanged because they are isotropic. However, the anisotropic diffusion coefficients have to be changed accordingly. Finally, the noise is introduced as η = (2Dη)1/2 q with q being a matrix of random numbers drawn from a normal distribution with mean 0 and standard deviation 1, which is obtained using the internal random number generator of MATLAB. Note that the integration of the noise has to be realized by multiplication with the square-root of the time increment, Δt1/2 η (Barabási and Stanley 1995).
5.3 Results and Discussions In the following, we present some numerical results obtained using the software package “Ripples and Dots” described above. Here, we will focus on continuum equations for ion erosion. However, we will also assess the dynamic scaling behavior of some of the continuum equations for surface growth and erosion. 5.3.1 Anisotropic Case: Ripple Patterns Numerical integration of the continuum equations for surface erosion has been performed on a grid with 200 × 200 lateral nodes with Δx = Δy = 1 and Δt = 0.01. Figure 5.3 depicts the surface morphology in the deterministic BH equation (5.5) at four different times t. The coefficients used for the integration were νx = −1, νy = −0.3, and K = 1. For these parameters, Equation 5.6 yields a periodicity of l ∼ 9. The term γ∂h/∂x has been omitted since it just causes a lateral uniform motion of the pattern and can be eliminated by the transformation h(x, y, t) → h(x − γt, y, t). In the early stage of the simulation, short and disordered ripple-like structures appear. With time the number of pattern defects decreases, and these structures become more elongated. This is also reflected in the 2d-FFT of the topography (insets of Figure 5.3). Here, one can observe that the full width at half maximum of the peaks in the Fourier spectrum that relate to the periodicity of the ripple pattern decreases with time. This indicates an increase of the correlation length of the pattern. The evolution of the global interface width (Barabási and Stanley 1995) W = <[h(r, t) − ]2 >1/2 (where brackets denote space averages) is shown in Figure 5.4. The growth rate of the pattern as determined from the numerical results is in good agreement with the expected growth rate of R = 0.25, which follows from Equation 5.7. In the anisotropic stochastic KS equation (5.8) ripples form similar to the BH equation. However, at a time tc (Equation 5.9) the surface morphology enters the nonlinear regime.
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t = 10
t = 25
t = 50
t = 200
FIGURE 5.3 Surface morphology obtained by numer ical integration of the BH equation at four different times t. The insets show the FFT of the images. 50 40
In W
30
FIGURE 5.4 Evolution of the global interface width W for the numerical integration of the BH equation. The broken line corresponds to a growth rate R = 0.25 as obtained from Equation 5.7. All units are arbitrary.
20 10 0
–10
0
20
40
60
80
100 120 140 160 180 200 t
This is demonstrated in Figure 5.5, which depicts the evolution of the surface morphology obtained by numerical integration of Equation 5.8 using the same parameters as in Figure 5.3, with λx = 0.1, λy = 0.05, and Dη = 0.1. Here, at t = 25, the ripple pattern is more pronounced than in the BH case. At t = 50, however, the pattern is already lost and the surface morphology exhibits a cellular structure that is typical for the KS equation (Boghosian et al. 1999). Therefore, the FFT of the surface is isotropic and does not exhibit any peaks (insets of Figure 5.5). From the evolution of the global interface width W shown in Figure 5.6, one can determine that the transition to spatiotemporal chaos occurs at tc ∼ 30. At this time, after the initial exponential growth of the ripple amplitude, W saturates to a more or less constant value. In Figure 5.7, the morphology evolution in the GC equation is depicted for the same coefficients as in the KS equation, and the additional coefficient values ξx = 0.05, ξy = 0.02, Ω1 = 2, Ω2 = 4, Dxx = 0.5, Dyy = 0.1, and Dxy = 0.6. Obviously, the overall evolution of the ripple pattern is qualitatively similar to the anisotropic KS case shown in Figure 5.5 but
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t = 25
t = 50
t = 200
201
FIGURE 5.5 Surface morphology obtained by numerical integration of the anisotropic KS equation at four different times t. The insets show the FFT of the images. 5 4
In W
3 2 1 0
KS GC dKS
–1 –2
0
20
40
60
80
100 t
120
140
160
180
200
FIGURE 5.6 Evolution of the global interface width W for the numerical integration of the KS equation, the GC equation, and the dKS equation. All units are arbitrary.
exhibits different time dependence. From the evolution of the interface width given in Figure 5.6, one finds a smaller growth rate of the ripple amplitude and a delayed saturation of W. These observations result from the effective surface relaxation coefficient in the x-direction, K xeff = K + Dxx , which now enters Equations 5.7 and 5.9 and due to Equation 5.6 also causes a slightly higher ripple wavelength l ∼ 11. In the long time limit, the surface morphology in the GC equation lacks the rather well-defined cellular structure of the KS equation in Figure 5.5. Here, the additional anisotropies and nonlinearities result in a rather disordered morphology that exhibits features with a broad size distribution. When a linear damping term is added to the KS equation, the temporal evolution of the surface is slowed down due to the reduced growth rate Reff = R − α. Figure 5.8 shows the morphology evolution in the anisotropic dKS equation (5.11) with the same coefficients as in Figure 5.5 and a damping of α = 0.2. Thus, Reff = 0.05. This is obvious from Figure 5.8 where only short and shallow ripples are observed at t = 50, a time at which the surface
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t = 10
t = 25
t = 50
t = 200
t = 25
t = 50
t = 200
t = 400
FIGURE 5.7 Surface morphology obtained by numerical integration of the GC equation at four different times t. The insets show the FFT of the images.
FIGURE 5.8 Surface morphology obtained by numerical integration of the anisotropic dKS equation at four different times t. The insets show the FFT of the images.
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has already undergone kinetic roughening in the undamped case (cf. Figure 5.5). The evolution of W in Figure 5.6 shows that saturation is reached at t ∼ 130 in contrast to the KS equation where W saturates already at t ∼ 30. In addition, the saturation is not accompanied by a loss of the ripple pattern. Rather, at t = 200 a highly regular and well-ordered ripple pattern is observed (see Figure 5.8). The order of the pattern increases further with integration time, and a nearly perfect pattern with only a few defects is obtained at t = 400, similar to experimentally observed patterns formed under near-normal incidence sputtering of silicon surfaces (Ziberi et al. 2005a). 5.3.2 Isotropic Case: Dot Patterns The main motivation for adopting the dKS equation for ion erosion systems was its ability to yield stationary hexagonally ordered dot patterns under isotropic conditions (Paniconi and Elder 1997; Facsko et al. 2004). Figure 5.9 depicts the evolution of the surface morphology in the isotropic undamped KS equation for ν = −1, λ = 0.1, K = 1, and Dη = 0.1. At t = 10, an isotropic roughening of the surface is observed with the formation of shallow dotlike structures. At t = 20, when the nonlinearity is becoming important, dots of homogeneous size are the dominating features of the surface. In the FFT, this results in a ring-like structure. However, these dots exhibit a rather low degree of order and are only observed in a short time interval. At t = 50, the surface has already entered the kinetic roughening regime as in the anisotropic case (Figure 5.5), and the ring structure of the FFT has disappeared. In the presence of a damping term (α = 0.2), the temporal evolution of the surface is again slowed down (Figure 5.10). Here, at t = 50, the first pronounced dots appear accompanied by worm-like structures. These structures already display a high degree of homogeneity, t = 10
t = 20
t = 50
t = 200
FIGURE 5.9 Surface morphology obtained by numerical integration of the isotropic KS equation at four different times t. The insets show the FFT of the images.
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t = 10
t = 20
t = 50
t = 200
FIGURE 5.10 Surface morphology obtained by numerical integration of the isotropic dKS equation at four different times t. The insets show the FFT of the images.
which results in a well-defined ring in the FFT. With increasing integration time, however, the dots do not only get more pronounced but also their order increases drastically. At t = 200, domains of hexagonally ordered homogeneously sized dots are observed. The hexagonal order of the pattern is also reflected in the FFT in which the ring structure has now developed a hexagonal shape. These patterns are very similar to the ones obtained on semiconductor surfaces by normal incidence ion sputtering (Facsko et al. 2004). 5.3.3 Anisotropy and Rotation: Dot Patterns Similar to the isotropic case, which describes the pattern formation under normal incidence ion beam erosion, one expects dot patterns also in the case of off-normal ion incidence with simultaneous sample rotation. In this case the anisotropy is smeared out by the rotation and the pattern should also exhibit isotropic symmetry. Indeed, in sputtering experiments with rotation of semiconductor surfaces dot patterns have been observed experimentally (Frost et al. 2000; Ziberi et al. 2005b). For the simulation of the dynamics of surface morphology with simultaneous rotation the dKS equation with α = 0.1, γ = 1, νx = −1, νy = −0.3, λx = 0.1, λy = 0.05, K = 1, and Dη = 0.1 was used. To mimick rotation the ion beam is rotated continuously by changing the azimuthal angle every integration time step by 0.1°. The obtained morphology at four different times t is shown in Figure 5.11. The morphology is clearly dominated by an isotropic pattern at the beginning followed by a hexagonal dot pattern for later times, similar to the isotropic case shown in Figure 5.9. Using a higher value for the global damping α, that is, 0.2, the interface width saturates very fast at a very low value before a pattern can develop. It can therefore be concluded that the rotation reduces the effective growth rate Reff, thus suppressing the formation of dot or ripple patterns. This has also been observed experimentally and proposed for increasing the depth resolution in secondary ion mass spectrometry (SIMS) (Cirlin et al. 1991).
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t = 50
t = 200
t = 400
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FIGURE 5.11 Surface morphology obtained by numerical integration of the anisotropic dKS equation with simultaneous sample rotation at four different times t. The insets show the FFT of the images.
5.3.4 Nonlinearities In this section, we will focus on the role of the KPZ nonlinearities in the KS and the dKS equations. In the framework of the Sigmund approximation, λx changes its sign depending on the angle of incidence (Makeev et al. 2002), so that λxλy > 0 or λxλy < 0 for different incidence conditions. The case λxλy < 0 leads to an interesting new phenomenon and is shown in Figure 5.12, for the same parameters as in Figure 5.5 but with a negative λy = −0.05. At t = 25, one observes an ordered ripple pattern as in the λxλy > 0 case of Figure 5.5. At t = 50, the ripple pattern is lost and the surface undergoes kinetic roughening. However, the characteristic cellular structure is absent in these simulations. Instead, the surface displays elongated zigzag-like structures. Then, at longer times, a new ripple pattern forms, which is rotated with respect to the initial pattern. These ripples represent 1d solutions of the anisotropic KS equation (5.8) for which the nonlinearities precisely cancel each other and are, therefore, called cancellation modes (Rost and Krug 1995). The angle of rotation is obtained by moving to a rotated coordinate system that cancels the nonlinear terms in the transverse direction and is given by φc = tan−1 (−λx,y/λy,x)1/2. For the present parameters, this leads to an angle φc ∼ 54°, which is in good agreement with the one observed from the simulation in Figure 5.12. Since the cancellation modes represent 1d solutions in a coordinate system in which the nonlinearities are effectively zero, they behave like ripples in the BH equation, that is, their amplitude grows exponentially with time without saturation. This is shown in Figure 5.13, which gives the evolution of the interface width W. At t ∼ 350, W starts to increase again. This is due to the formation and exponential growth of the rotated ripple structures. However, up to now no experimental evidence for these cancellation modes has been found. More generally, the sign of the KPZ nonlinearity has a dramatic effect on the surface morphology. This is demonstrated in Figure 5.14 for the isotropic dKS equation with
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t = 50
t = 25
t = 400
t = 200
FIGURE 5.12 Surface morphology obtained by numerical integration of the anisotropic KS equation at four different times t for λxλy < 0. The insets show the FFT of the images.
6
In W
4
FIGURE 5.13 Evolution of the global interface width W for the numerical integration of the anisotropic KS equation for λxλy < 0. All units are arbitrary.
2 0
–2
0
40
80
120
160
200 t
240
280
320
360
400
λ = −0.1. The other coefficients are chosen as in the integration shown in Figure 5.10. At short times, that is, in the linear regime, there is no obvious difference to the case shown in Figure 5.10. Then, at t = 50, i.e., the onset of the nonlinear regime, one again observes worm-like structures but this time accompanied by holes instead of dots. With increasing time, a hole pattern forms exhibiting the same hexagonal short-range order as observed for the dot pattern with positive λ. Similar ordered hole patterns have recently been fabricated on germanium and silicon surfaces by normal incidence sputtering either with Ga ions (Wei et al. 2009) or with simultaneous co-deposition of metal atoms (Sánchez-García et al. 2008). Therefore, the sign of the nonlinearity seems to be correlated with the presence of metal atoms on the surface, an effect that is not considered in the BH model.
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t = 50
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t = 20
t = 200
FIGURE 5.14 Surface morphology obtained by numerical integration of the isotropic dKS equation at four different times t for λ < 0. The insets show the FFT of the images.
5.3.5 Dynamic Scaling In many growth and erosion processes, the global interface width W is observed to satisfy the dynamic scaling ansatz of Family and Vicsek (FV) (Family and Vicsek 1985; Barabási and Stanley 1995) W (L, t) = tβ f (L/t1/z )
(5.19)
with the system size L and the scaling function ⎧const. if u 1 f ( u) = ⎨ χ , if u 1 ⎩u
(5.20)
where χ is called roughness exponent z is the dynamic exponent β = χ/z is the growth exponent Based on the numerical values of these dynamic scaling exponents, the system can be assigned to a certain universality class and, therefore, to a certain type of continuum equation (Barabási and Stanley 1995). Alternatively, the scaling behavior of a surface can also be studied in Fourier space by evaluating its PSD: PSD ( k , t ) = hˆ ( k , t ) hˆ ( − k , t ) ,
(5.21)
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where hˆ (k, t) is the Fourier transform of the surface height. In the case of FV scaling, the 1d PSD can be expressed by the relation (Barabási and Stanley 1995) PSD(k , t) = k −( 2 χ + 1)s (kt1/z )
(5.22)
⎧const. if u 1 s ( u) = ⎨ 2 χ + 1 . if u 1 ⎩u
(5.23)
with the scaling function
In the following, we will evaluate the dynamic scaling properties of some of the continuum equations discussed above by analyzing the PSD functions of the simulated morphologies. For simplicity, we will restrict ourselves to the discussion of isotropic equations. In the case of anisotropic surfaces, the 1d PSD has to be evaluated independently in the x- and y-directions (Keller et al. 2009a). Figure 5.15 shows the 1d PSD functions in the x-direction for the EW (ν = 1) and the linear MBE equation (K = 1) at t = 500. The noise amplitude was Dη = 1 and the system size was 400 × 400 in all cases. The PSD of the EW morphology exhibits a slope (in the log–log plot) of m = −1. According to Equations 5.22 and 5.23, this corresponds to a roughness exponent χ = 0. This observation is in agreement with the analytically determined roughness exponent of the EW equation in 2 + 1 dimensions (Barabási and Stanley 1995). The linear MBE PSD has a higher slope of m = −3, corresponding to χ = 1. Again, this value agrees with the one expected for the linear (2 + 1)-dimensional MBE equation (Barabási and Stanley 1995). The (2 + 1)-dimensional KPZ equation has a more complicated scaling behavior. For the nonlinear coefficient λ smaller than a critical value λ c , the KPZ equation shows EW scaling in the asymptotic limit whereas λ > λc leads to KPZ scaling with χ = 0.38 (Canet and Moore 2007). To assess this issue, numerical integrations of the KPZ equation have been performed with ν = 1 and two different values of λ, λ = 1 and λ = 5. To provide numerical stability for the λ = 5 case, the integration step had to be reduced to Δt = 0.002. As is demonstrated in Figure 5.16, EW scaling with χ = 0 is indeed observed for λ = 1. For λ = 5, however, a larger roughness exponent is observed, which does not meet the predicted χ = 0.38. This is not surprising because it is known that finite difference methods tend to underestimate the nonlinear term and thus do not yield the correct KPZ scaling exponents EW Linear MBE
105
PSD
104 m = –3 103 FIGURE 5.15 One-dimensional PSD functions for the EW and the linear MBE equations at t = 500. The curves are shifted vertically for clarity. The straight solid lines correspond to km. All units are arbitrary.
m = –1
102 0.1
1 k
Numerical Integrator for Continuum Equations of Surface Growth and Erosion
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λ=1 λ=5
105 PSD
209
m = –1.76
104 103 102
m = –1 0.1
FIGURE 5.16 One-dimensional PSD functions for the KPZ equation with λ = 1 and λ = 5 at t = 500. The straight solid lines correspond to km. All units are arbitrary.
1 k
PSD
λ=1 λ=5
m = –1
105 104 103
m = –1.76
102 101
0.1
1 k
FIGURE 5.17 One-dimensional PSD functions for the KS equation with λ = 1 and λ = 5 at t = 500. The straight solid lines correspond to km. All units are arbitrary.
in general (Giacometti and Rossi 2001; Giada et al. 2002). In the case of the isotropic KS equation in (2 + 1) dimensions, which in the asymptotic limit is expected to behave like the KPZ equation (Cuerno and Lauritsen 1995), it is even harder to access the asymptotic exponents because its scaling behavior exhibits several crossovers and transition regimes. This is demonstrated in Figure 5.17, which shows the PSD functions for the KS equation at t = 500 with the coefficients ν = −1, K = 1, and two λ values, λ = 1 and λ = 5. For λ = 1, the observed behavior is close to EW scaling whereas λ = 5 leads to an increased roughness exponent that again does not match the KPZ exponent. Therefore, in order to study the asymptotic scaling behavior of the KPZ and the KS and related equations, one should consider using alternative numerical or approximation schemes, such as, for example, pseudospectral methods (Gallego et al. 2007; Nicoli et al. 2009). 5.3.6 Two and Multiple Ion Beam Patterning An interesting case that was discussed theoretically and studied experimentally is sputtering by two equal ion beams incident under different azimuthal angles onto the surface. If these ion beams are applied subsequently it results in erasing the pattern produced by the first ion beam and the emergence of the rotated pattern produced by the second ion beam (Kim et al. 2009). Therefore, Carter (2004) proposed the use of two and more ion beams simultaneously to create new types of patterns. In using two orthogonal ion beams he predicted “four-sided pyramidal features.”
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t = 10
t = 25
t = 50
t = 100
FIGURE 5.18 Surface morphology obtained by numerical integration of the anisotropic dKS equation for the simultaneous incidence of two perpendicular ion beams at four different times t. The insets show the FFT of the images.
In our program it is easy to include a second or a third ion beam, as it is only an additional integration step with a rotated ion beam. Figure 5.18 shows the morphology evolution with two perpendicular equal ion beams. The coefficients for both ion beams were the same as for the case shown in Figure 5.11, that is, α = 0.2, γ = 1, νx = −1, νy = −0.3, λx = 0.1, λy = 0.05, K = 1, and Dη = 0.1. It can be seen that the surface develops a dot pattern. The FFT functions in Figure 5.18 clearly show an isotropic pattern similar to the case of normal incidence or off-normal incidence with rotation. Actually this result can also be expected analytically because the continuum equation describing this case is simply 2
2 λ y ⎛ ∂h ⎞ ∂h ∂h ∂2h ∂ 2 h λ ⎛ ∂h ⎞ = −αh + γ + νx 2 + ν y 2 + x ⎜ ⎟ + − K∇4 h + η ∂t ∂x ∂x ∂y 2 ⎝ ∂x ⎠ 2 ⎜⎝ ∂y ⎟⎠ 2
− αh + γ
2 λ x ⎛ ∂h ⎞ ∂h ∂2h ∂ 2 h λ y ⎛ ∂h ⎞ + η. + ν y 2 + νx 2 + ⎜ ⎟ + 2 ⎝ ∂x ⎠ 2 ⎜⎝ ∂y ⎟⎠ ∂y ∂x ∂y
(5.24)
Simplifying this equation yields ⎛ ∂2h ∂2h ⎞ λ x + λ y ⎛ ∂h ∂h ⎞ ∂h = −2αh + γ ⎜ + ⎟ + ( νx + ν y ) ⎜ 2 + 2 ⎟ + ∂y ⎠ ∂t 2 ⎝ ∂x ⎝ ∂x ∂y ⎠
⎛ ⎛ ∂h ⎞ 2 ⎛ ∂h ⎞ 2 ⎞ 4 ⎜ ⎜ ⎟ + ⎜ ⎟ ⎟ − K ∇ h + 2η, ⎜⎝ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎟⎠ (5.25)
which is actually an isotropic dKS equation. However, the effective coefficients are different from the isotropic case with one ion beam. Especially, the growth factor Reff, the periodicity, and the saturation time will be different.
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t = 25
t = 50
t = 100
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FIGURE 5.19 Surface morphology obtained by numerical integration of the anisotropic dKS equation for the simultaneous incidence of two ion beams enclosing an angle of 40° at four different times t. The insets show the FFT of the images.
The case when the ion beams are not perpendicular to each other but enclose an arbitrary azimuthal angle is not so easy anymore. However, the analytical analysis shows that the resulting pattern is a ripple pattern with an orientation at half the angle between the ion beams. Also, in this case, effective coefficients different from the single ion beam are determined. In Figure 5.19, the surface morphology is shown for two ion beams enclosing an angle of 40° to each other at different times. The morphology now develops a ripple pattern with the wave vector pointing in the direction of the sum of the projected ion beams. In Figure 5.19, this results in the wave vector enclosing an angle of 20° with the x-axis. At long times t = 100, one can observe that the ripples are decaying into a dot pattern. This is caused by the effective KPZ nonlinearity, which is now stronger than in the case of just one ion beam. The complex case of two different ion beams with an arbitrary angle will not be presented here. However, also in this case no pattern with fourfold symmetry is expected.
5.4 Conclusions The MATLAB program package “Ripples and Dots” is designed to solve various linear and nonlinear continuum equations of surface growth and erosion in (2 + 1) dimensions. It is especially suited for the simulation of equations based on the BH model that features a linear instability leading to the formation of periodic nanopatterns during ion beam erosion. It has been demonstrated that the evolution of the surface morphology observed in the numerical integrations is in good qualitative agreement with experimental results for both normal and oblique ion incidence. In addition, also the special case of oblique ion
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incidence with simultaneous sample rotation and erosion with multiple ion beams can be treated in the software. The program package “Ripples and Dots” can be tried out as a web application and can be obtained from the Web site http://www.hzdr.de/RipplesandDots or from the authors via email.
Acknowledgments We thank M. Nicoli for helpful discussions. This work was partially funded by Deutsche Forschungsgemeinschaft FOR 845 and by MICINN (Spain) through Grant No. FIS200912964-C05-01. AK acknowledges financial support from the Alexander von Humboldt foundation.
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6 Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes of Carbon, Silicon-Carbide, and Carbon-Nitride W.-D. Cheng, C.-S. Lin, G.-L. Chai, and S.-P. Huang CONTENTS 6.1 Introduction........................................................................................................................ 218 6.2 Finite Open Single-Walled Carbon Nanotubes of (n, n) with n = 3–6........................ 220 6.2.1 Calculation Methods and Simulation Procedures ............................................ 220 6.2.1.1 Optimizations of Geometrical Configurations Based on the Density-Functional Theory ....................................................... 220 6.2.1.2 Calculations of Absorption Spectra ..................................................... 220 6.2.1.3 Calculations of Third-Order Polarizabilities ...................................... 221 6.2.1.4 Calculations of TPA Cross Section .......................................................222 6.2.2 Descriptions and Understandings of Results on Finite Open (n, n) SWCNTs ................................................................................................... 223 6.2.2.1 Geometries and Electronic Structures at Ground State ....................223 6.2.2.2 Electronic Absorption Spectrum of Ground State ............................. 227 6.2.2.3 Third-Order Nonlinear Optical Properties of Ground State............ 229 6.2.2.4 TPA Properties at Ground States .......................................................... 230 6.3 Silicon-Carbide and Carbon-Nitride Nanotubes .......................................................... 233 6.3.1 Theoretical Method and Computational Details .............................................. 233 6.3.2 Optimized Geometry Structures of SiCNTs and CNNTs ...............................234 6.3.3 Periodic Density Functional Theory Studies on Band Structure and Density of States.............................................................................................234 6.3.3.1 SiCNTs ......................................................................................................234 6.3.3.2 CNNTs ...................................................................................................... 237 6.3.4 Simulations of Linear Optical Properties of SiCNTs and CNNTs.................. 240 6.3.4.1 SiCNTs ...................................................................................................... 240 6.3.4.2 CNNTs ...................................................................................................... 243 6.4 Conclusions......................................................................................................................... 245 Acknowledgments ...................................................................................................................... 245 References...................................................................................................................................... 246
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6.1 Introduction Carbon nanotubes (CNTs) have aroused much interest in scientists and technicians since their discovery in 1991 [1–3], due to the unique physical properties that they exhibit. These properties could impact broad areas of science and technology, ranging from probe tips used for imaging to ultrasensitive gas sensors [4,5]. The remarkable electronic properties of CNTs offer exciting intellectual challenges and potential for novel applications. It was first predicted that single-walled carbon nanotubes (SWCNTs) could exhibit either metallic or semiconducting behavior depending on the theoretical calculations of their diameter and helicity [6–8]. This ability to display fundamentally distinct electronic properties without changing the local bonding, which was experimentally demonstrated through atomically resolved scanning tunneling microscopy (STM) measurements [9,10], sets nanotubes apart from all other nanowire materials [11,12]. At the beginning of this century, ultrasmall radius SWCNTs were successfully grown inside inert AlPO4–5 zeolite channels, and the nanotubes arranged in the channels had a narrow diameter distribution of about 4 Å [13]. The absorption spectra of these tubes can be investigated by experiment and theory because the possible nanotube chiralities are reduced to three: (3, 3), (5, 0), and (4, 2) [14–16]. Moreover, the nonlinear optical (NLO) properties of CNTs have been of considerable interest, not only because the nonlinear spectrum gives information on their electronic structure, but also because the nonlinear materials can be applied to optical devices [17–21]. NLO materials with large third-order nonlinear susceptibilities at the optical transparent region are indispensable for all-optical switching, modulating, and computing devices. The nonlinear absorption materials with large third-order optical susceptibility are used for optical limiters and tunable filters that can suppress undesired radiation. The CNTs used in NLO devices have promising features at frequencies greater than infrared frequencies of the lattice vibration, and the main contributions to optical nonlinearities originate from the one-dimensional motion of delocalized p-electrons at a fixed lattice ion configuration. Additionally, the designs and devices are based on the third-order NLO mechanisms of two-photon absorption (TPA) that offers several advantages. Primary among these is negligible initial linear absorption losses for weak signal. Second, the TPA has a very fast temporal response with respect to the variation of input laser pulse. The third advantage is its ability to retain high beam quality for the transmitted signal. Accordingly, TPA- based devices are suitable not only for optical limiting [22,23], but also for other application purposes such as optical power stabilization, optical pulse and spatial field reshaping, two-photon-excited fluorescence microscopy [24], threedimensional optical data storage [25], and two-photon-induced biological caging studies [26,27]. In these respects, it is the key research effort to develop new nonlinear absorption materials, which have large TPA cross sections and high physical and chemical stabilities under the action of high-intensity laser radiation. Fortunately, CNTs have large stabilizations of chemistry and will become two-photon materials [18,28]. Margulis and coworkers calculated the spectra of the nonlinear refractive index n2(ω) and the TPA coefficient α2(ω) for ensembles of diameter distributed and aligned SWCNTs, and the obtained results clearly show the importance of the nanotube diameter uniformity for the applicability of these materials to NLO devices [29]. In this respective subject, first, we report the optimized configurations of finite open SWCNTs of (n, n) with n = 3–6 based on the density functional theory (DFT). Then, we calculate the spectra of electronic absorption by the time-dependent density functional theory (TDDFT) and discuss their anisotropy. Third, we compute the dynamic third-order optical polarizabilities and TPA properties in terms
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of the sum-over-states (SOS) method combined with TDDFT and understand the electronic origins of NLO responses for this type of nanotubes. Except for pure CNTs, there are noncarbon or part-carbon nanotubes. In 2002, Sun et al. synthesized one-dimensional silicon-carbon nanotubes (SiCNTs) via the reaction of silicon (produced by disproportionation reaction of SiO) with multiwalled CNTs (as templates) at different temperatures [30]. SiCNTs are expected to have advantages over CNTs because they may possess high reactivity of exterior surface facilitating sidewall decoration and stability at high temperature [31]. Moreover, SiCNTs are predicted to be more suitable materials for hydrogen storage than pure CNTs at a theoretical level [32]. The structure and stability of SiC single-wall nanotubes have been investigated by ab initio theory in detail [33]. It was found that the SiCNTs with alternating Si–C bonds are energetically preferred over the forms that contain C–C or Si–Si bonds [33]. By using DFT, Zhao et al. studied the electronic structure of SiCNTs and suggested that all SiCNTs are wide bandgap semiconductors [31]. Theoretical studies have also been reported on the structural and electronic properties of native defects[34], substitutional impurities [35], the sidewall hydrogenation in SiCNTs [36], and SiCNTs decorated by CH3, SiH3, N, and NHx (x = 1, 2) groups [37,38]. Additionally, the properties of nanotubes can be changed by doping with foreign atoms, which provides opportunities to tune their properties and optimize their appliances. Carbon-nitride nanotubes (CNNTs) may have improved applications compared with traditional CNTs due to their high specific surface area and polar C–N bonds at the surface. Theoretical studies have indicated that CNNTs with CN or C3N4 stoichiometry may exist [39]. Thus, many research groups made efforts to synthesize CNNTs. A lot of N-doped CNTs have been synthesized by various methods [40,41], and it was found that N-doped CNTs have many novel properties such as electrocatalytic activity [42,43], magnetism [44], and field emission [45] and can be used as catalyst supporter [46]. However, both experimental and theoretical studies indicated that the doped concentration of the N in the CNTs is limited (no more than 10%). Accordingly, it is impossible to synthesize CNNTs through doping method. Guo and coworkers [47] reported that the syntheses of C3N4 nanotubes are via a benzene-thermal process without the use of any catalyst or template, and Minsik and coworkers reported that C3N4 nanostructures were synthesized by condensing cyanamide (CN–NH2) using colloidal silica as a template; they used the C3N4 nanostructures as supporter of Pt–Ru alloy catalyst [48]. Porous nanotubes with CN stoichiometry were synthesized by Cao et al. [49] through the reaction of cyanuric chloride (C3N3Cl3) with sodium at 230°C using NiCl2 as catalyst. However, the structures of synthesized CNNTs are different from that of the CNNTs predicted previously [39]. By using the same method, Li et al. synthesized three kinds of CN architectures: nanotube bundles, aligned nanoribbons, and microspheres [50]. The synthesized CNNTs with the stoichiometry of CN are multiwall nanotubes with diameters ranging from 50 to 100 nm and lengths of several micrometers. The C and N atoms in CNNTs show the character of sp2 hybridization, which indicate the graphitic structures in CNNTs. Generally, the CNNTs always show open ends [49]. The UV-vis absorption spectra of the CNNTs indicate that there is strong absorption around 260–280 nm, which may be due to the π → π* electronic transition in the aromatic 1,3,5-triazine units within the CNNTs [50]. Theoretically, Jose et al. investigated the C3N4 nanotubes systematically recently [51]. There are also reports on the studies of optical or thermodynamic properties of nanotubes with other N:C stoichiometry [52,53]. However, there are not theoretical and experimental studies addressing the optical properties of SiCNTs and CNNTs. Therefore, precise predictions for the fundamental optical properties of SiCNTs and CNNTs are strongly desirable. In these respective subjects, we investigate
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the optical properties of the zigzag (n, 0), armchair (n, n), and chiral (n, m) SiCNTs, and of single-walled CNNTs built from triazine units differing in chirality and diameter size by using first-principles calculations.
6.2 Finite Open Single-Walled Carbon Nanotubes of (n, n) with n = 3–6 6.2.1 Calculation Methods and Simulation Procedures 6.2.1.1 Optimizations of Geometrical Configurations Based on the Density-Functional Theory Initial geometries of the finite open SWCNTs of (n, n) with n = 3–6 can be constructed by cut strips and rolled up from an infinite graphite sheet, respectively. These tubes can be specified by their chiral vectors Ch = na1+ ma2, where a1 and a2 are graphite primitive lattice vectors with |a1| = |a2| = a = (C–C)(31/2), n = m = 3, 4, 5, and 6, C–C = 1.42 Å. The tube diameter and chirality are uniquely characterized by using d = a(n2 + m2 + nm)1/2/π and cos θ = a(n + m/2)/dπ, respectively [54–56]; here n = m = 3, 4, 5, and 6. The geometrical optimizations of the finite open SWCNTs of (n, n) with n = 3–6 are carried out at the B3LYP/6-31G* level using the DFT method of the GAUSSIAN03 program. During the optimized processes, a convergent value of RMS (root mean square) density matrix and the critical values of force and displacement are set by default of the GAUSSIAN03 program. The obtained values that are less than these criterions are omitted during the calculations. Accordingly, after the convergences of the maximum force, RMS force, maximum displacement, and RMS displacement are reached, the zero of the first derivatives and the positive of the second derivatives are obtained on a potential-energy surface for these SWCNTs. Zero length of all the gradient vectors (first derivatives) characterizes a stationary point, and a minimum corresponds to a point having positive second derivatives. Equivalently, all forces on the atoms in a finite open SWCNT are zero, and the force constant is positive. In classical mechanics, the first derivative of the potential energy for a particle is minus the force on the particle, and the second derivative is the force constant. Hence a stationary point of minimum on energy surfaces corresponds to equilibrium geometry of these SWCNTs. 6.2.1.2 Calculations of Absorption Spectra After the optimized geometries of the finite open SWCNTs of (n, n) with n = 3–6 are obtained, we employ the TDDFT [57–59] at the B3LYP/6-31G* level (TDB3LYP/6-31G*) and run in the program of GAUSSIAN03 in the calculations of the transition moments and excited-state energies of these configurations. Here, the B3LYP designs the Becke exchange function combined with the three parameters of Lee–Yang–Parr hybrid correlation function, which includes both local and nonlocal terms [60–62]. The ground state and all excited states were multiplicities of one. In the TDB3LYP calculations, the core electrons were frozen and the inner shells were excluded from the correlation calculations. The range of molecular orbitals for correlation is from orbital 49 to orbital 720, orbital 65 to orbital 960, orbital 81 to orbital 1200, and orbital 97 to orbital 1440, individually, for the (n, n) with n = 3–6 in all the calculations. The wave functions and energy eigenvalues of the excited states were determined by solving the time-dependent Kohn–Sham equation [58]. The SCF (self consistent field) convergence criterion of the RMS density matrix and the maximum density matrix is set at
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10−8 and 10−6, respectively, in the excited-state calculations. The iterations of excited states are continued until the changes on energies of states are no more than 10−7 a.u. between the iterations, and the convergence has been obtained in all the calculations of excited states. 6.2.1.3 Calculations of Third-Order Polarizabilities The expression of third-order polarizability γ is obtained by the application of timedependent perturbation theory to the interaction between electromagnetic field and microscopic system. Straightforward application of standard quantum mechanical time-dependent perturbation theory, however, leads to unphysical secular divergences in γabcd(−ωp; ω1, ω2, ω3) when any subset of the frequencies ω1, ω2, and ω3 sums to zero. Fortunately, the divergences are eliminated by employing damping factor iΓ [63–65], as described in the following: rabcd ( −ω p ; ω1 , ω 2 , −ω 3 ) = (2π/h)3 k( −ω p ; ω1 , ω 2 , −ω 3 )e 4 ⎧ ⎪ ⎪ ⎪ ×⎨ ⎪ ⎪ ⎪ ⎩
⎡ ʹ ⎤⎫ < o|ra|k >< k|rb*| j >< j|rc*|i >< i|rd|o > ⎢ ⎥⎪ ⎢ ( ω ko − ω p − iΓ ko )(ω jo − ω 1 − ω 2 − iΓ jo )(ω io − ω 1 − iΓ io ) ⎥ ⎪ ⎣ i, j,k ⎦⎪ ⎬ ⎡ ʹ ⎤⎪ < o|ra|j >< j|rb|o >< o|rc|k >< k|rd|o > ⎢ ⎥⎪ − ⎢ ( ω jo − ω p − iΓ jo )(ω jo − ω 1 − iΓ jo )(ω ko + ω 2 + iΓ ko ) ⎥ ⎪ p ⎣ j,k ⎦⎭
∑∑ p
(6.1)
∑∑
Here, o ra k is an electronic transition moment along the a axis of a Cartesian system, between the reference state o and excited state, k ; k rb* j denotes the dipole difference operator being equal to k rb j − o rb o δ kj ; ħωko is the energy difference between state k and reference state o. In this study, the transition moments and dipole moments will be obtained from the calculated results based on the TDB3LYP/6-31G*. ω1, ω2, and ω3 are the frequencies of the perturbating radiation fields; ωp = ω1 + ω2 + ω3 is the polarization response frequency. Σp indicates an average overall permutation of ωp, ω1, ω2, and ω3 along with associated indices a, b, c, and d; Σ′ indicates a sum over all states but reference state o. The factor K(−ωp; ω1, ω2, ω3) accounts for distinguishable permutations of the input frequencies and its value is given by 2−mD, where m is the number of nonzero input frequencies minus the number of nonzero output frequencies and D is the number of distinguishable orderings of the set {ω1, ω2, ω3}. For example, the value of K(−3ω; ω, ω, ω) = 1/4 for third harmonic generation (THG), K(−2ω; 0, ω, ω) = 6/4 for electric field-induced second harmonic generation (EFISHG), and K(−ω; ω, ω, −ω) = 3/4 for degenerate fourwave mixing (DFWM). Accordingly, when input and output frequencies are all zero, that is, static case, K(0; 0, 0, 0) = 1 for the THG, EFISHG, and DFWM optical processes. In practical calculations, if ω1, ω2, and ω3 (as well as their arbitrary linear combinations) can be chosen to be away from a resonant frequency, all the damping factors iΓ can be neglected. In this case, although the damping factors are not included in this equation, the resonant divergences can be avoided and the nonresonant third-order polarizability tensor γ can be calculated by SOS method. Throughout this chapter, the symbols γ(3ω), γ(2ω), and γ(ω) represent the third-order polarizability of THG γ(−3ω; ω, ω, ω), EFISHG γ(−2ω; 0, ω, ω), and DFWM γ(−ω; ω, ω, −ω), respectively. The prefactor K(−ωp; ω1, ω2, −ω3) is the relative magnitudes of the reference state nonlinear polarizability for each optical process at nonzero frequency. In the following calculations, we use the same prefactor K
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in order to make the remark to justify plotting curves for the nonlinear polarizabilities of three optical processes against common axes. From the SOS formula (1), we note that γ is dependent on the transition dipole moments and the transition energies between excited state–state and is sensitive to the treatment of electron–electron interactions (EEIs) or electron correlation. To obtain a reliable value of γabcd, the accurate values of transition energies and dipole moments must be used in formula (1). In the content of the ab initio or semiempirical HF (hartree fock) model, the SCF state functions at least single- and double-excitation configuration interactions (CISD) must be included to treat EEIs and obtain the correct γ value [65]. However, the treatments of electron correlation and exchange interactions are included in a natural way in the TDDFT calculations [66]. So, the TDDFT method is employed to perform the excited state and γ calculations in this chapter. The electric dipole moment <Ψg|r|Ψg> of ground state and the transition dipole moments <Ψg|r|Ψm> from the ground state to excited states can be directly obtained from the outputs of TDDFT calculations. The transition dipole moments <Ψm|r|Ψn> between different excited states are obtained from the off-diagonal electric dipole matrix elements while the electric dipole moments of the excited states are from the diagonal matrix element. Then, the obtained electric dipole moments, transition dipole moments, and the transition energies are taken as the input values of SOS formula in order to get the third-order polarizabilities. An average <γ> of the considered species is obtained from the following expression: < γ > = 1/5( γ xxxx + γ yyyy + γ zzzz + γ xxyy + γ xxzz + γ yyxx + γ yyzz + γ zzxx + γ zzzyy )
(6.2)
Here, our calculations of γ are concerned only with contributions from electric dipole transitions because they are the most intense as compared with vibrational and rotational transitions [67]. In the following discussions, we only give systematic comparisons of third-order NLO properties among the studied species and omit vibration and rotation contributions to hyperpolarizabilities. We have made code based on SOS formula (1) and successfully applied to hyperpolarizability calculations of isolate systems [68–71]. 6.2.1.4 Calculations of TPA Cross Section TPA cross section δ(ω) can be directly related to the imaginary part of the second hyperpolarizability γ (−ω; ω, ω, −ω) by [72]. δ(ω) = (32π 4 /n2λ 2 )L4 Im[γ ( −ω ; ω , ω , −ω)].
(6.3)
Here λ is the wavelength n denotes the refractive index of the medium L corresponds to the local-field factor In the calculations presented here, n and L are set to 1 for vacuum. The unit of δ(ω) will be cm4 s photon−1, if cgs units are used for ħ and λ, and esu unit is used for γ(−ω; ω, ω, −ω). Noted here γ(esu) = γ (cm6 erg−1). We calculate Im γ(−ω; ω, ω, −ω) using the SOS formula (1), in which the formula can be divided into the two terms of summations, the first summation involves the two-photon allowed states and the second summation involves one-photon
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allowed states. Accordingly, the third-order polarizability of two-photon resonance enhancement in DFWMs leaves only behind the first term and can be written as γ ( −ω ; ω , ω , −ω) =
3 1 ⎛ 2π ⎞ 4 ʹ (ω jo − 2ω − iΓ jo )−1 ⎜⎝ ⎟⎠ e 4 h j
∑
⎡ ʹ ⎢ (< o|ra|k >< k|rb*|j > + < o|rb|k >< k|ra*|j >)/(ω ko − ω) ⎢ ⎣ k ⎤ ʹ ( < j|rc*|i > + < j|rd*|i >)/(ω io − ω)⎥ × ⎥ i ⎦
∑
∑
(6.4)
Then, we calculate Im γ(−ω; ω, ω, −ω) by formula (6.4). To compare the calculated TPA cross-sectional value with the experimental value measured in solution, the orientationally averaged (isotropic) value of γ is evaluated, which is defined as =
1 15
∑ (Im γ
iijj
+ Im γ ijij + Im γ ijji ), i, j = x , y , z
(6.5)
i, j
While is taken into Equation 6.3, the TPA cross section δ is obtained. Generally, the position and relative strength of the two-photon resonance are to be predicted using the following simplified form of the SOS expression [73,74]. When only one excited state mainly contributes to all responses, the simplified form would be δ ∝ ( Mok2 Δμ 2on )/((E0 k /2)2 Γ )
(6.6)
If two excited states mainly contribute to all response, the simplified form would be 2 δ ∝ ( Mok2 Mkn )/((Eok − E0 n /2)2 Γ )
(6.7)
where Mij is the transition dipole moment from the state i to j Eij is the corresponding excitation energy E0n = 2ħω, the subscripts 0, k, and n refer to the ground state S0, the intermediate state Sk, and the TPA final state Sn, respectively Γ is the damping factor of excited state Δμ0n is the dipole moment difference between S0 and Sn In this chapter, all damping factors Γ are set to 0.01 eV; this choice of damping factor is found to be reasonable on the basis of the comparison between the theoretically calculated and experimental TPA spectra. We have made code based on formulas (6.3) and (6.4) and successfully applied to simulations of TPA cross section [28,75,76]. 6.2.2 Descriptions and Understandings of Results on Finite Open (n, n) SWCNTs 6.2.2.1 Geometries and Electronic Structures at Ground State Figure 6.1 shows the optimized geometrical structures of finite open SWCNTs of (n, n) with n = 3–6. It is found that their configurations are armchair, and that the (n, n) tube is separately consisted of 48, 64, 80, and 96 carbon atoms for n = 3–6, respectively. For the
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
22
35 32
31
29 38
37
71
69
11
9
24
77
64
57
44
17
76
68
67
79 74
25 26 3
1
4
5
73
65 66
56
48
59
47 54 53
26
27
50
28
24
25 29
49 53
23
30
7
21
8
9
20
19
46
45 33
34
11
42
17
12
13
16
41 37
14
15
FIGURE 6.1 (continued) (c) (5, 5), and (d) (6, 6).
49
1
18
2
(d)
51
20
50
33
18
41
58
27
10
61
43
60
70
39
22
52
38
48
47 31
32
44
43 35
36 40 39
54 70
69 57
58 66
3
12
78
28
6
63
80
2
15
42
55
30
36
34
62
72 21
40
(c)
75
23
45
8
7
19 14 13
51
74
52
73
71 55
56
68
67 59
4
16
46
72
10
77
5 6 75 76 96
78
95
94
79
93 81
80
92
82
91
90
83
65 61
60
62
64
89 85
63
86
84 88 87
225
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TABLE 6.1 Selected Bond Lengths (Å) and Angles (°) of Finite Open SWCNs (n, n) (3, 3) C1–C2 C5–C6 C1–C12 C2–C3 C3–C4 C3–C14 C13–C14 C14–C15 C15–C16 C15–C26
1.241 1.241 1.412 1.412 1.450 1.446 1.469 1.437 1.411 1.456
(4, 4)
C1–C2–C3 C2–C3–C4 C2–C3–C14 C4–C3–C14 C3–C14–C13 C3–C14–C15 C13–C14–C15 C14–C15–C16 C14–C15–C26 C16–C15–C26
124.5 20.9 109.2 116.4 118.3 117.4 114.8 117.5 119.3 115.7
C1–C2 C5–C6 C2–C3 C4–C5 C3–C4 C3–C18 C17–C18 C18–C19 C19–C20 C19–C34
1.238 1.238 1.413 1.413 1.427 1.447 1.453 1.431 1.415 1.445
(5, 5) C2–C3 C5–C6 C2–C15 C6–C13 C11–C20 C11–C50 C12–C41 C41–C52 C51–C60 C60–C63
1.238 1.238 1.412 1.412 1.419 1.447 1.447 1.429 1.416 1.440
C1–C2–C3 C2–C3–C4 C2–C3–C18 C4–C3–C18 C3–C18–C17 C3–C18–C19 C19–C18–C17 C18–C19–C20 C18–C19–C34 C20–C19–C34
125.9 124.6 109.5 118.1 119.9 117.9 117.0 118.8 119.5 117.6
(6, 6)
C2–C3–C20 C5–C6–C13 C2–C15–C42 C6–C13–C46 C11–C20–C43 C11–C50–C51 C12–C41–C52 C41–C52–C61 C51–C60–C63 C60–C63–C62
126.5 126.7 109.7 109.7 118.9 118.2 118.2 119.6 118.4 118.4
C1–C2 C5–C6 C2–C3 C4–C5 C3–C4 C3–C26 C25–C26 C26–C27 C27–C28 C27–C50
1.237 1.237 1.411 1.411 1.416 1.447 1.444 1.428 1.416 1.438
C1–C2–C3 C2–C3–C4 C2–C3–C26 C4–C3–C26 C3–C26–C25 C3–C26–C27 C25–C26–C27 C26–C27–C28 C26–C27–C50 C28–C27–C50
127.1 127.5 109.8 119.4 121.1 118.3 118.5 119.8 119.6 118.9
and C–C–C angles in the nanotube wall range from 116° to 119° in the (3, 3) tube with diameter of 4.140 Å; however, the C–C lengths are more average and C–C–C angles are more close to 120° (toward planar structure of sp2 hybridization) in the (6, 6) tube with diameter of 8.408 Å (lengths ranging from 1.411 to 1.447 Å and angles ranging from 119° to 121° in the tube wall). Table 6.1 lists the selected bond lengths and angles for the finite SWCNTs (n, n) with n = 3–6. The bond lengths and angles of the relaxed structures, especially at the tube ends, are substantially different from those of an ideal rolling of a grapheme sheet. The calculated electronic populations based on TDB3LYP/6-31G* level at ground state show that there are four types of charge distributions in the SWCNTs, as listed in Table 6.2. The atoms localized in SWCNT ends accept the electrons and have negative charges. The atoms directly connected to end atoms donate electrons and have positive charges. The other TABLE 6.2 Calculated Charge Distributions (e) and Gap (eV) of Finite Open SWCNs (n, n) Type
C-End
C-Second
C-Third
C-Fourth
LUMO–HOMO Gap
(3, 3) (4, 4) (5, 5) (6, 6)
−0.025 −0.033 −0.040 −0.046
0.014 0.022 0.030 0.036
0.007 0.009 0.010 0.012
0.004 0.001 0.000 −0.001
2.0726 1.7545 1.5825 1.4672
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
227
two typical charge distributions show that the atoms more toward the middle wall of tube are more close to electronic neutral. Here we note that the Mulliken population strongly depends on basis sets which are employed in the calculations, and only systematic comparisons are made to give a variation trend among our studied species. The calculated gap energies corresponding to the energy differences of the HOMOs (highest occupied molecular orbitals) and LUMOs (lowest unoccupied molecular orbitals) based on TDB3LYP/6–31G* level are listed in Table 6.2 for (n, n) tubes. The results indicate that the smaller the diameter, the larger the LUMO–HOMO gap in open SWCNTs, that is, the energy gap decreases as the length of chiral vector increases in finite open SWCNTs of (n, n) type. 6.2.2.2 Electronic Absorption Spectrum of Ground State The calculated electronic absorption spectra of the finite open SWCNTs of (n, n) with n = 3–6 are shown in Figure 6.2. Comparing the absorption spectra among these tubes, we find that the spectra of the (3, 3) and (5, 5) tubes give sharp absorption peaks and the spectra of (4, 4) and (6, 6) tubes make broad absorption bands at the low-energy region. Our calculations predict a variety of the spectral shapes and peak positions of these four tubes. It is found from Figure 6.2 that the spectra of these tubes exhibit long absorption tails, which extend deep into the region of the lower transition energies. The strong absorption peak of Sa is localized at about 558, 581, 593, and 603 nm for (n, n) tubes with n = 3–6, 1.0
4, 4 6, 6
0.8
Normalized absorption spectra (a.u.)
0.6 0.4 0.2 0.0 1.0
3, 3 5, 5
0.8 0.6 0.4 0.2 0.0 400
500
600
700
800
Wavelength (nm)
900
1000
1100
FIGURE 6.2 Normalized electronic absorption spectra of finite open SWCNs of (3, 3), (4, 4), (5, 5), and (6, 6).
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TABLE 6.3 Transition Moments, Energies, and States Contributing to Absorption Spectra Type (3, 3) sa sw (4, 4) sa sw (5, 5) sa sw (6, 6) sa sw
x (a.u.)
y (a.u.)
z (a.u.)
Energy (eV)
Configuration Components
0.0000 0.1421 0.0000 −1.0266 −0.0001 −0.3738 0.0000 0.0000
−0.0001 −0.4864 0.0000 0.0433 0.0008 −1.4801 0.0000 −1.3641
−2.6218 −0.0001 −2.6539 0.0000 −2.5593 −0.0003 −2.4489 0.0000
2.2205 2.7519 2.1339 3.0283 2.0875 2.3047 2.0562 2.3047
0.3723MO143->145 + 0.4505MO144->148 −0.1225MO137->142 + 0.6698MO144->153 0.4720MO191->193 + 0.3696MO192->196 0.5373MO188->193 − 0.2880MO192->204 0.5295MO239->241 + 0.3201MO240->242 −0.3743MO237->241 + 0.4700MO240->254 0.5669MO287->289 + 0.2844MO288->292 0.5848MO285->289 – 0.3076MO288->304
respectively, and the absorption edge is redshift as the length of chiral vector increases among the finite open SWCNTs of (n, n). From the transition moments in the axial (z) and radial (x, y) directions listed in Table 6.3, we can find that the absorptions of Sa are mostly originated from axial allowed electron transitions from the ground state to excited states S15, S12, S14, and S11 for the (n, n) tubes with n = 3–6, respectively. The weak absorptions of Sw are mostly originated from radical allowed electron transitions from the ground state to excited states for (n, n) tubes, respectively. An analysis in terms of the results calculated by the TDB3LYP/6-31G* level shows that the absorption peaks of the largest wavelength (Sa) are mainly contributions from the charge transfers from π-bonding to π*-antibonding orbitals. Table 6.3 lists the excitation states mostly contributing to Sa and Sw absorption bands, and their configuration components. The molecular orbitals contributing to these configuration components are mostly originated from pπ orbitals of the (n, n) SWCNTs. Figure 6.3 give the plots of the HOMO (MO144) and LUMO+3 (MO148) for the (3, 3) tube, and the HOMO–1 (MO287) and LUMO (MO289) for the (6, 6) tube as they represent pictures in the (n, n) tubes. They describe the pictures of π–π interactions. In view of the configuration structures and the orbital localized at deep energy-level positions, the weak absorptions of Sw are also assigned as the electron transitions between the σ–π hybridized orbitals. Accordingly, we can conclude that a strong absorption band occurs at a low-energy region for the case of light polarized parallel to the tube axis, and a weak absorption appears at a high-energy region for the case of light polarized perpendicular to the tube axis; and the
(a)
(b)
(c)
(d)
FIGURE 6.3 Molecular orbitals involved in charge-transfer processes for electronic absorption of SWCNs. (a) HOMO and (b) LUMO+3 for (3, 3) tube; (c) HOMO–1; and (d) LUMO+1 for (6, 6) tube.
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
231
FIGURE 6.6 Molecular orbitals mostly contributing to third-order polarizability of SWCNs. (a) HOMO–2 (MO142) and (b) LUMO+7 (MO152) for (3, 3) tube; (c) HOMO–1 (MO191) and (d) LUMO+9 (MO202) for (4, 4) tube; (e) HOMO–1 (MO239) and (f) LUMO+13 (MO253) for (5, 5) tube; (g) HOMO (MO288); and (h) LUMO+17 (MO305) for (6, 6) CN tube.
The largest δ(ω) value is localized at about 750–820 nm, and the δ(ω) value increases as n is on the order of 3 < 4 < 5 < 6. This implies that the largest TPA cross sections increase with the diameter of tube or chiral vector length under constant tube length of finite open SWCNs, as shown in Table 6.5. Furthermore, we will search the originations of a large TPA cross section δ(ω) in the finite open SWCNs. From Equation 6.3, it is found that the δ(ω) value only depends on the imaginary part of the second hyperpolarizability, Im γ(−ω; ω, ω, −ω), while the input frequency is given at the vacuum medium. Accordingly, the two-photon states contributing to Im γ(−ω; ω, ω, −ω) shown in Equation 6.3 also contribute to δ(ω) value. The calculations of Im γ(−ω; ω, ω, −ω) values, that is, the calculations of TPA cross section of δ(ω) vs. two-photon states are made at the characteristic wavelength (resonant wavelength). It is shown that the TPA cross sections are mostly contributed from one two-photon state. State 43 of (3, 3), state 20 of (4, 4), state 48 of (5, 5), and state 55 of (6, 6) tubes, which are the TPA resonant states, have the contributions of 83%, 70%, 100%, and 100% to δ(ω) values at wavelength of about from 750 to 820 nm, respectively. Table 6.6 lists the largest configuration components of these TABLE 6.5 Calculated TPA Properties Tube
D (Å)
L (Å)
λm (nm)
Im <γ>
δ(ω)
(3, 3) (4, 4) (5, 5) (6, 6)
4.214 5.665 7.018 8.408
8.441 8.438 8.435 8.432
752 835 841 816
7452 10345 16998 24122
433 488 791 1192
Note: Unit: 10−36 esu for γ; 1 GM = 10−50 cm4 s photon−1 for δ.
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TABLE 6.6 Two-Photon State Construction and Contribution to δ Tube
State
Largest Configuration
(3, 3) (4, 4) (5, 5) (6, 6)
43 20 48 55
0.4380MO140->147 − 0.4258MO142->146 0.5864MO190->193 + 0.3018MO192->201 0.6710MO239->253 + 0.1312MO233->242 0.4583MO287->302 − 0.3563MO285->298
Contributions (%) 83 70 100 100
TPA resonant states. The configuration interaction functions are formed by the occupied and unoccupied molecular orbitals. For example, the TPA resonant state of the (6, 6) tube has the greatest contributions from the configuration of MO287 -> MO302 and configuration of MO285 -> MO298, respectively, where the MO285 and MO287 are occupied molecular orbitals and the MO298 and MO302 are unoccupied molecular orbitals. These orbitals are mostly constructed by the C–p orbitals. The other (n, n) tubes have the same situations, that is, the resonant state is mostly the contribution from p-charge-transfer excited states. Accordingly, enhancement of TPA cross section δ(ω) ascribes the dispersion effect of tube radius of curvature and large p-electron delocalizations in (n, n) tubes with large radius. Additionally, for three-state model of TPA process, the TPA state can be constructed by one-electron promotion from ground state to excited state i (onephoton state) by absorbing one-photon energy, and the electron promotion from state i to state j (two-photon state) by absorbing again one-photon energy. Table 6.7 lists the states involved in TPA processes based on three-state model shown in formula (6.7). From this formula, we can deduce two factors: (i) δ(ω) value increases with the moment products (Mok 2 M kn2) and (ii) δ(ω) value increases as the one-photon detuning term (E 0k − E 0n/2)2 decreases. Table 6.7 also lists the transition moments from ground state to one-photon states and from one-photon states to two-photon states, together with transition energies and incident photon energies. Comparing the listing data among them, we find that the three-state model is simply approximate description of two-photon resonant transition processes. TABLE 6.7 Transition Moment (a.u.) and Energy (eV) Tube
->State
(3, 3)
G->8 8->43 G->43 G->4 4->20 G->20 G->4 4->48 G->48 G->4 4->55 G->55
(4, 4)
(5, 5)
(6, 6)
Moment
Energy
ħω (eV)
0.2932 0.5685
1.6514
1.676
3.2128 1.4827
1.485
2.5971 1.4892
1.474
2.9465 1.4965
1.520
0.3018 3.9372 0.6736 1.0130 1.5144 0.4224
3.0440
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
233
6.3 Silicon-Carbide and Carbon-Nitride Nanotubes 6.3.1 Theoretical Method and Computational Details For zigzag (n, 0), armchair (n, n), and chiral (n, m) SiCNTs, the coordinates of all the atoms are optimized without any symmetry constraint using Broyden–Fletcher–Goldfarb–Shanno (BFGS) scheme [77]. Ultrasoft pseudopotentials [78,79], local-density approximation (LDA) functional [80], and the total energy and properties are calculated within the framework of the Perdew–Burke–Ernzerhof generalized gradient approximation (GGA-PBE). The interactions between the ionic cores and the electrons are described by the norm-conserving pseudopotentials. All the above calculations are carried out by using the total energy code CASTEP [81,82], which employs pseudopotentials to describe electron–ion interactions and represents electronic wave functions using a plane-wave basis set. The CNNTs are rolled up from corresponding g–CN (graphite-like (g–)CN). The geometrical structures of zigzag and armchair CNNTs, namely, (n, 0) CNNTs with n = 6, 9, 12, 15 and (n, n) CNNTs with n = 6, 9, 12 are optimized by GGA-PBE functional and double numerical basis sets with polarization function (DNP) in DMol3 code [83]. The calculations of electronic and optical properties were carried out by GGA-PBE functional and norm-conserving pseudopotentials in CASTEP code. The orbital electrons of C–2s22p2, Si–3s23p2, and N–2s22p2 are treated as valence electrons. We have used approximately 10 Å vacuum in the lateral directions to avoid artificial tube–tube interaction. This should be a sufficient distance since the basis functions do not overlap. The supercells of the armchair and zigzag SiCNTs contain four layers of atoms along the tube axis (z axis). The CNNTs are settled in an orthogonal unit cell, and the tube axis of CNNTs is along the c-direction of the unit cell. The units are all periodic in the direction of the tube axis. Considering the balance of the computational cost and precision, we choose a cut-off energy of 600, 700 eV, and the Monkhorst–Pack grid is used for the Brillouin zone sampling [84] for SiCNTs and CNNTs, respectively. The linear response of the system to an external electromagnetic field with a small wave vector is measured through the complex dielectric function ε(ω). ε(ω) is connected with the interaction of photons with electrons. The real part and imaginary part of ε(ω) are often referred to as ε1(ω) and ε2(ω), respectively. ε2(ω) can be thought of as detailing the real transitions between occupied and unoccupied electronic states. The imaginary part ε2(ω) of the dielectric function ε(ω) is given by the following equation: εij2 (ω) =
i pcv (k )pvcj (k ) 8π2 2e 2 Σ Σ ( f − f ) δ[Ecv (k ) − ω] k cv c v 2 m2Veff Evc
(6.8)
where Ecv(k) = Ec(k) – Ev(k). Here, fc and fv represent the Fermi distribution functions of the i (k ) denotes the momentum matrix element tranconduction and valence band. The term pcv sition from the energy level c of the conduction band to the level v of the valence band at the kth point in the BZ and Veff is the effective unit cell volume. The real part ε1(ω) of the dielectric function ε(ω) follows from the Kramer–Kronig relationship. All the other optical constants may be derived from ε1(ω) and ε2(ω) [85,86]. For example, the loss function L(ω) can be calculated using the following expression: L(ω) = Im( −1/ε(ω ))
(6.9)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
(a)
(b)
(c)
FIGURE 6.7 Optimized cells: (a) for (12, 0), (b) for (6, 6), and (c) for (6, 3) SiCNTs. Gray and black balls represent carbon and silicon atoms, respectively. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
The loss function L(ω) is an important optical parameter describing the energy loss of a fast electron traversing in the material. The peaks represent the characteristic associated with the plasma oscillation, and the corresponding frequencies are the so-called plasma frequencies [87]. 6.3.2 Optimized Geometry Structures of SiCNTs and CNNTs The equilibrium configurations are obtained for SiCNTs with different diameters and chiralities. The Si and C atoms are placed alternatively without any adjacent Si or C atoms in the optimized configurations of SiCNTs. For example, the optimized cells of zigzag (12, 0), armchair (6, 6), and chiral (6, 3) SiCNTs are shown in Figure 6.7. After the relaxation, Si atoms move toward the tube axis and C atoms move in the opposite direction. The calculated average Si–C bond length of these tubes is about 1.775 Å. The optimized structures of the zigzag and armchair CNNTs are obtained and the top view of their structures is shown in Figure 6.8. From Figure 6.8, it is shown that the armchair CNNTs with the chiral index (n, n) appear as a polygon with (2/3)n edges since it requires (2/3)n triazine units to wrap the tube, and the (6, 6), (9, 9), and (12, 12) CNNTs appear like square, hexagon, and octagon, respectively; however, all (n, 0) CNNTs appear like circles. There are three types of C–N bonds and two types of C–C bonds in (n, 0) CNNTs. The bonds parallel to the tube axis are named as vertical (v) bonds and other bonds are called as tilt (t) bonds. The tC–N bonds can be in two different positions. The bonds near to tC–C and vC–C are named as tC–N and t′C–N bonds, respectively. So, they have vC–N, tC–N, t′C–N, vC–C, and tC–C bonds. In the (n, n) CNNTs, the bonds perpendicular to the tube axis are named as horizontal (h) bonds and other bonds as tilt (t) bonds. Similarly, they have hC–N, tC–N, t′C–N, hC–C, and tC–C bonds. Figure 6.9 plots the average bond lengths of optimized CNNTs as a function of the tube index. The vC–N bond length increases while the tC–N and t′C–N bond length decreases with the increase of tube index in the (n, 0) CNNTs. The tC–N and t′C–N bond length increases while the hC–N bond length decreases with the increase of tube index in the (n, n) CNNTs. The behavior results from the different σ or π characters in the σ–π hybridization effect. However, all the C–C bond lengths decrease with the increase of tube diameter. 6.3.3 Periodic Density Functional Theory Studies on Band Structure and Density of States [88,89] 6.3.3.1 SiCNTs Table 6.8 lists the calculated band gaps of SiCNTs, and it shows that the zigzag SiCNTs are direct-gap semiconductors, while the armchair and chiral tubes are indirect-gap ones. For the zigzag SiCNTs, the band gap increases with the tube diameter (n value), which is due
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Computational Nanotechnology: Modeling and Applications with MATLAB®
TABLE 6.8 Band Gaps and Static Dielectric Functions for SiCNTs SiC Tube
Band Gap
εx(0)
εz(0)
(8, 0) (9, 0) (10, 0) (11, 0) (12, 0) (13, 0) (14, 0) (3, 3) (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (10, 10) (6, 3)
1.299(d) 1.514(d) 1.713(d) 1.763(d) 1.860(d) 1.980(d) 2.001(d) 2.145(ind) 1.897(ind) 2.192(ind) 2.168(ind) 2.301(ind) 2.296(ind) 2.373(ind) 2.365(ind) 1.824(ind)
3.836283 3.711828 3.592737 3.468935 3.358378 3.266671 3.159909 4.340275 4.072665 3.813181 3.525262 3.310900 3.132527 2.936466 2.777178 3.859515
6.663484 6.083632 6.025104 5.745599 5.517879 5.313320 5.089013 6.440032 6.079181 5.755652 5.395955 5.064952 4.747626 4.483205 4.229498 6.083363
Note: The “d” and “ind” in parentheses denote direct and indirect band-gap semiconductors, respectively.
to smaller curvature leading to smaller π–σ hybridizations and larger repulsions between π and π* states. However, for the armchair ones, the band gap increases with the odd or even n. For a given n, the (n, n) tube has a bigger band gap than the (n, 0) tube because it has a larger radius. The ratio between the (n, n) and (n, 0) tube radius is 31/2, independent of n. The orbitals localized at G (0, 0, 0) point of the topmost valence band (HOMO) and lowest conduction band (LUMO) of (12, 0), (6, 6), and (6, 3) tubes as the representatives of zigzag, armchair, and chiral tubes are plotted in Figure 6.10a and b. It is found that the HOMO is constructed by anti-phase p-orbital interactions between the two neighboring carbon atoms along the tube radius. The smaller the n value, the larger the curvature and the stronger is the anti-phase interactions of the SiCNTs. Accordingly, the HOMO energy decreases as the SiCNT radius increases. The LUMO is formed by nonbonding or weak antibonding interactions between Si atoms. Figure 6.11a and b give the plots of the energies of the top of the highest occupied valence band (HOVB) and the bottom of the lowest unoccupied conduction band (LUCB) against the n values from the CASTEP calculations. It is shown that the energy of the top of HOVB decreases as the SiCNT radius increases, whereas the variations of tube radius, except for the smallest n value, have not substantial effects on the energy of the bottom of LUCB, in particular for armchair SiCNTs. These results tell us why the band gap between the topmost valence band and the lowest conduction band increases with the SiCNT radius. The band structures, total density of states (DOS), and partial DOS (PDOS) projected on the constitutional atoms for the zigzag (12, 0), the armchair (6, 6), and the chiral (6, 3) tubes are plotted in Figures 6.12 and 6.13. Here, we must note that the energy of the top of the HOVB is taken as a reference and always set to zero in drawing band structures. The band structures of (12, 0) and (6, 6) tubes are more oscillating than that of the (6, 3) tube, which indicates that the states of (6, 3) tube are more localized. For (12, 0) and (6, 6) tubes, the degeneracy of energy levels at Z (0, 0, 0.5) point is higher than that of energy levels at G (0, 0, 0) point. The DOS and PDOS are similar for all of the SiCNTs being studied.
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
(a)
(12, 0)
(6, 6)
(6, 3)
(b)
(12, 0)
(6, 6)
(6, 3)
237
–2.1
–1.5
–2.4
–2.0
–2.7 –3.0
Energy (eV)
Energy (eV)
FIGURE 6.10 (a) The orbitals of the topmost valence band (HOMO) and (b) the orbitals of the lowest conduction band (LUMO) both localized at G (0, 0, 0) point for (12, 0), (6, 6), and (6, 3) tubes. (The absolute values of the isosurfaces of the wavefunctions are 0.03.) (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
The top of HOVB (at G point) The bottom of LUCB (at G point)
–3.3 –3.6
(a)
–3.0
The top of HOVB The bottom of LUCB
–3.5 –4.0
–3.9 –4.2
–2.5
8
9
10
11 n
12
13
14
–4.5
(b)
3
4
5
6
n
7
8
9
10
FIGURE 6.11 The variations of the energies of the top of HOVB and the bottom of LUCB as a function of n for (a) zigzag and (b) armchair SiCNTs. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
The regions below the Fermi level (the Fermi level is set at the top of the valence band) can be divided into two regions. The bottom valence band regions are mainly composed of C-2s states, with a nonnegligible contribution from Si-3s,3p states. The top of the valence band region mainly arises from C-2p states with a small mixing of Si-3s,3p states. The conduction bands above the Fermi level are mainly due to Si-3p states. 6.3.3.2 CNNTs Figure 6.14a and b show the band structures of the (9, 9) and (9, 0) CNNTs as representation of the studied CNNTs. The zigzag and armchair CNNTs are direct-gap semiconductors.
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4
3
3
Energy (eV)
2
Energy (eV)
4
(12, 0)
1 0 –1 –2
(a)
2
(6, 6)
1 0
–1 G F
Q Z
G
–2 (b)
G
F
Q Z
G
3 Energy (eV)
2 (6, 3)
1 0 –1 –2
(c)
G
F
Q
Z
G
C-2s
C-2p
Si-3s
Si-3p
(12, 0)
Total
–10
–5 Energy (eV) 100 80 60 40 20 0 40 30 20 10 0 120 90 60 30 0 –15
DOS (electrons/eV)
(a)
40 30 20 10 0 45 30 15 0 51 34 17 0 –15
DOS (electrons/eV)
DOS (electrons/eV)
FIGURE 6.12 SiCNT band structures of (a) zigzag (12, 0), (b) armchair (6, 6), and (c) chiral (6, 3) types. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
(c)
5
0
(b)
50 40 30 20 10 0 33 22 11 80 64 48 32 16 0 –15
C-2s
C-2p
Si-3s
Si-3p
Total
–10
Si-3s
–5 Energy (eV)
0
5
(6, 3)
C-2p
C-2s
(6, 6)
Si-3p
Total
–10
–5 Energy (eV)
0
5
FIGURE 6.13 SiCNT density of states of (a) zigzag (12, 0), (b) armchair (6, 6), and (c) chiral (6, 3) types. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
3
3
2
2 Energy (eV)
Energy (eV)
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
1
0
–1 (a)
239
1
0
G
F
Q
Z
G
–1 (b)
GF
QZ
G
FIGURE 6.14 Calculated band structures (a) for (9, 0) and (b) for (9, 9) CNNTs.
The bottom of the conduction band energy and the top of the valence band energy are located at the G point and Q point for zigzag and armchair CNNTs, respectively. The calculated band gaps are from 1.819 to 1.998 eV or from 1.675 to 1.942 eV for armchair or zigzag CNNTs, respectively, as shown in Table 6.9. For the armchair CNNTs, the band gap decreases as the tube size increases and converges to that of the g-CN (1.683 eV) when the diameter of the tube becomes very large. However, with the increase of the tube size, the band gap increases firstly (the (12, 0) CNNT gets the largest band gap of 1.942 eV) and then the band gap decreases and converges to that of the g-CN (1.683 eV) for the zigzag CNNTs. Figure 6.15a and b show the DOS and PDOS of the (9, 9) and (9, 0) CNNTs. For both armchair and zigzag CNNTs, the top of valence band originates from the N-2p states with a small mixing of C-2p states. The bottom of conduction band originates from the C-2p and N-2p antibonding states. Table 6.9 lists the calculated relative energies per CN unit of the optimized CNNTs (the total energy of g-CN is set to be 0 eV). The total energies of calculated armchair CNNTs are smaller than those of zigzag CNNTs for the same tube index, and the difference is more obvious for the small tubes, which is due to the stronger curve effect in small size nanotubes. The energies decrease as the size of the tube increases for TABLE 6.9 Relative Energies (per CN Unit) and the Energy Gap for the CNNTs and Graphitic (g-) CN CNNTs
Energy (eV)
Band Gap (eV)
(6, 0) (9, 0) (12, 0) (15, 0) (6, 6) (9, 9) (12, 12) g-CN
0.32 0.127 0.066 0.040 0.084 0.034 0.017 0
1.675 1.926 1.942 1.921 1.998 1.920 1.819 1.683
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DOS (states/eV)
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50 40 30 20 10 0 12 9 6 3 0 30 20 10 0
DOS (states/eV)
(a) 50 40 30 20 10 0 15 12 9 6 3 0 30 20 10 0
(b)
Total
C-s
C-p
N-s
N-p
–20
–15
–10
–5 Energy (eV)
0
5
–10
–5 Energy (eV)
0
5
Total
–20
C-s
C-p
N-s
N-p
–15
FIGURE 6.15 Density of states and partial density of states (a) for (9, 0) and (b) for (9, 9) CNNTs.
both armchair and zigzag CNNTs, and they converge to that of the corresponding g-CN. This finding shows that the stability increases with increasing tube size, since the strain is gradually decreasing. 6.3.4 Simulations of Linear Optical Properties of SiCNTs and CNNTs 6.3.4.1 SiCNTs Table 6.8 lists the zero-frequency dielectric constants εx(0)and εz(0) of SiCNTs. It is noted here that no scissors-operator shift is applied for the calculations of optical properties of SiCNTs. The zero-frequency dielectric constants εx(0) and εz(0) decrease monotonically as the diameter of SiCNTs increases. For a given SiC nanotube, εz(0) is larger than εx(0), that is, the zero-frequency dielectric constant along the tube axis is larger than that perpendicular to the tube axis. It appears that the optical anisotropy becomes smaller as the diameter becomes larger. Figure 6.16 shows the dielectric function and loss function under different polarizations for chiral (6, 3) tube. The peaks under parallel polarization (E ∙ z) are stronger
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
Perpendicular polarization Parallel polarization
Loss function
9
ε2
6
3
0 (a)
0
5
10 Energy (eV)
15
20
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
(b)
241
Perpendicular polarization Parallel polarization
0
5
10 Energy (eV)
15
20
FIGURE 6.16 Imaginary part of dielectric function (a) and loss function (b) under different polarizations for chiral (6, 3) SiCNT. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
than the ones under perpendicular polarization (E ⊥ z), because the optical transition probability for parallel polarization is about half of that for perpendicular polarization. And the ε2 ∙ (ω) is larger than the ε2 ⊥ (ω) at almost all frequencies. The first peak below 5.0 eV of dielectric function and loss function under parallel polarization is redshifted when compared with the one under perpendicular polarization. The prominent peak at ∼12.5 eV of loss function under parallel polarization is slightly blueshifted when compared with the one under perpendicular polarization. The imaginary parts of the dielectric functions of armchair and zigzag SiCNTs are plotted in Figure 6.17 when the electric field of the light is parallel to the tube axis. For these two types of SiCNTs, the spectra can be divided into two regions of the low-energy range from 0 to 5 eV and the high-energy range from 5 to 20 eV, and it shows that the peak at the low-energy region is stronger than the one at the high-energy region. The first peak, located at about 3.1 eV, is mainly due to the electronic transitions from C-2p bonding orbitals to Si-3p nonbonding orbitals (inter-π band transition) for the armchair SiCNTs, and it is slight blueshift (from 3.05 to 3.22 eV) with Parallel polarization
Parallel polarization
(10, 10)
(14, 0) (9, 9)
(13, 0)
(7, 7) (6, 6)
ε2
ε2
(8, 8)
(11, 0)
(5, 5)
(10, 0)
(4, 4)
(9, 0)
(3, 3) (a)
(12, 0)
0
(8, 0) 5
10 Energy (eV)
15
(b)
0
5
Energy (eV)
10
15
FIGURE 6.17 Imaginary parts of the dielectric functions under parallel polarization: (a) armchair SiCNTs and (b) zigzag SiCNTs. (Adapted from Huang, S.-P., et al., Opt. Express, 15, 10947, 2007.)
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
243
frequency becomes small as the tube diameter increases. Under parallel polarization, the first peak located around 5.0 eV is blueshift with the increase of the tube diameter. In the case of perpendicular light polarization, the first peak located around 4.5 eV remains where it is as the size of the tube increases. For zigzag SiCNTs, the prominent peak is suppressed while n increases from 8 to 11; however, it gains strength while n increases from 11 to 14 under parallel polarization. The prominent peak becomes intense while n increases from 8 to 10 and is suppressed as n varies from 10 to 14 under perpendicular polarization. For armchair SiCNTs, the prominent peak becomes intense with n from 4 to 6 and is suppressed with n from 6 to 10 under perpendicular polarization, while in the case of parallel polarization, the prominent one is suppressed with n from 3 to 6, gains strength with n from 6 to 8, and is suppressed again from 8 to 10. The peaks below the energy of 10.0 eV represent the characteristics associated with the oscillation of the plasma of losing π-electrons and correspond to plasma frequency of losing π-electrons. And π + σ plasmons are mainly responsible for the peaks above 10.0 eV. From the simulations of optical properties, we can infer that the peaks localized at low-energy region in the dielectric function are blueshift and the peaks localized at high-energy region in the energy loss function are redshift with the increase of tube diameter for a given chirality of SiCNTs. 6.3.4.2 CNNTs In order to make comparison, we first calculate the dielectric functions of triazine unit and g-CN (shown in Figure 6.19). It is seen from Figure 6.19a that the imaginary parts of the dielectric functions ε2(ω) of triazine unit has a strong peak centered at 5.7 eV which mostly originate from π → π* electronic transition. It also has a broaden adsorption from 12 to 18 eV. We considered two directions for g-CN: the electric field of the light is parallel and perpendicular to the g-CN. For the ε2 ∙ (ω), the strong peak is centered at 4.2 eV, which is 1.5 eV redshift compared with triazine unit, and there is only a weak peak located at around 8 eV. For the ε2 ⊥ (ω), there is a broad and weak peak in the visible right region centered at 2.5 eV shown in Figure 6.19b. Figures 6.20a through d show the calculated imaginary parts of dielectric functions ε2(ω) of the CNNTs, which correspond to electronic absorption spectra. The spectra of CNNTs can be divided into two parts, namely, the low-energy region (0–5 eV) and the high-energy region (5–20 eV). For all the spectra of the CNNTs, the peaks in the low-energy region are stronger than the ones in the high-energy region, and the peaks under parallel polarization are stronger than the ones under perpendicular polarization. 1.0
2.0
Triazine
0.8
1.5 ε2
ε2
0.6 0.4
(a)
1.0 0.5
0.2 0.0
Parallel Perpendicular
0
5
10
15
Energy (eV)
20
0.0
25 (b)
0
5
10
15
20
25
Energy (eV)
FIGURE 6.19 Imaginary parts of the dielectric functions: (a) triazine and (b) parallel and perpendicular polarization for g-CN.
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1.2 1.0 0.8
2.1 1.8
0.6
0.3 0
1.2
5
10 15 Energy (eV)
Perpendicular polarization
1.0
20
25
(b)
0.6
5
10 15 Energy (eV)
Parallel polarization
2.0
20
25
(6, 6) (9, 9) (12, 12)
1.5 1.0
0.4
0.5
0.2
(c)
0
2.5
ε2
ε2
0.0
(6, 6) (9, 9) (12, 12)
0.8
0.0
1.2 0.6
0.2
(a)
1.5 0.9
0.4
0.0
(6, 0) (9, 0) (12, 0) (15, 0)
Parallel polarization
2.4
ε2
ε2
2.7
(6, 0) (9, 0) (12, 0) (15, 0)
Perpendicular polarization
0
5
10 15 Energy (eV)
20
25
0.0 (d)
0
5
10 15 Energy (eV)
20
25
FIGURE 6.20 Imaginary parts of the dielectric functions: (a) perpendicular polarization, (b) parallel polarization for zigzag CNNTs, (c) perpendicular polarization, and (d) parallel polarization for armchair CNNTs.
For zigzag CNNTs, ε2 ⊥ (ω) have one pronounced peak located at about 4.5 eV, which is 1.2 eV redshift compared with triazine unit, and one weak peak located at about 2.5 eV in the low-energy region, which originates from the electronic transitions from N-2p nonbonding states to antibonding states between N-2p and C-2p orbitals, and assigned as n–π* electronic transitions. The difference of the intensity is due to the structures of the molecular orbitals of CNNTs. For the π–π* electronic transitions, the π and π* orbitals are all perpendicular to the tube surface. For the n–π* electronic transitions, the n orbitals are parallel while the π* orbitals are perpendicular to the tube surface. In the highenergy region, there is a broadened absorption from 8 to 15 eV which is responsible for the electronic transitions between inner nonbonding p states to antibonding π* states. For the ε2 ∙ (ω) of zigzag CNNTs, the absorption peak located at about 4.5 eV is stronger while the peak located at about 2.5 eV is weaker than that of the ε2 ⊥(ω) due to optical anisotropy. In the high-energy region, the broadened absorption of ε2 ∙ (ω) is weaker than that of the ε2 ⊥ (ω). There are also a pronounced peak located at about 4.5 eV and a weak peak located at about 2.5 eV at the low-energy region of armchair CNNTs. In the highenergy region, the ε2 ⊥ (ω) has a broadened adsorption from 8 to 15 eV. The variations of dielectric functions ε(ω) for armchair CNNTs are dependent on their structures and optical anisotropy, which is similar to that of zigzag CNNTs. By the comparisons, we find that the absorption peaks of CNNTs are stronger than those of g-CN.
Configuration Optimizations and Photophysics Simulations of Single-Wall Nanotubes
245
6.4 Conclusions In this chapter, we simply introduce the calculation methods and simulation procedures of geometrical structures, optical properties for finite open SWCNTs, and non-finite long silicon-carbide and CNNTs. First, the B3LYP/6-31G* level was employed to optimize the geometrical structures and calculate the electronic structures of ground state, and the TDB3LYP/6-31G* level was employed to calculate the excitation state properties and absorption spectra (one-electron absorption) based on the optimized configurations of finite open SWCNTs of (n, n) type (n = 3–6). After following the calculations of excitation state properties, the SOS method was used to simulate the third-order polarizability and TPA spectra. The optimized configurations show that the bond lengths and angles, especially at the tube ends, are substantially different from those of an ideal rolling of a grapheme sheet, and the atoms localized in SWCNT ends accept the electrons and have negative charges and the atoms more toward the middle wall of tube are more close to electronic neutral. There are some variation trends in the finite open SWCNTs, (1) the tube diameter increases, the gap between LUMO–HOMO reduces, electronic absorption spectrum is redshift, and TPA cross section enhances as chiral vector length increases at constant axis length for (n, n) tubes; (2) the third-order polarizability along with the tube–axis direction decreases, and that along with the bisector direction between the parallel and perpendicular to the tube axis increases while chiral vector length of (n, n) tube increases, that is, the smaller diameter and the larger anisotropy of polarizabilities are for finite open SWCNTs with the same length of tube axes. It is found that the electronic originations of nonlinear polarizability and TPA spectra at low-energy region result from the π–π charge transfers within (n, n) tubes, in view of state-dependent third-order polarizability and configuration interaction contributions. Second, the periodical DFT with GGA-PBE functional was employed to investigate geometrical structures and linear optical properties of SiC and CN nanotubes with different types. The optimized configurations show that the Si and C atoms are placed alternatively without any adjacent Si or C atoms, and Si atoms move toward the tube axis and C atoms move in the opposite direction in the SiCNTs, and that the armchair CN nanotubes appear as polygons while zigzag ones are circles. The band gap between the topmost valence band and lowest conduction band increases with the SiCNT radius, and the top of the valence band region mainly arises from C-2p states with a small mixing of Si-3s,-3p states, and the conduction bands above the Fermi level are mainly due to Si-3p states in the SiCNTs. The optical properties of the zigzag, armchair, and chiral SiCNTs are dependent on the diameter and chirality, and the zero-frequency dielectric constants εx(0) and εz(0) decrease monotonically as the diameter of SiCNTs increases. It is indicated that both the armchair and zigzag CNNTs are direct band-gap semiconductors, and their band gaps and dielectric functions of CNNTs are dependent on tube size and chirality.
Acknowledgments These investigations were based on work supported by the National Natural Science Foundation of China under project 20773131, the National Basic Research Program of China (No. 2007CB815307), the Science Foundation of the Fujian Province (No. E0210028), and the Funds of the Chinese Academy of Sciences (KJCX2-YW-H01).
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47. Guo, Q. X., Y. Xie, X. J. Wang, S. Y. Zhang, T. Hou, and S. C. Lv. 2004. Synthesis of carbon nitride nanotubes with the C3N4 stoichiometry via a benzene-thermal process at low temperatures. Chem. Commun. 1: 26–33. 48. Kim, M., S. Hwang, and J. S. Yu. 2007. Novel ordered nanoporous graphitic C3N4 as a support for Pt–Ru anode catalyst in direct methanol fuel cell. J. Mater. Chem. 17: 1656–1665. 49. Cao, C. B, F. L. Huang, C. T. Cao, J. Li, and H. S. Zhu. 2004. Synthesis of carbon nitride nanotubes via a catalytic-assembly solvothermal route. Chem. Mater. 16: 5213–5215. 50. Li. J, C. B. Cao, J. W. Hao, H. L. Qiu, Y. J. Xu, and H. S. Zhu. 2006. Self-assembled one-dimensional carbon nitride architectures. Diam. Relat. Mater. 15:1593–1600. 51. Gracia, J. and P. Krollab. 2009. First principles study of C3N4 carbon nitride nanotubes. J. Mater. Chem. 19: 3020–3026. 52. Jana. D., A. Chakraborti, L. C. Chen, C. W. Chen, and K. H. Chen. 2009. First principles calculations of the optical properties of CxNy single walled nanotubes. Nanotechnology 20: 175701–175712. 53. Hales, J. and A. S. Barnard. 2009. Thermodynamic stability and electronic structure of small carbon nitride nanotubes. J. Phys. Condense Mat. 21: 144203–144207. 54. Ding, J. W., X. H. Yan, and J. X. Cao. 2002. Analytical relation of band gaps to both chirality and diameter of single-wall carbon nanotubes. Phys. Rev. B 66: 073401–073404. 55. Dresselhaus, M. S. 1998. Nanotechnology—New tricks with nanotubes. Nature (London) 391: 19–20. 56. Dresselhaus, M. S. 2001. Nanotubes—Burn and interrogate. Science 292:650–651. 57. Casida, M. E., C. Jamorski, K. C. Casida, and D. R. Salahub. 1998. Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: Characterization and correction of the time-dependent local density approximation ionization threshold. J. Chem. Phys. 108: 4439–4449. 58. Bauernschmitt, R. and R. Ahlrichs, 1996. Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory. Chem. Phys. Lett. 256: 454–464. 59. Stratman, R. E., G. E. Scuseria, and M. J. Frisch. 1998. An efficient implementation of timedependent density-functional theory for the calculation of excitation energies of large molecules. J. Chem. Phys. 109: 8218–8224. 60. Becke, A. D. 1993. Density-functional thermochemistry. 3. The role of exact exchange. J. Chem. Phys. 98: 5648–5652. 61. Lee, C. T., W. Yang, and R. G. Parr. 1988. Development of the Colle–Salvetti correlation-energy formula into a functional of the electron-density. Phys. Rev. B 37: 785–789. 62. Miehlich, B., A. Savin, H. Stoll, and H. Preuss. 1989. Results obtained with the correlationenergy density functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett. 157: 200–206. 63. Bredas, J. L., C. Adant, P. Tackx, A. Persoons, and B. M. Pierce. 1994. 3rd-order nonlinear-optical response in organic materials—Theoretical and experimental aspects. Chem. Rev. 94: 243–278. 64. Orr B. J. and J. F. Ward. 1971. Perturbation theory of nonlinear optical polarization of an isolated system. Mol. Phys. 20: 513–518. 65. Pierce, B. M. 1989. A theoretical analysis of 3rd-order nonlinear optical-properties of linear polyenes and benzene. J. Chem. Phys. 91: 791–811. 66. van Gisbergen, S. J. A., J. G. Snijders, and J. E. Baerends. 1999. Implementation of time-dependent density functional response equations. Comput. Phys. Commun. 118: 119–138. 67. Atkins, P. W. 1983. Molecular Quantum Mechanics, 2nd edn. New York: Oxford University Press, Chap. 11. 68. Hu, H., W.-D. Cheng, S. P. Huang, Z. Xe, and H. Zhang, 2008. Size effect of encased atom on absorption and nonlinear optical properties of embedded metallofullerenes M@C82 (m + Sc, Y, La). J. Theor. Comput. Chem. 7: 737–749. 69. Shen, J., W.-D. Cheng, D. S. Wu, S. P. Huang, H. Hu, and Z. Xie. 2007. First-principles determinations and investigations of the electronic absorption and third-order polarizability spectra of electron donor–acceptor chromophores tetraalkylammonium halide/carbon tetrabromide. J. Phys. Chem. A 111: 9249–9254.
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7 MATLAB® Applications in Behavior Analysis of Systems Consisting of Carbon Nanotubes through Molecular Dynamics Simulation Masumeh Foroutan and Sepideh Khoee CONTENTS 7.1 Introduction........................................................................................................................ 252 7.2 Molecular Dynamics Simulation and Analytical Studies of Nano-Oscillators: Applications with MATLAB®........................................................................................... 252 7.2.1 Molecular Dynamics Simulation Study of Nano-Oscillators.......................... 253 7.2.1.1 Molecular Dynamics Simulation of Oscillatory Behavior of Fullerene Inside CNTs ....................................................................... 253 7.2.1.2 Practical MATLAB Applications for Molecular Dynamics Simulation Studies of Nano-Oscillators ..............................................254 7.2.2 Analytical Studies on Nano-Oscillators: Mechanics of Oscillators Like Fullerene in (Single-, Double-, and Multi-) Walled CNTs and Bundles......... 256 7.2.2.1 Analytical Studies on Nanotube Bundle Oscillators......................... 257 7.2.2.2 Practical MATLAB Applications for Analytical Studies of Nano-Oscillator: Hypergeometric Functions................................. 259 7.3 Molecular Dynamics Simulations of Functionalized Carbon Nanotubes in Water: Effects of Type and Position of Functional Groups ..................................... 259 7.3.1 Behavior of FCNTs with Two Different Functionality Groups in Aqueous Solution .............................................................................................. 261 7.3.1.1 Computational Procedure...................................................................... 261 7.3.1.2 Results and Discussion .......................................................................... 262 7.3.1.3 Practical MATLAB Applications for Solvation Studies of CNTs or FCNTs: Trapezoidal Numerical Integration.................... 271 7.4 Investigation of the Interfacial Binding between Single- Walled CNTs and Heterocyclic Conjugated Polymers.......................................................................... 272 7.4.1 Simulation Method ................................................................................................ 274 7.4.2 Results and Discussion ......................................................................................... 275 7.4.2.1 Intermolecular Interaction between Polymers and Single- Walled CNTs ........................................................................ 275 7.4.2.2 Influence of the Nanotube Radius on the Intermolecular Interaction ................................................................................................ 278 7.4.2.3 Influence of Temperature on the Intermolecular Interaction ........... 278 7.4.2.4 Morphology of the Polymers on the Single-Walled CNT Surface.................................................................................. 279 7.4.3 Practical MATLAB Applications for Nanocomposite System......................... 281 251
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7.5
Molecular Dynamics Simulation Study of Neon Adsorption on Single- Walled Carbon Nanotubes.............................................................................. 283 7.5.1 Simulation Method ................................................................................................ 283 7.5.2 Results and Discussion ......................................................................................... 285 7.5.2.1 Adsorption Isotherms ............................................................................ 286 7.5.2.2 Heat of Adsorption ................................................................................. 286 7.5.2.3 Self-Diffusion Coefficient....................................................................... 288 7.5.2.4 Activation Energy ................................................................................... 289 7.5.2.5 Radial Distribution Function ................................................................ 289 7.5.3 Practical MATLAB Applications in Adsorption of Ne Atoms on CNTs ....... 290 7.6 Conclusion .......................................................................................................................... 291 Acknowledgments ...................................................................................................................... 292 References...................................................................................................................................... 292
7.1 Introduction In this chapter, we present the applications of MATLAB® in the calculation of nanosystems, which mostly contain carbon nanotubes (CNTs). The present chapter consists of five sections dealing with nano-oscillators, aqueous solution containing CNTs, nanocomposites, and gas adsorption on CNTs. As a high-performance language for technical computing, MATLAB can be used in these cases. The recently published papers show that the programming environment of MATLAB provides useful aids for algorithm development, data analysis, visualization, and numeric computation in nano-researches in several fields. In Section 7.2, we present some specialized applications of MATLAB for studying the behavior of nano-oscillators. Section 7.2 contains two sections relating to molecular dynamics simulation and analytical studies of nano-oscillators. For each section, we present some applications of MATLAB in calculating different functions. In Section 7.3, we introduce the application of MATLAB in the study of the behavior of water molecules surrounding the CNT. In the first stage, we performed molecular dynamics simulation and then obtained the radial distribution function (RDF) for water molecules around CNT. Then, the number of water molecules gathering around the CNT is obtained using MATLAB. In Sections 7.4 and 7.5, we refer to some applications of MATLAB to calculate and solve the equations relating to nanocomposites and adsorption of gases on CNTs, respectively. Using the output of molecular dynamics simulation of some programs written in MATLAB, we calculated the gyration radius of polymers (in Section 7.4) and diffusion coefficients of gases (in Section 7.5).
7.2 Molecular Dynamics Simulation and Analytical Studies of Nano-Oscillators: Applications with MATLAB ® Recent achievements in nanotechnology are promising to produce technologies that utilize nano-devices with new and fundamentally unique properties.1–3 The high efficiency of these devices has generated a considerable interest over the recent years in many theoretical
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and experimental research groups so that designing such nano-devices and their applicability have become a specific field in nano-science and nanotechnology.4–8 CNTs and fullerenes are two important types of nanostructures, which have unique mechanical, physical, and chemical properties. These unusual properties of nanostructures promise new nano-devices based on CNTs and fullerenes. It has been shown that CNTs in their different forms, including single-walled CNTs, double-walled CNTs, multi-walled CNTs, CNT bundle, and also fullerenes oscillating inside a nanotube like a single-walled CNT, can be considered as nano-oscillators. Results of recent studies have led researchers to conclude that these oscillators are able to generate high frequencies, mainly in the gigahertz range. The useful applications of these gigahertz oscillators have been of value to many researchers, especially electrical engineers. The nano-oscillators were first introduced by Cumings and Zettl.9 Two years later, Zheng and Jiang showed that sliding of inner shell inside the outer shell of a multi-walled CNT can generate oscillatory frequency up to several gigahertz.10 Cumings and Zettl demonstrated the controlled and reversible telescopic extension of multi-walled CNTs. They believed that since repeated extension and retraction of telescoping nanotube segments revealed no wear or fatigue on the atomic scale, these nanotubes constitute near perfect, wear-free surfaces. There have been classical applied mathematical investigations into the mechanics of nano-oscillators, which have been done mainly by Hill et al.11–57 They studied analytically the oscillators involving atoms, spherical and spheroidal fullerenes, and CNTs oscillating in a (single-, double-, and multi-) walled CNTs. Also different studies concerning the interesting field of nano-oscillators and molecular dynamics simulations have revealed some new characteristics of them.58–63 In this section, we present some important functions in MATLAB to study the behaviors of nano-oscillators. This section contains two sections, one for molecular dynamics simulation and another for analytical studies of nano-oscillators. In each section, we introduce MATLAB applications for calculating different functions. 7.2.1 Molecular Dynamics Simulation Study of Nano-Oscillators Molecular dynamics simulation has been used extensively to investigate the behavior of nano-oscillators. Cummings et al. have studied damped oscillating behavior between the centers of mass of inner and outer tube of double-walled CNTs using molecular dynamics simulation.59,60 Lee has investigated the oscillatory behavior of double-walled boron nitride nanotubes.61 He showed that boron nitride nanotube oscillators have higher frequency than the CNT oscillators. Hai-Yang and Xin-Wei have investigated the effects of radius and defect of single-walled CNTs on oscillating behavior of C60 oscillating inside a single-walled CNT using molecular dynamics simulation.62 Kang et al. studied the couple oscillation of multi-walled CNT oscillators.63 They showed that the frequencies of the multi-walled CNT oscillators are higher than those of the double-walled CNT oscillators. Now, we refer to some applications of MATLAB for calculating the power spectral and spectrogram of a fullerene oscillating as a nano-oscillator inside a CNT. 7.2.1.1 Molecular Dynamics Simulation of Oscillatory Behavior of Fullerene Inside CNTs We performed molecular dynamics simulation for an oscillating fullerene C60 inside a singlewalled CNT (15, 15) and also inside a nanotube bundle (10, 10), separately.
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FIGURE 7.1 Fullerene C60 oscillating inside a nanotube (15, 15).
FIGURE 7.2 Fullerene C60 oscillating inside a nanotube bundle (10, 10).
AMBER99 force field and van der Waals force Lennard–Jones potential function as non bonding interaction was used in simulation. We used a canonical (constant NVT [number, volume, and temperature]) ensemble and the Nose–Hoover thermostat algorithm. The Beeman integration scheme was used with a time step size of 1.0fs. The cut-off distance was 12.5 Å and the temperature was fixed at 100K. The ε parameter values for C60 and CNTs were equal to 0.1051 and 0.0860kcal/mol, respectively. Also the σ parameter values for C60 and CNTs were equal to 3.85 and 3.40 Å, respectively. Figures 7.1 and 7.2 show two snapshots of fullerene C60 oscillating inside CNT (15, 15) and nanotube bundle (10, 10), respectively. The curves of the oscillatory position versus the simulation time of C60 oscillators inside CNT (15, 15) are shown in Figure 7.3. 7.2.1.2 Practical MATLAB Applications for Molecular Dynamics Simulation Studies of Nano-Oscillators After running the molecular dynamics simulation for the mentioned nano-oscillator, the positions of center of mass of nano-oscillator versus time is obtained and using spectrum and spectrogram functions of MATLAB, we can present the frequency analysis of the oscillatory behavior of fullerene. The obtained spectrum shows that the frequency of oscillating fullerene is in gigahertz range.
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Position (Å)
290 280 270 260 250 300
400
500
600
700
800
900
1000
Time (ps) FIGURE 7.3 Variation of the oscillatory position of the C60 with oscillatory time.
7.2.1.2.1 Calculation of Frequency Analysis Using Spectrum Function in MATLAB Figure 7.4 shows the spectrum the C60 oscillating inside a nanotube (15, 15) obtained from molecular dynamics simulation. This spectrum has been obtained from Welch’s method using MATLAB software. Welch’s method is used for estimating the power of a signal versus frequency. There are some syntaxes for pwelch in MATLAB toolbox like including [Pxx, w] = pwelch(x). Description of all syntaxes can be found in MATLAB toolbox. For example, [Pxx, w] = pwelch(x) estimates the power spectral density Pxx of the input signal vector x using Welch’s averaged modified periodogram method of spectral estimation. The spectral density, power spectral density, or energy spectral density is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function Welch power spectral density estimate
–110
Power/frequency (dB/Hz)
–120 –130 –140 –150 –160 –170 –180
0
50
100
150
200
250
300
Frequency (GHz) FIGURE 7.4 Frequency analysis of the fullerene C60 inside a nanotube (15, 15).
350
400
450
500
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–140 –145 –150 –155 18
16
×10–10
14
12
10 Time
8
6
4
2
0
1
3 z) 2 y (H uenc q e r F
5
4
×1011
FIGURE 7.5 Spectrogram of the fullerene C60 inside a nanotube (15, 15).
of time, which has dimensions of power per Hz, or energy per Hz. It is often called simply the spectrum of the signal.64,65 Welch’s method is used by MATLAB’s pwelch command to calculate spectral density. 7.2.1.2.2 Frequency–Time Plots Using Spectrogram Function in MATLAB In Figure 7.5, we present the frequency of oscillator for oscillating C60 inside CNT (15, 15) as a function of time. The frequency–time plot has been obtained using spectrogram’s method. Spectrogram’s method is used by MATLAB spectrogram command to calculate the short-time Fourier transform or spectrogram of a signal. MATLAB 7 and MATLAB 2006a and newer versions contain the function spectrogram. The spectrogram function plots the spectrogram as a 3D surface, using the surf command. 7.2.1.2.3 Related Program The below mail program asks the user to input the data, calls the functions pwelch.m and spectrogram.m, and finally plots the spectrum and spectrogram in the 2D and 3D graphs, respectively. clear; clc; Fs = 1/1e-12; x = xlsread('D:\my data.xls'); x1 = zeros(1,length(x(:,1))); for i=1:length(x(:,1)) if ~isnan(x(i)) x1(i) = x(i); end end y = x1 - mean(x1); close all; pwelch(y(1:end),[],[],[],Fs, 'onesided') hold off figure; spectrogram(y(1:end),128,120,128, Fs)
7.2.2 Analytical Studies on Nano-Oscillators: Mechanics of Oscillators Like Fullerene in (Single-, Double-, and Multi-) Walled CNTs and Bundles Now, we refer to some analytical studies of Hill et al. on nano-oscillators.11–57 Following systems have been investigated by Hill et al.: certain fullerene-nanotube bundle oscillators, namely, C60-CNT bundle, C60-boron nitride (BN) nanotube bundle, B36N36-CNT bundle, and B36N36-boron nitride nanotube bundle.
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Hill et al. have presented the formulations of interaction energy between nanotubes in nano-oscillators. Equations describing the interaction energy are based on hypergeometric functions, which have been introduced in MATLAB. 7.2.2.1 Analytical Studies on Nanotube Bundle Oscillators Now we review the work of Thamwattana and Hill56 entitled “Nanotube bundle oscillators: Carbon and boron nitride nano-structures” and summarize the essential mechanics of the gigahertz nano-oscillators. Hill et al. introduced the Lennard–Jones potential for the nonbonded interaction energy between two molecules. Using the continuum approach, they determined the interaction energies of a fullerene located inside a bundle. From the potential energies, the van der Waals restoring forces were obtained, which were then used to describe the oscillatory behaviors of nano-oscillators. The nonbonded interaction energy E is obtained by summing the interaction energy for each atom pair, E=
∑ ∑ Φ(r ),
(7.1)
ij
i
j
where Φ(rij) is a potential function for atoms i and j at distance rij apart. In the continuum approximation, it has been assumed that carbon atoms are uniformly distributed over the surface of the molecules. Therefore, we can replace the double summation in Equation 7.1 by a double integral, which averages over the surfaces of each entity. E = n1n2
∫∫ Φ(r)d∑ ∑ 1
(7.2)
2
where n1 and n2 are the mean surface density of atoms on each molecule r is the distance between two typical surface elements dΣ1 and dΣ2 on each molecule There are a number of models of potential in the literature for Φ(r), for example, the Lennard–Jones potential with the following equation: Φ(r ) = −
A B + , r 6 r 12
(7.3)
where A and B are the attractive and the repulsive constants, respectively r is the distance between two atoms Equation 7.3 can be written in the form ⎡ ⎛ A ⎞6 ⎛ B ⎞12 ⎤ Φ(r ) = 4ε ⎢ − ⎜ ⎟ + ⎜ ⎟ ⎥ , ⎢⎣ ⎝ r ⎠ ⎝ r ⎠ ⎥⎦
(7.4)
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where σ and ε are the van der Waals diameter and the well depth, ε = A2/(4B). The equilibrium distance r0 is given by ⎛ 2B ⎞ r0 = 21/6 σ = ⎜ ⎟ ⎝ A⎠
1/6
,
(7.5)
The total interaction energy of a fullerene radius r0 located at the center of a bundle containing N tubes of infinite length is given by W = − NE(R)
(7.6)
where R, the bundle radius, is the distance from the center of the fullerene to the axis of the nanotubes in the bundle E is the interaction energy between the fullerene and a single nanotube ⎡ B ⎛ 315 ⎤ 12155 8 ⎞ A 1155 2 9009 4 6435 6 E(R) = 4π 2 r02 r η f ηt ⎢ ⎜ J5 + r0 J 6 + r0 J 7 + r0 J 8 + r0 J 9 ⎟ − (3 J 2 + 5r02 J 3 )⎥ , ⎝ ⎠ 256 8 64 128 64 ⎣ 5 256 ⎦
(7.7) where ηf and ηt are the mean atomic densities of the fullerene and CNT, respectively Jn is defined in terms of hypergeometric function as Jn =
2π 2
2 n + 1/2 0
⎡⎣(r − R) − r ⎤⎦
⎛1 ⎞ 1 4rR F ⎜ , n + ; 1; − 2 (r − R)2 − r02 ⎟⎠ ⎝2
(7.8)
The parameters of total energy for C60-CNT bundle are η f = 0.3789 Å−2,
ηt = 0.3812 Å−2, A = 15.41 eVÅ6, B = 22534.75 Å12
(7.9)
The velocity of the oscillating fullerene inside a nanotube bundle is given by ⎛ 2W ⎞ ν=⎜ + ν02 ⎟ ⎝ M ⎠
1/2
(7.10)
where M is the mass of the fullerene (=1195.2 × 10−27 kg) ν0 denotes the initial velocity of the fullerene, which here is assumed to be zero Also the oscillatory frequency f is determined from f =
ν ( 4 L)
(7.11)
In the next section, we present the total interaction energy W and frequency for fullerenenanotube bundle oscillators.
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2E–17
f (GHz)
1.8E–17 1.6E–17 1.4E–17 1.2E–17 1E–17 25
30
35
40
L
45
50
55
60
FIGURE 7.7 Frequency for fullerene-nanotube bundle oscillators varying the half-length of the bundle L.
and therefore they need to be functionalized. The insolubility of CNTs in water is generally considered as a significant barrier in technological development toward the practical uses of CNTs. Despite rapidly growing interest in solvation CNTs and functionalized CNTs (FCNTs), there has been no virtually systematic study for them. As an initial attempt to gain theoretical understanding of the solvation of single-walled CNT in solvents, Grujicic and Cao investigated the solubilization of (10, 10) single-walled CNT in toluene.70 They showed that the solvation Gibbs free energy for CNT in toluene is small but positive, suggesting that a suspension of these nanotubes in toluene is not stable and that the nanotubes would fall out of the solution. The structure properties of water surrounding a (16, 0) CNT and energy changes that occur by the process of introducing the CNT in water have been studied by performing fully atomistic molecular dynamics simulation by Walther et al.71 In another investigation, the solvation of CNTs in a room-temperature ionic liquid has been studied via molecular dynamics simulations.72 A lot of efforts have been done to make CNTs soluble in aqueous solutions.73 Recently, experimental studies have shown that chemical functionalization is a proper choice to improve the solubility of single-walled CNTs in water.74 In particular, the fictionalization of CNTs to introduce aqueous solubility has received much recent attention for investigations targeting potential biological applications of CNTs.75 Many techniques have been proposed for the functionalization of CNT including defect and covalent sidewall functionalization, as well as noncovalent exo- and endohedral functionalization.76 Chemical functionalization of CNTs by carboxyl (COOH) groups has been investigated by some researchers from both theoretical77–81 and experimental approaches.82–86 In the present work, we study the behavior of FCNTs with two different functionality groups in water. Two types of functional groups are considered: glutamine as a longchain functional group with many hydrophilic parts and carboxyl group as a short-chain functional group. It is reminded that while the obtained hydration structures of CNT and FCNTs in water provides us useful insights, it does not directly address the solubility of CNTs and FCNTs. As mentioned before, there are several experimental and theoretical studies concerning FCNT with carboxyl functional group, but there are not any reported studies for FCNT with glutamine functional group in literatures. Before the synthesis of FCNTs with
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glutamine functional group experimentally, the molecular dynamics simulation can be used for predicting the behavior of FCNTs with glutamine functional group in water. 7.3.1 Behavior of FCNTs with Two Different Functionality Groups in Aqueous Solution We present the obtained results regarding the behavior of FCNTs with carboxyl functional group in aqueous media via molecular dynamics simulations.87 7.3.1.1 Computational Procedure Computer simulations of the structure of water-CNT system was carried out using a 2.8 × 2.8 × 3.3 nm3 cubic computational cell with periodic boundary conditions applied in all three principal directions at 300 K. In all simulated systems, 864 water molecules were placed in the cell to obtain the average density of water essentially equal to its experimental counterpart (1 kg/m3). For water–FCNT systems, the volume of the system was adjusted to match the desired density of water. In all simulated systems, a (2, 8) single-walled CNT with fixed length of 1.954 nm has been used. Also, in all simulations, the CNTs were surrounded by a water bath in order to let the water spontaneously enter the nanotubes. All simulations were repeated using TIP3P of water model and nanotube flexibly for simulation of flexible water and CNT or FCNT. Energy minimization was performed to find the thermal stable morphology and achieve a conformation with minimum potential energy for all molecules. We used a canonical ensemble with undefined boundary conditions for considering the simulated volume were equals to infinite. The velocity form of Verlet algorithm method and the Nose–Hoover thermostat algorithm was used to integrate the equations of motion with a time step of 1.0 fs and temperature control of 300 K, respectively. A cut-off distance of 9 Å was used for the van der Waals potentials, and Lorentz-Berthelot mixing rules were used for cross interactions.29 The systems were initially equilibrated for 500 ps ns (nano seconds) and then simulated for 3 ns. Nonbonded van der Waals’ interactions were modeled by a Lennard–Jones potential with a cut-off distance of 0.9 nm. The values of σc–c and εc–c used in the simulations were, respectively, 1.9080 Å and 0.086 kcal/mol (from AMBER force field). Calculations have been performed for atomic NBO charges on atoms for several systems, utilizing the AM1 semiempirical quantum chemistry. Now we introduce six systems, which have been considered in the present work naming A, B, C, D, E, and F systems. In all systems, in addition to single-walled CNT or FCNT, there were 864 water molecules. The system containing the pristine single-walled CNT and 864 water molecules is denoted as system A. Four functional groups in any FCNT can localize at two positions: (i) All four functional groups are in the sidewalls of nanotube; (ii) Two functional groups are at the ends and two functional groups are in the sidewalls of nanotube. In the systems B and C, the glutamine functional groups are localized at positions (i) and (ii) of CNTs, respectively. Also in the systems D and E, the carboxyl functional groups are localized at positions (i) and (ii) of CNTs, respectively. In system F, there are 16 carboxyl functional groups around the FCNT. The closed structure of glutamine is NH2–C (COOH) C2H4–CO–NH2. The numbers of all atoms of glutamine functional group were indicated in Figure 7.8. As Figure 7.8 shows, the number of carbon atom of CNT, which is connected to
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0
Interaction energy (kcal/mol)
–50 –100 –150 –200 –250 –300 A
–350 –400
B C 0
500
1000
1500
2000
2500
3000
Time (ps) FIGURE 7.11 Interaction energy evolution during 3 ns of simulations for systems A, B, and C.
(in systems B and C) versus CNT-water (in system A) emphasize that there are strong interactions between FCNTs and water. Also different values of interaction energies for systems B and C indicate that the interaction energy between FCNTs and water is influenced by the position of functional group. Interaction energy evolution for systems A, D, and E during 3 ns of simulations is shown in Figure 7.12. 0
A D E
Interaction energy (kcal/mol)
–50 –100 –150 –200 –250 –300 –350
0
500
1000
1500 Time (ps)
2000
2500
FIGURE 7.12 Interaction energy evolution during 3 ns of simulations for systems A, D, and E.
3000
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TABLE 7.1 The Integration Numbers of Oxygen Atoms of Water for Simulated Systems B, C, D, and E 1B
3B
1C
3C
D
E
7.63
8.74
8.86
10.18
8.275
8.379
Note: For more details, refer to Figures 7.14, 7.15, and 7.18 and their explanations.
TABLE 7.2 The Integration Numbers of Hydrogen Atoms of Water for Simulated Systems B, C, D, E, and F 2B
18B
2C
18C
4D
4E
4F
8.30
9.83
8.87
10.48
9.64
10.05
9.00
Note: For more details, refer to Figures 7.16, 7.17, and 7.19 and their explanations.
The integration numbers of oxygen and hydrogen atoms of water for different simulated systems have been collected in Tables 7.1 and 7.2, respectively, and are explained in detail below. The plots of RDF for oxygen atoms of the water molecules versus the carbon atoms of CNT (denoted by number 1 in Figure 7.8) and the hydrogen atom of glutamine functional group (denoted by number 3 in Figure 7.8) are shown in Figure 7.14. These RDFs have been identified as 1(B) and 3(B) in Figure 7.14. Two sharp peaks appeared at 1.75 and 6.25 Å for 3(B) plot and two peaks at 3.75 and 7.5 Å for 1(B) plot. These results indicate that water molecules more tend to localize closer to the hydrogen atoms of glutamine group, rather than to carbon atoms of CNT. Therefore, the probability of finding water molecules surrounding the functionalized tube is higher than that of CNT. As Table 7.1 summarizes, the running integration numbers obtained from integrating the g(r) for 1(B) and 3(B) RDFs were equal to 7.63 and 8.74, respectively. The plots of RDF for oxygen atoms of the water molecules versus the carbon atoms of CNT (denoted by number 1 in Figure 7.8) and the hydrogen atom of glutamine functional group (denoted by number 3 in Figure 7.8) are shown in Figure 7.15. These RDFs have been 1.2
1(B) 3(B)
1
g (r)
0.8 0.6 0.4
FIGURE 7.14 RDFs for oxygen atom of the water with respect to the atom numbers 1 and 3 of glutamine group in system B.
0.2 0
0
5
10 r (Å)
15
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1.2
1(C) 3(C)
1
g (r)
0.8 0.6 0.4 0.2 0
0
2
4
6
8 r (Å)
10
12
14
16
FIGURE 7.15 RDFs for oxygen atoms of the water with respect to the atoms numbers 1 and 3 of glutamine group in system C.
identified as 1(C) and 3(C) in Figure 7.15. The running integration number for 1(C) and 3(C) RDFs were equal to 8.86 and 10.18, respectively. The obtained results emphasized that the number of water molecules gathering around the glutamine functional groups is greater than the number of molecules gathering around carbon atom of CNT, and we know that this is because of the high hydrophobicity character of CNT. For investigating the effect of functional group position on the numbers of water molecules gathering around the FCNTs in systems B and C, we compare the running integration number obtained from 3(B) and 3(C), that is, 8.74 and 10.18. Indeed, the results from our comparison indicate that position (ii) provides better conditions for the water molecules to gather around hydrophilic group of FCNT. For system C, the RDFs between the hydrogen atoms of water molecules and two kinds of nitrogen atoms of glutamine are shown in Figure 7.16; atom numbers 2 and 18 in Figure 7.8, which are the nearest and the farthest nitrogen atoms of glutamine group to carbon atom of CNT, respectively. These RDFs have been identified as 2(C) and 18(C) in Figure 7.16. In Figure 7.16, one sharp peak was found for atom number 18 at 3.25 Å while no sharp peak was observed for atom number 2 at 5.75 Å. Therefore, water molecules more tend to localize around the hydrophilic groups at the end of the glutamine functional group, 1.2
18(C) 2(C)
1
g (r)
0.8 0.6 0.4 0.2 0
0
5
10 r (Å)
15
FIGURE 7.16 RDFs for the hydrogen atoms of the water with respect to the atom numbers 2 and 18 of glutamine functional group in system C.
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1.2
18(B) 2(B)
1
g (r)
0.8 0.6 0.4 0.2 FIGURE 7.17 RDFs for hydrogen atoms of the water with respect to the atom numbers 2 and 18 in system B.
0
0
5
10
15
r (Å)
rather than nearer hydrophilic groups. In other words, these results tell us that water molecules tend to localize around the hydrophilic groups with less space constraints. The running integration numbers obtained for plots 2(C) and 18(C) in Figure 7.16 were equal to 8.87 and 10.48, respectively. Figure 7.17 shows the RDF between hydrogen atoms of water molecules and atom numbers 2 and 18 of glutamine group for system B. These RDFs have been identified as 2(B) and 18(B) in Figure 7.17. The sharp peak in RDF 18(B) has appeared at 3.25 Å, which refers to migration of hydrogen atoms of water molecules around the farthest nitrogen atom of glutamine group to atom carbon of nanotube. Also the first and second peaks in RDF 2(B) have appeared at 3.75 and 5.75 Å. The running integration numbers obtained for plots 2(B) and 18(B) in Figure 7.17 were equal to 8.30 and 9.83, respectively. The running integration numbers obtained for systems B and C show that the probability of finding water molecules surrounding the hydrophilic group of glutamine is more for those that are located near CNT than those that are located far from this tube. To determine the most efficient type of functional groups in the movement of water molecules toward the FCNTs, we studied the plot of the RDFs of water molecules versus FCNTglutamine and FCNT-carboxyl. Evidently, glutamine functional group has several hydrophilic parts containing two amine groups, one carboxyl group and one carbonyl group. Therefore, several hydrogen bonding can be established between the water molecules and atoms of glutamine functional group. Here we consider one hydrophilic group of glutamine functional group: amine groups at the end of glutamine groups far from CNT. The RDFs for oxygen atoms of the water molecules versus the hydrogen atoms connected to nitrogen atom of glutamine group (atom number 19 in Figure 7.8) for systems B and C are given in Figure 7.18. Figure 7.18 also shows the RDFs for oxygen atoms of the water molecules against the hydrogen atom of hydroxyl group for systems D and E. The running integration numbers obtained for FCNTs in systems B, C, D, and E were 8.74, 10.18, 8.275, and 8.379, respectively. It means that the number of water molecules surrounding the amine group of glutamine functional group is more than that for carboxyl functional group. In order to investigate the effect of the number and position of carboxyl functional group on the accumulation of water molecules around the FCNT, the RDFs of hydrogen atoms of water molecules versus oxygen atom of carboxyl group for systems D, E, and F were calculated and shown in Figure 7.19 and have been identified as 4(D), 4(E), and 4(F) plots, respectively. The running integration numbers calculated for RDFs 4(D), 4(E), and 4(F) were 9.637,
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1.2 1
g (r)
0.8 A B C D E F
0.6 0.4 0.2
FIGURE 7.20 RDFs for oxygen atoms of the water with respect to the carbon atom of nanotube for systems A, B, C, D, E, and F.
0
(a)
0
5
10
15
r (Å)
(b)
FIGURE 7.21 The simulated systems B (a) and D (b).
Figure 7.21a and b show the simulation snapshots of FCNTs in systems B and D, respectively. These figures show that the number of water molecules gathering around the functional groups is greater than that gathering around carbon atom of CNT. It is evident that due to the high hydrophobicity character of CNT, the water molecules tend to gather around hydrophilic groups of FCNTs. 7.3.1.2.3 Diffusion Coefficients of CNT and FCNTs in Water We have used molecular dynamics simulations to investigate the dynamics of CNT and various FCNTs in aqueous solution. The diffusion coefficient (D) of CNT or FCNT can be calculated from the following equation: 3D = lim
[MSD (t)] 2t
(7.14)
t→∞ where MSD (t) = ri (t) − ri (0)2 . The angular brackets indicate an ensemble average over atoms in the system and over time origins. ri is the center of mass coordinates of the CNTs or FCNTs. Figure 7.22 shows mean square displacement (MSD) versus time of CNT and FCNTs in aqueous solution.
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400
A C B D E F
350 300 MSD
250 200 150 100 50 0
0
200
400
600 Time (ps)
800
1000
1200
FIGURE 7.22 The MSD of CNT and FCNTs in all systems A, B, C, D, E, and F.
TABLE 7.3 The Diffusion Coefficients of CNT and FCNTs in Water for Studied Systems System A B C D E F
MSD (Å2/ps) 0.051300 0.025083 0.018467 0.026883 0.046167 0.009058
The diffusion constants of CNT in system A and FCNTs in systems B, C, D, E, and F are listed in Table 7.3. As Table 7.3 summarizes, the FCNT in system F has the minimum value of diffusion coefficient. It means that the water molecules have the most attractive interaction with FCNT in system F so that nanotube moves slowly in aqueous media compared with other nanotubes. System F has the smallest diffusion constants and the largest interaction energy, and these are consistent with each other. As interaction energy between water molecules and hydrophilic groups of FCNT increases, the movement of FCNT decreases. 7.3.1.3 Practical MATLAB Applications for Solvation Studies of CNTs or FCNTs: Trapezoidal Numerical Integration Z = trapz (Y) computes an approximation of the integral of Y via the trapezoidal method (with unit spacing). To compute the integral for spacing other than one, multiply Z by the spacing increment. Input Y can be complex.66 The below mail program asks the user to input the data, calls the function trapz.m for integration, and displays the results. clear; clc; x = xlsread('D:\my data.xlsx’); y = trapz(x(:,1),x(:,2));
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7.4 Investigation of the Interfacial Binding between Single- Walled CNTs and Heterocyclic Conjugated Polymers In this section, we review recent studies on the behavior of heterocyclic conjugated polymers when they are close to the CNTs and the mechanism of polymer wrapping of CNTs in such systems.88 The exploration of CNTs in 199189 caused extensive investigation of their properties and potential applications in diverse fields of science. An important application of these novel materials is in composites via their usage as fiber-like reinforcers and hence has induced much interest both in experimental and in theoretical research groups. The extraordinary specifications of CNTs make them ideal candidates as ultrastrong reinforcers for composites, specifically, polymer composites. Recent experiments have shown that the addition of small amounts of CNTs to polymers can change their important properties such as mechanical,90–96 electrical,97,98 thermal,99–101 and optical102 properties. Qian et al.103 showed that by applying only 1wt% of multi-walled CNT, the elastic stiffness of polymer matrix nanocomposites was increased up to 40%, compared to the pristine polymer matrix. Mamedov and his coworkers104 created a revolutionary material by sandwiching CNTs between polymer layers, in which its strength was six times stronger than conventional carbon-fiber composites and ultra values of hardness, which matches of some ultrahard ceramic materials. It is well known that the structure and properties of the interface between CNTs and polymer matrix play a major role in determining the mechanical performance and structural integrity of such nanocomposites. In order to take advantage of the high Young’s modulus and strength of CNTs in such composites, we need a good interfacial binding as a necessary condition for an efficient load transfer from the polymer matrix to CNTs. Bower et al.105 analyzed the morphology of fracture surface of CNT–epoxy composites. After the fracture test, they came across unbroken CNTs implying that the interfacial binding has not been strong enough to cause a sufficient load transfer. This experiment reveals the significance of the interfacial binding in such materials. An obvious strategy for improving this interfacial binding is chemical functionalization by attaching side chains to the CNTs. For instance, Kuzmany et al.106 have attached different functional groups (–OH, –COOH, –F) directly to CNTs and then changed these simple groups by performing substitution reactions. Since the extraordinary mechanical properties of CNTs depend on their specific chemical structure, chemical functionalization causes an important disadvantage; as this process (i) is a nontrivial processing step and (ii) introduces atomic defects and internal stresses into the nanotubes, which thereby deteriorates their mechanical properties.107,108 It is therefore desirable to explore the possibility of optimizing noncovalent intermolecular interactions between CNTs and polymers in view of achieving strong interfacial binding in a composite. For example, Wong et al.109 have investigated the local fracture morphologies of CNT/polystyrene (PS) rod and CNT/epoxy film composites by transmission and scanning electron microscopy. Their observations indicated that these polymers adhered well to CNT at the nanometer scale. Since it is difficult to study the CNT–polymer interface by experimental methods, molecular mechanics (MM) and molecular dynamics simulations can be used effectively in the investigations of reinforcement mechanisms in CNT–polymer composites. Gou et al.110 using MM and molecular dynamics simulations showed that individual nanotubes have stronger interactions with the epoxy resins than the nanotube rope. The interfacial characteristics of a CNT–PS composite system through MM simulations and elasticity calculations have been
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studied.111 The results showed that the fiber/matrix adhesion arises from the electrostatic and van der Waals interactions. Frankland et al.112 in a molecular dynamics work suggested that load transfer between nanotube–polymer composites can be effectively increased by adding chemical cross-links between single-walled CNTs and polymeric matrix. Recently Yang et al.113 studied the interactions between PS/polyphenylacetylene (PPA)/ poly (p-phenylenevinylene) (PPV)/PmPV and a single-walled CNT. Based on their simulations, they suggested that polymers with backbones containing aromatic rings are promising candidates for the noncovalent binding of CNTs into composite structures. In a similar work, Zheng et al.114 used molecular dynamics simulations to study the interaction between polyethylene/polypropylene/PS/polyaniline and a single-walled CNT. The influence of temperature, single-walled CNT radius and chirality on polymer adhesion was investigated. The results showed that the interaction between the single-walled CNT and the polymer is strongly influenced by the specific monomer structure such as an aromatic ring. Furthermore, they conducted simulations of “filling” into the single-walled CNT cavity by a polymer molecule. They thought that the possible extension of a polymer into a single-walled CNT cavity that structurally bridged the single-walled CNT and the polymer could be used to improve load transfer from the polymer to the single-walled CNT. Liu et al.115 studied the effect of polymer’s different repeat unit arrangements and different conformations on the interaction energies. They found that the interaction strength between the PPA molecules and singlewalled CNTs is obviously influenced by these factors, and the degree of the PPV wrapping around the single-walled CNT is associated with its repeat unit arrangement. This behavior was expected since the interaction strength in polymer systems is sensitive to polymer’s different repeat unit arrangements. Recently the effect of copolymer’s different repeat unit arrangements and polydispersity of multiblock copolymers on their phase behavior has been calculated by one of us116,117 in the framework of the mean field theory. Chen et al.118 investigated the influence of nanotube chirality and chemical modification on the interfacial binding between CNTs and PPA. They demonstrated that the armchairtype nanotube may be the best nanotube for reinforcement and indicated that some specific chemical modifications of single-walled CNTs by methyl or phenyl groups can be wellwrapped by PPA, while the single-walled CNTs modified by other types of groups (–OH, –F) cannot. Kang et al.119 in an combined experimental- molecular dynamics work indicated that amphiphilic, linear conjugated poly [p-{2,5-bis(3-propoxysulfonicacidsodiumsalt)} phenylene] ethynylene (PPES) efficiently disperses single-walled CNTs into the aqueous phase. Transmission electron microscopy (TEM) data reveal that the interaction of PPES with single-walled CNTs gives rise to a self-assembled superstructure in which a polymer monolayer helically wraps the nanotube surface; the observed PPES pitch length (13 nm) confirms the structural predictions made via their molecular dynamics simulations. Although these studies cannot give bulk nanocomposite properties directly, they do present to us a clear and vivid view of the interface between the polymer and the CNT. From the literature mentioned above, we know that the efficiency of load transfer from polymer matrix to CNTs is in positive correlation with the interfacial binding energy (intermolecular interaction energy), and the well-ordered coating of polymers on the CNT surface is vital for effective load transfer. In this research, molecular dynamics simulations were conducted to explore the interaction between single-walled CNTs and three heterocyclic conjugated polymers including PT, PPy, and PPyPV and their comparison with the most investigated homocyclic conjugated polymer, that is, PmPV. To the best of our knowledge, there has not been reported any simulation results regarding the aforementioned heterocyclic conjugated polymers with CNTs;
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thereby, we bring here some experimental works regarding their composites with CNTs. A polymer that has been studied extensively in optoelectronic applications as a CNT dopant is PmPV,120,121 and electron microscopy studies have demonstrated heavy coating of singlewalled CNTs by PmPVs.122–124 Raman and absorption studies suggested that the polymer wraps preferentially with nanotubes possessing a specific range of diameters. In a subsequent work, the same researchers studied for comparison the chemical interactions of CNT with PmPV and PPyPV.125 In both cases, they observed the dispersion of the tubes in the organic media. Nanotube-PPy composites have been engineered by in situ chemical126 or electrochemical polymerization.127 These types of composites have been used as active electrode materials in the assembly of a supercapacitor,128 for the selective detection of glucose129 and for selective measurement of DNA hybridization.130 Recently, nanotube-PPy composites have been studied as gas sensors for NO2.131 Blends of nanotube-PT have been fabricated132 and their electrical properties were studied.133,134 The enhanced photovoltaic behavior of the composites makes them ideal candidates as solar cells for energy conversion.135 Our simulation is related to the behavior of these heterocyclic conjugated polymers when they are close to the CNT in an “ideal” bad solvent and the mechanism of polymer wrapping of CNTs in such systems. 7.4.1 Simulation Method Molecular dynamics simulations were performed in Tinker molecular modeling package (version 5.0)136 using the MM3 force field.137 The MM3 force field has been considered to calculate the structures and energies, including heats of formation, conformational energies, and rotational barriers for hydrocarbons more accurately than was possible with earlier force fields.137 The total potential energy (Etotal) of our simulated systems is calculated as the sum of the eight individual energy terms including Etotal = Es + Eθ + Eω + ESθ + EωS + Eωθ + Eθθʹ + Evdw
(7.15)
where ES, Eθ, Eω, ESθ, Eωs, Eωθ, Eθθ′, and Evdw are the energies corresponding to bond stretching, angle bending, torsion, stretch–bend interaction, torsion–stretch interaction, torsion– bend interaction, bend–bend interaction, and van der Waals interaction terms, respectively. The electronic structures of carbon atoms in the single-walled CNT models were sp2 hybridization. The unsaturated boundary effect was avoided by adding hydrogen atoms at the ends of the single-walled CNTs. Each C–C bond length was 1.42 Å, and each C–H bond length was 1.14 Å. We considered all single-walled CNTs in the armchair configuration with fixed length of 110 Å and diameters ranging from 5.49 to 27.13 Å. Since CNTs exhibit an even charge distribution, there is hardly any electrostatic interaction between nanotubes and polymers, and therefore in the absence of chemical functionalization, the polymer–CNT interaction is solely due to van der Waals forces; thus, the neutrally charged single-walled CNTs were constructed. PT, PPy, PPyPV, and PmPV polymer molecules were considered in this work. From the two possible structures of PT, that is, aromatic and quinoid structures, only aromatic structure was investigated. Figure 7.23 shows the chemical structure of investigated polymers. Since the number of atoms and monomers used is small (about 150 atoms per molecule, which corresponds to 20 monomers for PT, 18 monomers for PPy, and two monomers for PPyPV and PmPV), our molecules are better described as oligomers than polymers; however, we use the term “polymer” throughout the chapter for simplicity, with the understanding that the results of our simulation describe the behavior of a small part (block) of a “long” polymer. Energy minimization was performed to find the thermal
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Intermolecular energy (kcal/mol)
0 PT
–20
PPy
PmPV
PPyPV
–40 –60 –80 –100 –120 –140
0
200
400
600 Time (ps)
800
1000
1200
FIGURE 7.24 Intermolecular interaction energy evolution for (10, 10) single-walled CNT–polymer composites at 300 K.
chains can be illustrated by tracking the intermolecular interaction energy between the single-walled CNT and polymers within the simulation time of wrapping process. The intermolecular interaction energy is defined as the sum of all the van der Waals energy terms that is computed only between the atoms of two opposite molecules (not among the atoms within single molecules). Molecular dynamics simulations were first carried out on a (10, 10) armchair singlewalled CNT of 13.58 Å in diameter interacting with individual molecules of PT, PPy, PPyPV, and PmPV, respectively. The simulation system was equilibrated for 500 ps to stabilize the interactions. After this, the total intermolecular interaction energy between the single-walled CNTs and the polymers in equilibrium was recorded for 1 ns with interval of 1 ps for further analysis. In Figure 7.24, we present the intermolecular interaction energy evolution for the four simulation systems as a function of simulation time. The simulated polymers have comparable numbers of atoms, that is, 142, 146, 156, and 158 atoms for PT, PPy, PPyPV, and PmPV, respectively; hence, the magnitude of the intermolecular interaction energy gives us a direct measure of the strength of their binding to the single-walled CNT. As Figure 7.24 shows, although all of the polymers have an obvious attractive interaction with the nanotube, their values are found to differ profoundly. It is seen that PT exhibits the strongest interaction with the single-walled CNT, followed by PPy, PmPV, and PPyPV. The high values of these interactions energies are directly attributed to their strong π–π interactions of their aromatic rings with the nanotube surface,113,139 as the atomic arrangement of carbon atoms in aromatic rings is isomorphic to their hexagonal arrangement in graphitic sheets. The same kind of isomorphism is responsible for the strong tendency of CNTs to “stick together,” which causes a major obstacle to their processing; also this phenomenon has been called as “π-stacking.”140 A logical explanation for the difference in their interaction energies is the difference in their specific monomer structures, which has been proposed previously by the other research groups.113,114 In order to elucidate the role of the specific monomer structure on intermolecular interactions in our simulations, we focus on the ε parameters of the specific atoms (atoms that differ mostly in similar polymer structures), which is a determinant factor in this regard. According to this hypothesis, the higher intermolecular
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interaction of PT compared to PPy is caused by its higher energy parameter of sulfur atom (ε = 0.202 kcal/mol, in PT) than nitrogen atom (ε = 0.043 kcal/mol, in PPy) and the difference in the intermolecular interaction between PmPV and PPyPV is due to their different monomer structures in which one carbon atom in PmPV has been replaced by one nitrogen atom in PPyPV (Figure 7.23). Since this nitrogen atom has lower value of depth of the potential well (ε = 0.043 kcal/mol, in PPyPV) compared to carbon atom (ε = 0.056 kcal/ mol, in PmPV), this results in lower van der Waals interaction, and consequently lower intermolecular interaction; however this difference is negligible. Another reason for this difference is the better isomorphism of PmPV113 backbone with single-walled CNT surface rather than PPyPV. Since our work is fairly similar to Yang’s work113 in simulation details, we can compare a wider variety of polymers containing their investigated polymers including PS, PPA, and PPV with us. From their data at 300 K, we can find the intermolecular interaction energies of −100, −54, and −50 kcal/mol for PPV, PPA, and PS, respectively. So according to their data and their comparison with our work, in an overall comparison, we observe that PT owns the strongest interaction with single-walled CNT, followed by PPV, PPy, PmPV, PPyPV, PPA, and finally PS. The aforementioned role of specific monomer structure completely shows its determinant role in this comparison. The lateral-view snapshots of wrapped polymers on the CNT surface are given in Figure 7.25.
Single-walled CNT + PT
Single-walled CNT + PPy
Single-walled CNT + PPyPV
Single-walled CNT + PmPV
FIGURE 7.25 Molecular dynamics simulation snapshots of the wrapping of a single-walled CNT (10, 10) by different polymers (lateral view).
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Computational Nanotechnology: Modeling and Applications with MATLAB®
Intermolecular energy (kcal/mol)
0 PT
–20
PPyPV
PPy
PmPV
–40 –60 –80
–100 –120 –140 –160
0
5
10
15
20
25
CNT diameter (Å)
30
Infinity (Graphite surface)
FIGURE 7.26 Intermolecular interaction energy as a function of single-walled CNT diameter. (From Curatola, G. and Iannaccone, G., J. Appl. Phys., 95, 1251, 2004.)
7.4.2.2 Influence of the Nanotube Radius on the Intermolecular Interaction To determine the influence of the nanotube radius on polymer adhesion, molecular dynamics simulations were carried out on single-walled CNTs with different diameters. The single-walled CNT diameter in these simulations was varied from 5.49 to 27.13 Å. Figure 7.26 shows the intermolecular interaction energy versus CNT diameters at constant temperature of 300 K. When the single-walled CNT radius is increased, the surface contact area between CNT and polymer increases accordingly that causes the increase in the attractive interaction between the polymers and the single-walled CNT monotonically toward an asymptotic value given by the interaction of the polymer with a flat graphitic surface. This indicates that the curvature of the graphitic sheets that constitute the CNTs diminishes polymer adhesion; this effect also has been reported by other research groups25,26 for their investigated polymers—the more curvature the CNTs have, the lower the interface interaction energy will be. Thus, the CNTs with larger diameters are suggested as better candidates compared to CNTs with smaller diameter for nanocomposite reinforcement applications.
7.4.2.3 Influence of Temperature on the Intermolecular Interaction To assess the temperature dependence of the intermolecular interaction between the polymers and the single-walled CNTs, molecular dynamics simulations were carried out at different temperatures including polymers and single-walled CNT (10, 10). The temperature was varied between 300 and 500 K in steps of 25 K. Figure 7.27 shows the temperature dependence of the intermolecular interaction. It is seen that for PPyPV and PmPV polymers, the attractive interaction with the CNT decreases only weakly with increasing temperature; even for PT and PPy polymers this effect becomes so weaker such that it is negligible. It is due to the strong intermolecular interaction between CNT and polymer that cannot be influenced by the temperature in the range we reported; however, this effect is more significant for PT and PPy.
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Intermolecular energy (kcal/mol)
–40 –50
PT
PPy
PPyPV
PmPV
–60 –70 –80 –90
–100 –110 –120 –130 –140 275
325
375 425 Temperature (K)
475
525
FIGURE 7.27 Intermolecular interaction energy as a function of temperature.
7.4.2.4 Morphology of the Polymers on the Single-Walled CNT Surface Since the strength of interfacial binding in such systems depends on the geometrical conformation of the polymer with respect to the nanotube, we investigated the morphological aspects of polymer–CNT interactions both on the local level, where we focus on the arrangement of the aromatic rings of the polymer with respect to the longitudinal nanotube axis, and on the level of the global conformation of the polymer, which we characterize in terms of its radius of gyration. We observed that nearly all polymer backbones, that is, the aromatic rings in it, align parallel to the nanotube surface, and this significantly increases the overall CNT–polymer interaction. Such parallel π-stacking has been experimentally confirmed by atomic force microscopy for a complex of the conjugated polymer and single-walled CNTs.122 Thereby to show this parallel alignment, we investigated the relative arrangement of polymer aromatic rings with respect to the CNT longitudinal axis (X axis in our systems) and then calculated the angles between the longitudinal axis and the vector representing the plane of the aromatic rings in xy plane, in which we name this angle the “alignment angle θ.” The schematic view of the alignment angle is given in Figure 7.28. For each aromatic ring on the polymer, we fit a vector function as an indicative of aromatic ring orientation. Polymer hexagon aromatic plane
Polymer pentagon aromatic plane
(x2, y2)
(x2, y2)
θ θ (x1, y1) Nanotube longitudinal axis (x1, y1)
y
x
θ = Alignment angle
FIGURE 7.28 Definition of the alignment angle (θ) between the nanotube longitudinal axis and the plane of a polymer aromatic ring.
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TABLE 7.4 Time-Averaged Alignment Angle (θ) of Four Simulated Polymers Adsorbed to the Single-Walled CNT (10, 10) Surface at 300 K, during 1 ns of Wrapping Process System Type Time-averaged alignment angle (°)
Single-Walled CNT + PT
Single-Walled CNT + PPy
Single-Walled CNT + PPyPV
Single-Walled CNT + PmPV
0.22
0.32
7.35
4.31
The atomic configuration of the polymer atoms was recorded every 1 ps during the simulations after the equilibration, and the alignment angles were determined. Then we averaged this angle over the total simulation time for each system. The time-averaged values of θ are given in Table 7.4. As the table summarizes, the highest degree of parallel alignment to the nanotube surface is observed for PT, followed by PPy, PmPV, and PPyPV, which are in agreement with the arrangement of their intermolecular energies (Figure 7.24). To characterize the overall size of polymer chains on the CNT surface, we calculated their Rg. Rg is defined as follows: ⎛ 1 Rg = ⎜ ⎜N ⎝
N
∑(r − r i
cm
)
⎞ ⎟ ⎟ ⎠
2
i 1
1/ 2
(7.16)
where ri and rcm denote the position vector of each atom in a chain and the vector of centerof-mass for the whole chain, respectively. If Rg of a polymer chain increases, it shows the expansion of the polymer chain, while its reduction shows the collapse of polymer chain.141 Using molecular dynamics simulation of the individual polymers without CNTs, we determined the Rg values of 20.77 Å for PT, 17.71 Å for PPy, 8.05 Å for PPyPV, and 8.73 Å for PmPV. We then computed the changes of Rg of the polymers when adhered to the singlewalled CNTs of different diameters. Figure 7.29 shows the changes of Rg values versus the different single-walled CNTs in diameter. 25
Rg (Å)
20 15 10 5 PT 0
0
5
10
15
20
25
CNT diameter (Å)
PPy 30
PPyPV
PmPV
Infinity (Graphite surface)
FIGURE 7.29 Change of the radius of gyration (Rg) of the polymers after the interaction with different single-walled CNTs in diameter.
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23 21 19
Rg (Å)
17 15 13 11 9 7 5
PT 0
200
400
Time (ps)
PPy 600
PPyPV 800
PmPV 1000
FIGURE 7.30 The radius of gyration (Rg) evolution for polymers after the interaction with single-walled CNT (10, 10) at 300 K.
It is clearly seen that the Rg does not change whatsoever for PT and PPy. The reason is the high rigidity of their polymer backbones, which arises from the double-conjugated bonds within their structure. So, this specific structure does not let the polymer to twist or cause any obvious change in their total size. The Rg of PmPV and PPyPV molecules changes and shows a fluctuation as the flexible octyloxy (–OC8H17 alkane chain) chains wrap around the nanotube, which means their contribution to the changes of Rg; however, no distinctive trend is observed for them. In order to demonstrate the above description on the rigidity of PT and PPy polymers and the flexibility of PmPV and PPyPV, we calculated the changes of Rg of these polymers during the wrapping process for the same systems. Figure 7.30 shows the changes of Rg of investigated polymers interacting with single-walled CNT (10, 10) at 300 K. As it is observed, the Rg values of PT and PPy are stable and maintain their steady state, while for PPyPV and PmPV, the Rg fluctuates considerably. We also investigated the dependence of Rg of polymer chains on the applied temperature. Figure 7.31 shows the computed Rg values of polymers versus the temperature range of 300–500 K for single-walled CNT (10, 10). As Figure 7.31 shows, the higher temperature and consequently higher kinetic energy does not have any significant effect on the total polymer size due to (i) the high rigidity of polymer structures and (ii) high intermolecular interaction between polymers and CNTs. However, we observed a fluctuation in Rg values of PPyPV and PmPV, which is due to their flexible octyloxy side chains. Molecular dynamics simulation snapshots of the wrapping of a single-walled CNT (10, 10) by different polymers (cross-sectional view) are given in Figure 7.32. 7.4.3 Practical MATLAB Applications for Nanocomposite System As Equation 7.16 shows, the vector of center of mass for the whole chain is needed for calculating the gyration radius of polymers. The center of mass is equal to the sum of the atom positions times their masses all divided by the total mass. The output of molecular dynamics simulation of the above system and the basic operations and commands in MATLAB were used to calculate the gyration radius of polymers.
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25
Rg (Å)
20 15 10 5
PT
0 275
325
PPy
375 425 Temperature (K)
PPyPV
PmPV
475
525
FIGURE 7.31 The change of radius of gyration (Rg) of polymers after the interaction with single-walled CNT (10, 10) as a function of temperature.
Single-walled CNT + PT
Single-walled CNT + PPy
Single-walled CNT + PPyPV
Single-walled CNT + PmPV
FIGURE 7.32 Molecular dynamics simulation snapshots of the wrapping of a single-walled CNT (10, 10) by different polymers (cross-sectional view).
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7.5 Molecular Dynamics Simulation Study of Neon Adsorption on Single-Walled Carbon Nanotubes In this section, we review our recent molecular dynamics simulation study of neon adsorption on single-walled CNTs.142 The adsorption of various gases on CNTs is nowadays an extremely active research field of both experimental and theoretical approach due to their role as novel storage gas media. A large number of experimental studies have been carried out on the adsorption of diverse gases on various CNTs—single- or multi-walled, closedor open-ended.143–148 It has to be pointed out that the study of the adsorption behavior via experiments is difficult for such systems because the microscopic properties of such systems are often hard to determine and many properties of fluids in these media become inaccessible to experimental measurements. Therefore, molecular simulations can be considered as a good candidate for understanding and predicting the adsorption behavior of various gases on CNTs. Many molecular simulation studies have been performed so far to understand gas adsorption on CNTs, including adsorption of nitrogen and oxygen,149 xenon,150 krypton,150 methane,151 ethane,152 organic compounds,153 helium,154 hydrogen,155,156 and neon.157 Physisorption of noble gases has been investigated using theoretical methods and experimental techniques extensively.146–148,150,154,157 It has been found that opening the ends of the nanotube by chemical cutting increases both the kinetic rate and the saturation capacity of the nanotubes for rare gases.150,157 Also, adsorption isotherms of several gases on various CNTs have been investigated experimentally so far.143,146,147,158,159 All obtained Langmuir adsorption isotherms are shown to be type II at bulk adsorbate subcritical and type I or IV at supercritical temperatures. The adsorption isotherms, which have been predicted theoretically, are type I regardless of temperature, or in a few cases type IV. Physically, prediction of a type I isotherms is expected for bulk adsorbate at supercritical temperatures since multilayer adsorption is not likely to occur. However, at subcritical temperatures, multiple adsorption layers may form and wetting might occur, producing dissimilarities between theoretical predictions and experimental data (type II isotherm).149 Recently, there have been a few simulations examining the effect of the external surface of a nanotube bundle on the adsorption of hydrogen at supercritical temperatures160 and the adsorption of neon, argon, krypton, xenon, and methane at subcritical temperatures.161,162 In the latter, the adsorption isotherms were predicted to be of type II, consistent with experiment. This suggests that, to correctly predict adsorption on a finite-sized nanotube bundle, the external surface must be taken into account. In this section, neon adsorption is investigated using molecular dynamics simulations at several supercritical temperatures of neon (T = 50, 70, and 90 K) on a small finite isolated single-walled CNT with an external surface. Then using the output of molecular dynamics simulation, analytical studies of the mentioned system are presented. Adsorption isotherms, heat of adsorption, activation energy, and structural and transport properties of such a system are investigated in detail and analyzed in a different way compared to previous similar works150,157,162. 7.5.1 Simulation Method We employed an empirical force field scheme using Tinker molecular modeling package,136 similar to our recent paper,88 to describe molecular interactions to allow largescale, long-time molecular dynamics simulations.
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Since it has been investigated that the chirality of the nanotubes has no significant effect on the adsorption,155 and as the nanotubes prepared by carbon arc163 and laser ablation164 have predominantly an armchair architecture, we have selected an open-ended (10, 10) single-walled CNT with diameter and length of 13.56 and 37 Å, respectively. The nanotube is assumed to be flexible without exerting any constraint, within a periodic rectangular parallelepiped of 80.0 × 80.0 × 80.0 Å3. The lengths in the x, y, and z directions are sufficiently large to eliminate the nearest neighbor interactions with periodic images, ensuring that the finite nanotube is truly isolated. The simulation boxes contain 100 up to 1500 neon atoms, until the saturation condition provided. The saturation condition is defined as a condition in which all adsorption sites of CNT (adsorption capacity) are occupied by the first layer of fluid and increasing neon atoms (pressure) does not lead to more adsorption in this layer. Nonbonded van der Waals’ interactions were modeled by a Lennard–Jones potential with a cut-off distance of 10 Å as follows: ⎡⎛ σ ij U (rij ) = 4ε ⎢⎜⎜ ⎢⎝ rij ⎣
12 6 ⎞ ⎛ σij ⎞ ⎤ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥ ⎠ ⎝ rij ⎠ ⎥⎦
(7.17)
The values of σC–C (σNe–Ne) and εC–C (εNe–Ne) used in the simulations were, respectively, 3.40 (2.78) Å and 0.086 (0.069) kcal/mol.162,165 Interatomic Lennard–Jones potentials were calculated according to the Lorentz–Berthelot mixing rule138: (σ ii + σ jj ) 2
(7.18)
ε ij = (ε ii ⋅ ε jj )1/2
(7.19)
σ ij =
Simulations involving flexible nanotubes also require the modeling of intermolecular forces. The relative potentials of these forces are approximated on the basis of the following equation (AMBER force field)165: U intramolecular = U stretch + U angle + U dihedral
(7.20)
Each contribution to Uintramolecular is further modeled according to
and
U stretch = K bond (r − r0 )2
(7.21)
U angle = K angle (θ − θ0 )2
(7.22)
U dihedral = k dihedral ⋅ (1 + cos(nφ − φ0 ))
(7.23)
Ustretch represents the force applied when the bond is stretched from its initial position r0 to the new position r; Uangle models the force exerted when the angle θ between two bonds
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TABLE 7.5 Intramolecular Force (Equations 7.18 through 7.21) Parameters
CNT
kstretch (kcal/ mol Å−2)
r0 (Å)
kangle (kcal/ mol rad−2)
θ0 (°)
kdihedral (kcal/mol)
n
Φ0 (°)
469
1.4
63
120
3.625
2
180
FIGURE 7.33 Snapshot of initial configuration of 1500 neon atoms around a (10, 10) single-walled CNT (front view).
changes with respect to its initial angle θ0; Udihedral describes the force that atoms separated by three covalent bonds exert when they are subject to a torsion angle Φ. The n is the periodicity of torsional motion; n = 2 term describes a rotation that is periodic by 180°. The values of these parameters for the flexible CNT are reported in Table 7.5. A snapshot of a possible initial configuration is shown in Figure 7.33, consisting of 1500 neon atoms. The distance between neon atoms and single-walled CNT was chosen in a way to start the dynamics in a situation where the gas atoms were not sitting right on the Lennard– Jones minimum but still within its attractive range. Starting from this initial configuration, the molecular dynamics simulations are performed within the canonical NVT ensemble; each molecular dynamics trajectory is equilibrated during 100 ps and then propagated for another 200 ps, with a time step of 1 fs. The velocity form of Verlet algorithm method138 and the Nose–Hoover thermostat algorithm139 was used to integrate the equations of motion and temperature control, respectively. Starting from the external surface, the attractive energy decreases with increasing distance from the nanotube. Sufficiently far from the nanotube, bulk gas behavior is found, as the gas atoms do not interact with the nanotube. Consequently, the number of adsorbed atoms is the total number of gas atoms in the simulation cell with the adsorbent as corrected by subtracting the number of gas atoms behaving as bulk gas. To do so, a cut-off distance from the center of the isolated nanotube is selected, within which the gas atoms are considered adsorbed. 7.5.2 Results and Discussion For investigating the adsorption behavior of neon atoms on single-walled CNTs, we evaluate the adsorption isotherms, heat of adsorption, self-diffusion coefficient, activation energy, and RDF.
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7.5.2.1 Adsorption Isotherms In order to obtain adsorption isotherms, gravimetric storage capacity (absolute value adsorption per mass of adsorbent), ρw, was calculated as follows168: ρw =
N Ne ⋅mNe N Ne ⋅mNe + N c ⋅mc
(7.24)
where NNe is the number of adsorbed Ne atoms Nc is the number of carbon atoms in the simulation box mNe and mc (g/mol) are the corresponding molar mass Keeping temperature, diameter, and length of single-walled CNT constant and varying pressure (number of neon atoms per simulation box), adsorption isotherms were obtained at supercritical temperatures of neon, that is, T = 50, 70, and 90 K (see Figure 7.34). The exo- and endo-adsorption terms refer to the adsorption of neon on the external and internal surface of nanotube, respectively. Results reveal that gravimetric storage capacity of single-walled CNT (total, endo, and exo, according to Equation 7.24) increases with gas pressure, and on the other hand it decreases by increasing the applied temperature, which confirms that again the adsorption is a more favorable process at lower temperatures and high pressures. These observations are in good agreement with the experimental measurements for gas adsorption on single-walled CNTs144,146 and previous theoretical works149,150,168 and also the most recent work on neon adsorption.157 The amount of neon adsorbed on the outer surface of single-walled CNT is more than that on the inner side (see Figure 7.34); this also has been observed previously.150,157 Langmuir shape isotherms are predicted to be of type I at supercritical temperatures (critical temperature of neon is 44.4 K), which is in a good agreement with previous results.149,150,168 These isotherms are indicative of enhanced solid–fluid interactions, implying that condensation is prohibited in small single-walled CNT. The result is consistent with previous reports143,168 that indicate capillary condensation occurs when the pore is large enough to hold more than four layers of molecules. That is to say, in a pore with less than two layers of adsorption, capillary condensation is prohibited. Furthermore, at supercritical conditions, no capillary condensation is observed.149,168 Figure 7.35 shows snapshots of neon adsorption on the internal (endo) and external (exo) surface of nanotube at a temperature of 50 K. 7.5.2.2 Heat of Adsorption For computation of the heat of adsorption (adsorption energy),155 the total energy for each molecular dynamics run is obtained by time averaging the sum of energies in the entire simulation course and is then utilized to define the heat of adsorption using Equation 7.25. ΔEadsorption = Etube + Ne − Etube − ENe
(7.25)
where Etube and ENe are derived from running the system with nanotube (without neon atoms) and neon atoms (without nanotube), respectively. The calculated values of heat
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FIGURE 7.35 Snapshots of neon adsorption on the internal and external surfaces of a (10, 10) single-walled CNT at 50 K, (a) lateral and (b) front view.
(a)
(b)
TABLE 7.6 Heat of Adsorption Energies and Self-Diffusion Coefficients of Neon in (10, 10) Single-Walled CNT at Various Temperatures Temperature (K) Adsorption energy (kcal/mol) Self-diffusion coefficient (Å2/ps)
50
70
90
−3.913 24.74
−2.683 39.94
−2.030 52.99
of adsorption are given in Table 7.6 for saturation conditions. We assume that the highest average heat of adsorption represents the thermodynamically most favorable gas adsorption. Noteworthy that increasing the operating temperature decreases the heat of adsorption. This is fundamentally consistent with the fact that higher temperatures give the adsorbates more kinetic energy, and this, in turn, results in less chance of being adsorbed. This result is in agreement with the previous theoretical and experimental reports,155–157,169 which demonstrate that, when the gas molecules are physisorbed on the nanotube surface, increasing the operational temperature makes the adsorbed system unstable and therefore decreases the heat of adsorption. 7.5.2.3 Self-Diffusion Coefficient Various methods are available to calculate the self-diffusivity from the molecular dynamics simulations. Here, we employed the Einstein relation,169 which relates the long-time limit of the MSD of the particles to the self-diffusivity, D, through r(t + Δt) − r(t) 1 D = lim 6 t →∞ Δt
2
(7.26)
The self-diffusion coefficients were evaluated from the limiting slope of the MSD curve with time (excluding both the initial, transient ballistic motion as well as the statistically noisy final region). The corresponding MSD plots are given in Figure 7.36.
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100,000
50 K 70 K 90 K
90,000 80,000 MSD (Å2)
70,000 60,000 50,000 40,000 30,000 20,000 10,000 0
0
50
100
150 Time (ps)
200
250
300
FIGURE 7.36 MSD for neon in (10, 10) single-walled CNT at different temperatures.
It is obvious that with the increase in the temperature, the slope of MSD plots is increased, which results in higher values of self-diffusion coefficients. Therefore, the temperature shows its significant role in adsorption systems in this section, in which, the higher amounts of self-diffusion coefficient at higher temperatures cause considerable reduction in adsorption, that is, lower value of ρw (see Figure 7.34), and subsequent decrease in heat of adsorption. The values of neon self-diffusion coefficients are also given in Table 7.8. However, a large diffusion coefficient is advantageous in this regard, as it assists in the loading and unloading of gas during the duty cycle of the material.170 7.5.2.4 Activation Energy Using the Arrhenius equation171 ⎛ −E ⎞ D = D0 exp ⎜ a ⎟ ⎝ RT ⎠
(7.27)
Ea or activation energy is the potential barrier for translational motion of gas atoms. We can calculate the activation energy for the diffusion process of neon fluid. When ln D is plotted versus 1/T, calculated results show the characteristic Arrhenius behavior (linear behavior), suggesting that diffusion is an activated process as it has been shown in Figure 7.37. Indeed, we have estimated the potential barrier for translational motion of neon in such a system that is equal to 0.17 kcal/mol. 7.5.2.5 Radial Distribution Function In analyzing the structural characteristics of the adsorption systems, the RDF, g(r), provides a better understanding of the quality of the adsorption process. This function is defined as the probability of finding neon atoms at a distance r from the nanotube surface, relative to the probability expected for a completely random distribution at the same density of neon.156 The corresponding g(r) plots at saturation conditions are given in Figure 7.38. The sharp rise near 3 Å represents the distance of closest approach of neon atoms to the single-walled CNT, demonstrating purely physisorption behavior. As deduced
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–14 y = –85.474 × –13.505 R2 = 0.9997
ln D
–14.5
–15
–15.5
–16 0.009
0.011
0.013
0.015
0.017
0.019
0.021
1/T
FIGURE 7.37 Plot of ln D versus 1/T for diffusion process of neon at T = 50, 70, and 90 K. 14
50 K 70 K 90 K
12
gc-Ne (r)
10 8 6 4 2 0
0
5
10
15
20
25
30
r (Å)
FIGURE 7.38 g(r) plots of carbon–neon for (10, 10) single-walled CNT at 50, 70, and 90 K.
from the adsorption isotherms, heat of adsorption, and self-diffusion coefficients, once again, the RDF plots emphasize that the lower the temperature applied, the more is the neon adsorbed. This is in agreement with the other experimental159 and computational151,156,157,172 works that report a monotonically increase in adsorption amount while decreasing the temperature. This function approaches a value of unity in the limit of no correlation between investigated particles. 7.5.3 Practical MATLAB Applications in Adsorption of Ne Atoms on CNTs In this section, we use some applications of MATLAB to calculate and solve the equations relating to the diffusion coefficient. We used the output of molecular dynamics simulation of the above system and of some programs written in MATLAB. Basic operation and
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commands in MATLAB were used to calculate the diffusion coefficients of neon atoms, which can be written easily in MATLAB environment.
7.6 Conclusion In Section 7.2, we reviewed some important functions in MATLAB to study the behaviors of nano-oscillators from two points of view: molecular dynamics simulation and analytical studies. In Section 7.3, we studied the behavior of single-walled CNT and FCNTs with glutamine and carboxyl functional groups in water using the molecular dynamics simulation. For these systems, three methods have been considered: the intermolecular interaction energies between CNTs or FCNTs and water molecules, the RDFs, and the diffusion coefficients of CNT and FCNTs. The obtained results from the three methods are consistent with each other. The obtained RDF shows that water molecules more tend to localize around the hydrophilic groups at the end of the glutamine functional group, rather than nearer hydrophilic groups to CNT. Also, the number of water molecules surrounding the amine group of glutamine functional group is more than that for carboxyl functional group. The obtained results for FCNTs with 16 functional group carboxyl show that the relatively high compression caused by 16 functional groups prevents the water molecules from gathering around FCNT with 16 carboxyl groups. Also our results indicate that large numbers of functional groups prevent water molecules from moving toward the hydrophilic groups. Therefore, it would be better to use lower number of long-chain functional groups containing large numbers of hydrophilic groups. In Section 7.4, on the basis of obtained values of the intermolecular interaction energies from molecular dynamics simulations, we can conclude that the PT and PPy polymers adhere better to nanotubes than PPyPV and PmPV polymers. We have shown that the total polymer sizes are not altered from their lowest energy configuration by varying the CNT curvature and applied temperature, which is a direct consequence of their specific rigid structures; however different parallel alignment of their aromatic rings to the nanotube surface was observed, in which PT owned the highest one. In the end, our results suggest that CNT–polymer interactions are very strong for such heterocyclic-conjugated polymers and therefore provide strong interfacial adhesion and consequently improve the load transfer between the single-walled CNTs and polymers, which is of an essential significance for polymeric nanocomposite reinforcement. In Section 7.5, we present the adsorption isotherms of neon adsorption on single-walled CNT; all of them are of Langmuir shape type I and agree well with previous experimental and theoretical works at supercritical temperatures. More properties of neon gas such as heat of adsorption, self-diffusion coefficient, and activation energy were computed compared to previous works. Results showed that the adsorption is a process, which is very sensitive to the applied temperature, that is, increasing the temperature results in lower adsorption. Our results confirm this fact in a compatible way together. Previous works155,156 revealed that nanotube curvature may directly affect the single-walled CNT adsorbate atomistic potentials, which is not considered herein but will be the subject of future research based on the intermolecular potentials obtained from quantum chemical calculations.
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Acknowledgments The authors are grateful to Amir Taghavi Nasrabadi from the University of Tehran for his useful helps in preparing Sections 7.4 and 7.5. The authors also acknowledge Mahshad Moshari and Roghayeh Hadidi M. from the University of Tehran for performing molecular dynamics simulations.
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163. Journet, C., Maser, W. K., Bernier, P., Loiseau, A., Lamy de la Chapelle, M., Lefrant, S., Deniard, P., Leek, R., and Fischer, J. E. 1997. Large-scale production of single-walled carbon nanotubes by the electric-arc technique. Nature 388: 756–758. 164. Thess, A., Lee, R., Nikolaev, P., Dai, H., Petit, P., Robert, J., Xu, C. et al. 1996. Crystalline ropes of metallic carbon nanotubes. Science 273: 483–487. 165. Cornell, W. D., Cieplak, P., Bayly, C. I., Gould, I. R., Merz, K. M., Ferguson, Jr., D. M., Spellmeyer, D. C., Fox, T., Caldwell, J. W., and Kollman, P. A. 1995. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc. 117: 5179–5197. 166. Hoover, W. G. 1985. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 31: 1695–1697. 167. Ziff, R. M. 1998. Four-tap shift-register-sequence random-number generators. J. Comput. Phys. 12: 385–385. 168. Rafati, A. A., Hashemianzadeh, S. M., Nojini, Z. B., and Naghshineh, N. 2010. Canonical Monte Carlo simulation of adsorption of O2 and N2 mixture on single walled carbon nanotube at different temperatures and pressures. J. Comput. Chem. 31: 1443–1449. 169. Schlapbach, L. and Zuttel, A. 2001. Hydrogen storage materials for mobile applications. Nature 414: 353–358. 170. Arora, G., Wagner, N. J., and Sandler, S. I. 2004. Adsorption and diffusion of molecular nitrogen in single wall carbon nanotubes. Langmuir 20: 6268–6277. 171. Demontis, P. D., Suffritti, G. B., Fois, E. S., and Quartieri, S. 1992. Molecular dynamics studies on zeolites, temperature dependence of diffusion of methane in silicate. J. Phys. Chem. 96: 1482–1490. 172. Foroutan, M. and Taghavi, N. A. 2010. Adsorption behavior of ternary mixtures of noble gases inside single-walled carbon nanotube bundles. Chem. Phys. Lett. 497: 213–217. 173. Curatola, G. and Iannaccone, G. 2004. Two-dimensional modeling of etched strained-silicon quantum wires. J. Appl. Phys. 95: 1251–1257.
8 Device and Circuit Modeling of Nano-CMOS Michael L.P. Tan, Desmond C.Y. Chek, and Vijay K. Arora CONTENTS 8.1 Introduction........................................................................................................................ 301 8.2 Quantum Confinement..................................................................................................... 302 8.3 Transport in the Channel Electric Field ......................................................................... 303 8.4 I–V Characteristic of Nano-MOSFET..............................................................................305 8.5 MATLAB® Implementation .............................................................................................. 310 8.6 Ohmic (Linear) to Sub-Linear Transport........................................................................ 315 8.7 Non-Ohmic Circuit Behavior........................................................................................... 320 8.8 Power Consumption.......................................................................................................... 322 8.9 RC Circuit Delay ................................................................................................................ 324 8.10 Conclusion and Further Reading .................................................................................... 330 Acknowledgments ...................................................................................................................... 330 References...................................................................................................................................... 330
8.1 Introduction The metal-oxide-semiconductor field effect transistor (MOSFET) of Figure 8.1 is at the core of the design of integrated circuits, both for digital and analog applications. It has a long history of channel length scaled down, which is now in deca-nanometer regime. The fundamental processes that control the performance of a MOSFET channel continue to elude physicists and engineers alike. There is an ongoing debate over the interdependence of mobility controlled by momentum-randomizing scattering events and saturation velocity dependent on the streamlined motion of electrons [1]. Most investigators tend to converge on the theme that the saturation velocity does not sensitively depend on the low-field mobility. However, there is no consensus on the factors limiting current and velocity. The separation zQM of electrons from the interface due to quantum confinement, as shown in Figure 8.1, cannot be ignored. It is of the same magnitude as the oxide thickness. The gate electric field Et does not heat electrons as it is not an accelerating field rather it is a confining electric field that makes an electron a quantum entity described by wave character with discrete (digitized) energy levels. The wavefunction vanishes at the Si/SiO2 interface and peaks at a distance approximately zQM away from the interface. This alters the gate capacitance and hence the carrier density in the channel. Section 8.2 describes the mobility degradation due to this quantum confinement. Section 8.3 describes the charge transport driven by the longitudinal electric field E that is distinctly high in nanoscale channels. This channel electric field degrades the mobility 301
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5 4
E/εo
3 2 1 0
0
0.5
1.0
1.5
2.0
2.5 3.0 z/zo
3.5
4.0
4.5
5.0
FIGURE 8.2 Electron distribution in the first two quantized levels.
not reside at the Si/SiO2 interface as wavefunction vanishes there due to the quantumconfinement effect shown in Figure 8.2. In the nano-MOSFET the gate oxide measures a few nanometers (tox = 1.59 nm in our case). The distance zQM depends on the gate voltage. Effective oxide thickness of the gate after correction for difference in permittivity of SiO2 (εox) and Si (εSi), is given by toxeff = tox +
ε ox 1 zQM ≈ tox + zQM ε Si 3
(8.2)
where tox is the thickness of the gate oxide. The gate capacitance CG per unit area is lesser than Cox and is given by CG =
ε ox Cox = toxeff 1 + 1 zQM 3 tox
(8.3)
8.3 Transport in the Channel Electric Field The carrier drift with electric field in the channel driving the electron has been explored in a number of works. For example, Monte Carlo simulations, energy-balance theories, Green’s functions, and path integrals, are all extensively in use. However, the fundamental issues relating velocity saturation and its relation to carrier mobility and scattering interaction continue to elude those doing modeling and simulations. A number of fitting parameters are used in compact modeling of the data. The number of these parameters is increasing as technology is entering the quantum domain. The ballistic transport was predicted in the archival work of Arora [1], although not specifically named ballistic. The theory developed was for nondegenerate bulk semiconductors that gave saturation velocity comparable to the thermal velocity. Recently, the theory has been extended to embrace all dimensions under both degenerate and nondegenerate conditions [5]. The ballistic nature of the velocity is apparent from Figure 8.3, which streamlines the randomly oriented velocity vectors in a very high electric field. In the absence of
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vi
FIGURE 8.3 Random motion in zero-field transforming to a streamlined one in an extremely high electric field.
=0
≈∞
vi
2vi
any external stimulation, the carrier velocity vectors are randomly oriented; their dipole energy ±qE o in a mean free path ℓo is zero. However, as electric field is applied, the electron energy decreases by qE o for electrons drifting opposite to the electric field and increases for those drifting parallel to the electric field. This creates two quasi Fermi levels (electrochemical potentials) EF ± qE o within one mean free path on each side of the location of a drifting electron (or a hole), where EF is the Fermi energy related to carrier concentration. As conduction and valence bands and associated Fermi and intrinsic levels tilt in an electric field, electrons in antiparallel directions are favored over those in the parallel direction, finding it difficult to surmount the potential barrier with the net result that all carriers are drifting with the ballistic velocity in an infinite electric field. The associated traveling quantum waves are reflected by the apparently insurmountable barrier in an intense driving field. The limiting intrinsic velocity for three-dimensional (bulk) nondegenerate electrons is 2 2kBT/≠m* . The saturation velocity is the weighted average of the magnitude of the carrier velocity |ν| with weight equal to the probability given by the Fermi–Dirac distribution multiplied by the two-dimensional density of states. This results in saturation velocity being intrinsic velocity vi2 for a two-dimensional distribution given by vi 2 = vth 2
ℑ1/2 ( ηF 2 ) ℑo ( ηF 2 )
(8.4)
with vth 2 = vth
Γ(3/2) π vth = = Γ(1) 2
vth =
πkBT 2mt*
2kBT m*t
(8.5)
(8.6)
∞
1 x j dx ℑj ( η) = Γ( j + 1) 1 + e x − η
∫
(8.7)
0
ηF 2 =
EF 2 − Ec 2 kBT
(8.8)
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where ℑj (ηF 2 ) is the Fermi integral of order j with ηF 2 = (ε F 2 − ε c 2 )/kBT as the normalized Fermi energy with respect to the quantized bandedge ε c 2 = ε co 2 + ε o 2 εco2 is the bulk conduction bandedge that is lifted by the zero-point energy εo2 due to the quantum-confinement (QC) effect mt* is the transverse effective mass in the ellipsoidal band structure of silicon In (100) configuration, only the lower two valleys with conductivity effective mass m*t are occupied in the quantum limit.
8.4 I–V Characteristic of Nano-MOSFET The carrier density in the inverted MOSFET channel is normally degenerate. Hence, the drain velocity vD at the onset of current saturation is limited by vD =
2 3 m*t
2πns =
2 3 m*t
2πCG (VG − VT − VDsat )/q
(8.9)
where CG is the gate capacitance VG is the gate voltage V T is the threshold voltage VDsat is the drain voltage saturation The drain velocity is always smaller than that expected from infinite electric field at the drain (VDsat = VDsat1 given below for α = 1). The ultimate saturation velocity appropriate for the infinite electric field at the drain end (α = 1) is given by vsat =
2 3 mt*
2πns =
2 3 mt*
2πCG (VGT − VDsat1 )/q
(8.10)
VDsat1 itself depends on vsat requiring an iterative solution with the nondegenerate saturation velocity vsat = vth2 = 1.96 × 105 m/s as the seed value. Figure 8.4 shows the calculated vsat as a function of gate voltage. As the carrier concentration in the channel increases with the gate voltage, so does the saturation velocity. The drain-to-saturation-velocity ratio α = νD/νsat is plotted in Figure 8.5 as a function of drain voltage VD starting from the saturation point VDsat to VD = 1.0 V. In the developed theoretical framework and described elsewhere [3,5], the velocity-field characteristics in the presence of a driving electric field E of an arbitrary magnitude are given by ⎛E⎞ ⎛V⎞ v = vsat tanh ⎜ ⎟ = vsat tanh ⎜ ⎟ ⎝ E c⎠ ⎝ Vc ⎠
(8.11)
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2
Vsat (105 m/s)
1.8 1.6 1.4 1.2
FIGURE 8.4 The ultimate saturation velocity as a function of gate voltage. The saturation velocity increases as carrier concentration increases with the gate voltage.
1 0.7
0.8
0.9
VG (V)
1.2
1.1
1
0.9 0.8
α
0.7 0.6
FIGURE 8.5 The ratio α = νD/νsat of the drain velocity as a function of drain voltage beyond the onset of saturation VD > VDsat for VGS = 0.7 (topmost curve), 0.8, 0.9, 1.0, 1.1, and 1.2 (bottom curve).
0.5 0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VD (V)
with Vc =
vsat μb L , μ f = ≈ μ b e −VGT /VQG VGT μ f 1+ VQG
(8.12)
where E c is the critical electric field Vc the critical voltage that defines the onset of nonlinear behavior beyond which Ohm’s law is not valid E = V/L when voltage V is applied across the channel μℓ f is the low-field mobility that is lower than the bulk mobility because of change in the density of states VQG = 1.33 V defines the effect of quantum confinement in degrading low-field mobility The current–voltage characteristics for a MOSFET are given by [6] 1 ⎞ ⎛ V − V V CGμ f W ⎜⎝ GT 2 D ⎟⎠ D ID = V L 1+ D Vc
0 ≤ VD ≤ VDsat
(8.13)
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Device and Circuit Modeling of Nano-CMOS
where VD is the drain voltage with respect to source W is the width of the channel VGT = VG − V T . Equations 8.10 and 8.11 transform Equation 8.13 to 1 ⎞ ⎛V ⎞ ⎛ I D = CGWvsat ⎜ VGT − VD ⎟ tanh ⎜ D ⎟ ⎝ ⎝ Vc ⎠ 2 ⎠
(8.14)
The origin of VDsat1 (see below) at the onset of current saturation is normally ascribed to the velocity reaching saturation at the drain end, which is possible only if the electric field is infinitely large. In fact, with this assumption, the driving electric-field profile along the channel at the onset of current saturation is [6] E =
1
VDsat1 2L
1−
x L
(8.15)
Equation 8.15 is consistent with the assumption that the electric field is infinite at the drain end (x = L). The average electric field in the channel obtained from Equation 8.15 is VDsat/L. To ameliorate this apparent contradiction of unattainable infinite drain electric field, we take the electric field linearly rising from VDsat/2L at x = 0 as given by Equation 8.15 reaching the ultimate value 3VDsat/2L at x = L, giving an average field of VDsat/L (same as for the full saturation profile) but VDsat1 replaced with VDsat. No infinite-electric-field assumption at the drain end with onset of saturation is necessary with this formalism (Figure 8.6). The electric field in the regime VD ≥ VDsat is similarly written as E D = E ( L ) = 3
VD 2L
VD ≥ VDsat
(8.16)
The drain velocity (vD = αvsat) is now a fraction α of vsat with α given by VD 2L α= V 1+ 3 D 2L 3
5.0
(8.17)
Infinite Finite
4.0 E/(VDsat/2L)
VD ≥ VDsat
3.0 2.0 1.0 0
0
0.1
0.2
0.3 0.4
0.5 x/L
0.6
0.7
0.8
0.9
1.0
FIGURE 8.6 Electric-field profile (solid) with partial saturation (α ≤ 1) and that (dotted) with full saturation (α = 1).
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With the fraction estimated as in Equation 8.17, the current at the onset of current saturation is (8.18)
I Dsat = αCG (VGT − VDsat )Wvsat
Equations 8.13 and 8.18 must reconcile at VD = VDsat. This reconciliation gives for VDsat and IDsat the expressions VDsat =
I Dsat =
1
(2α − 1)
[(s − α)Vc − (1 − α)VGT ]
(8.19)
CGμ f W α Vc [ αVGT − (s − α )Vc ] (2α − 1) L
(8.20)
with 2
V ⎤ V ⎡ s = ⎢α + (1 − α ) GT ⎥ + 2α(2α − 1) GT Vc ⎦ Vc ⎣
(8.21)
The small-signal channel conductance gch and transconductance gm are now obtained as (Figures 8.7 and 8.8)
gch =
∂I D ∂VDS
∂I D gm = ∂VGT
= VGT const
VDS const
1 CGμ f W 2 L
VD ⎞ ⎛ ⎜⎝ 1 + V ⎟⎠ c
VD Vc = g mo , V 1+ D Vc
Transconductance (mS)
2.5
FIGURE 8.7 The channel transconductance as a function of gate voltage for channel current at full saturation with α = 1 (dashed line) and partial saturation α < 1 (solid curve).
2(VGT − VD ) − 2
VD2 Vc
(8.22)
(8.23)
g mo = CGWvsat
α<1 α=1
2.0
1.5
1.0
0.5 0.7
0.8
0.9
1.0 VG (V)
1.1
1.2
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Device and Circuit Modeling of Nano-CMOS
Conductance (mS)
8
6 4
2 0
0
1.4
IDsat (mA)
1.2
0.1
0.2
0.3 VD (V)
0.4
0.5
0.6
FIGURE 8.8 The channel conductance as a function of drain voltage 0 ≤ VD ≤ VDsat for VGS = 0.7, 0.8, 0.9, 1.0, 1.1, and 1.2. The conductance is finite at the onset of saturation (solid line). The dash lines cover the region VDsat ≥ VD ≥ VDsat1.
α<1 α=1
1.0 0.8 0.6 0.4 0.2 0.7
0.8
0.9
1.0
1.1
VG (V)
1.2
FIGURE 8.9 The drain saturation voltage for channel current at full saturation with α = 1 (dashed line) and partial saturation with α < 1 (solid curve).
With α ≠ 1, no carrier has reached the ultimate intrinsic velocity at the saturation point giving gch ≠ 0 and gm is below its limiting value gmo. In a model where α = 1 the carriers are traveling at saturation velocity with an intrinsic assumption that the drain electric field is infinite (Figures 8.9 and 8.10). Equations 8.19 and 8.20 then simplify to give ⎤ ⎡ 2V VDsat1 = Vc ⎢ 1 + GT − 1⎥ Vc ⎦ ⎣ I Dsat1 =
1 CGμ f W 2 VDsat 2 L
(8.24)
(8.25)
With α = 1, the channel conductance gch1 = 0. The transconductance in the saturation regime gm1 is 1 ⎤ ⎡ g m1 = g mo ⎢1 − 1/2 ⎥ ⎢ ⎧⎨1 + ⎛ 2VGT ⎞ ⎫⎬ ⎥ ⎢⎣ ⎩ ⎜⎝ Vc ⎟⎠ ⎭ ⎥⎦
(8.26)
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Device and Circuit Modeling of Nano-CMOS
0.8
VG = 0.5
0.7
VG = 0.9
VG = 0.7
0.6 ID (mA/µm)
VG = 0.8
VG = 0.6
VG = 1.0
0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VD (V) FIGURE 8.11 Simulated (solid lines) 45 nm MOSFET drain characteristic verified against TSMC experimental data (filled boxes) at VG = 0.6, 0.8, and 1.0 V.
FIGURE 8.12 Matrix row versus matrix column plotting.
This command runs the conventional matrix row versus matrix column iteration shown in Figure 8.12. A symmetric p-type MOSFET (PMOS) is also generated by subtracting VD and VG in Equation 8.13 from VDD of 1 V, and the maximum drain and gate voltage, respectively, are VDP = 1 − VD
(8.27)
VGP = 1 − VG
(8.28)
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Computational Nanotechnology: Modeling and Applications with MATLAB®
30 25
VG = 0.66 VG = 0.64 VG = 0.62 VG = 0.60 VG = 0.58 VG = 0.56 VG = 0.54 VG = 0.52
ID (μA)
20 15 10 5 0
0
0.1
0.2
0.3
0.4
0.5 0.6 VD (V)
0.7
0.8
0.9
1
FIGURE 8.13 Drain characteristic of 45 nm NMOS (line approaching right y-axis) and PMOS (line approaching left y-axis) from VG = 0.34–0.66 V in 0.02 V spacing. The filled circles represent the circuit current in CMOS during switching.
This will bring the PMOS drain characteristic onto the same x-axis as NMOS as a mirrored image as showed in Figure 8.13. On the other hand, the drain current versus gate voltage transfer characteristic (VTC) depicted in Figure 8.15 is generated by plotting the x-axis matrix row against y-axis matrix row as shown in Figure 8.14. h1=plot(Vd,ID([6,11,16,21,26,31,36,41,46,51],:));
where ID ([6,11,16,21,26,31,36,41,46,51],:)) points to the VD = 0.1–1.0 V. Figure 8.15 is used to calculate the DIBL and SS from the gate characteristics. The following MATLAB script is for an automated calculation of DIBL and SS as illustrated in Figure 8.16.
FIGURE 8.14 Matrix row versus matrix row plotting.
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Device and Circuit Modeling of Nano-CMOS
10–3
VD = 0.1 VD = 0.2 VD = 0.3 VD = 0.4
10–4
ID (A/µm)
VD = 0.5
VD = 0.6 VD = 0.7 VD = 0.8
10–5
VD = 0.9
VD = 1.0
10–6
0.2
0.4
0.6
0.8
1
VG (V) FIGURE 8.15 Drain current versus gate voltage characteristic of 45 nm NMOS from VD = 0.1–1.0 V in 0.1 V spacing. 0.70 0.65
VD = 1.0 VD = 0.1
0.60
VG (V)
0.55 0.50 0.45 0.40 0.35 –7
–6.5
–6 –5.5 Exponential power (base 10)
–5
–4.5
FIGURE 8.16 Gate voltage versus exponential power characteristic.
Two curves of VD = 1.0 and 0.5 V are selected. Ie1=log10(ID(6,:)); %select Vds = 1.0 V Ie2=log10(ID(51,:)); %select Vds = 0.1 V plot (Ie1,new_x, '--r', Ie2,new_x, '--b');
The first and second points of a linear plot are chosen. u=-6; %region where curve is almost linear v=u+0.301; %current where ID at v is twice of u log (2x/x) is 0.301
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Computational Nanotechnology: Modeling and Applications with MATLAB®
vv1=interp1(Ie1,new_x,u,'cubic'); vv2=interp1(Ie1,new_x,v,'cubic'); vv3=interp1(Ie2,new_x,u,'cubic');
Calculation of DIBL and SS in mV/V and mv/decade, respectively, is done by the following commands in the source code: DIBL=abs((vv3-vv1)/(1-0.1)*1000) %for Vds=1V minus Vds=0.1V. Unit is mV/V SS=abs((vv2-vv1)/(v-u)*1000) %for Vgs=0.1. Unit is mv/decade
The filled circles in Figure 8.13 represent a CMOS drain and current VTC for VG = 0.66 V (first circle) to 0.34 V (last circle). Both magnitudes are plotted versus gate or input voltage to yield the results depicted in Figure 8.17. The easiest way to do this is by using fzero to determine the VD that satisfies both NMOS and PMOS equations as depicted below. r=1; for kVg=35:67; VGS=Vin(kVg); VGSp=VGS-Vdd; VSGp=-VGSp;
%This is the switching region as both MOS saturated %Vgs for NMOS is from 0.34V to 0.66V %Vgs for PMOS is from −0.66v to −0.34V %Make VGSp positive
f=@(VDS)(A*sqrt(VGS-0.32)).*VDS./(1+VDS./0.1739))... -(B*sqrt(VSGp-0.32)).*(Vdd-VDS)./(1+(Vdd-VDS)./0.1739)); [VDS]=fzero(f, [0 1]); Vout(kVg)=VDS; r=r+1;
1.0
10.0
0.9
9.0
0.8
8.0
0.7
7.0
0.6
6.0
0.5
5.0
0.4
4.0
0.3
3.0
0.2
2.0
0.1
1.0
0.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iout (µA)
Vout (V)
end
0.0
Vin (V) FIGURE 8.17 Voltage transfer curve (left axis) and current transfer curve (right axis) with dc a gain of 15 for a 45 nm CMOS. A peak current of 9.5 μA is drawn at the gate when switching from “low” to “high” and vice versa. The inset shows a CMOS circuit used to obtain the results.
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Computational Nanotechnology: Modeling and Applications with MATLAB®
of Equation 8.29 transforms to I = ns qvdW (−ve sign for electrons and +ve sign for holes) for a sheet resistor and to I = ∓nℓqvd for a Q1D nanowire. nv is the carrier concentration per unit volume, ns per unit area, and nl per unit length. Circuit designers use the Q2D planar properties with the W/L ratio to design, characterize, and assess the electronic response of a resistor to an applied stimulus. This is considered in the prototype analysis that follows. A semiconducting sample has both electrons and holes that can contribute to the current. They move in opposite directions in response to the applied electric field, but since they are of opposite polarities, the current due to both of them is in the same direction. The total current, I = In + Ip, is thus the sum of current due to the electrons (In) and current due to the holes (Ip). The total current (I) response to the applied voltage (V) is thus given by I = In + I p =
V Ro
Ro = ρv
L Ac
ρv =
1 1 = σ (nvμ on + pvμ op )q
(8.32)
where σ = σn + σp = nvμonq + pvμopq is the sample conductivity and the sum of electron and hole conductivities (σn and σp). n(p) in the subscript denotes electrons (or holes). Ohmic resistance Ro (in ohms) is the inverse slope of the linear I–V characteristics in the Ohmic L model. Ro = ρs for a sheet resistor and Ro = ρℓL for a nanowire resistor. W Ohm’s law enjoyed its superiority in performance assessment of all conducting materials until it was discovered that the velocity cannot increase indefinitely with the increase of electric field, and eventually saturates to a value vsatn(p). A number of empirical relations to relate drift response to the uniform electric field have been tried to fit the experimental data, the most prominent being that given by [11] vDn( p ) =
−μ on( p )ε ⎡ ⎛ ε ⎞ ⎢1 + ⎜ ⎟ ⎢⎣ ⎝ ε cn ⎠
γ n ( p ) 1/γ n ( p )
⎤ ⎥ ⎥⎦
with ε cn( p ) =
vsatn( p ) μ on( p )
(8.33)
This empirical equation agrees very well with the theoretical formalism of the velocity response to the uniform electric field developed by Arora [1,3,6,12]: ⎛ ε ⎞ vDn( p ) = vsatn( p ) tanh ⎜ ⎟ ⎝ ε cn( p ) ⎠
(8.34)
The critical electric field, εc = vsat/μo for both electrons and holes, defines the boundary between the Ohmic and non-Ohmic behavior. In most operational devices, the electric field is not uniform and the present paradigm needs modification when applied to, for example, a MOSFET [13]. When the uniformly applied electric field ε = V/L is lower than its critical value εc (ε < εc = vsat/μo or ε/εc < 1), it is reasonable to establish the validity of Ohm’s law. However, as ε substantially exceeds εc , the drift velocity saturates to vdn(p) = vsatn(p) with a typical value in the neighborhood of 105 m/s. It is worth mentioning that the true saturation is unattainable as it requires an infinitely large electric field. Device breakdown will occur at a finite electric field and hence the saturation velocity can only be determined indirectly. The value determined depends on the algorithm or the theoretical framework,
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Device and Circuit Modeling of Nano-CMOS
and hence multiple values are quoted in the literature and can be modified in simulation programs according to the requirements of a designer. Simulation programs include, in addition to drift, the diffusion processes through the continuity equations are advanced to different levels of modeling (for instance SPICE levels 1–6). As devices become smaller, the importance of drift processes outweighs the diffusion processes. Diffusive transport in high electric field is discussed in [14,15]. As feature sizes approach the nanoscale and device lengths become smaller than the scattering-limited mean free path, ballistic transport prevails, as discussed in [8]. For the microresistor of Figure 8.1, Arora’s theoretical formalism [1,3,6,12] gives
⎛ V ⎞ Vcn ⎛ V ⎞ V tanh ⎜ I = I sat tanh ⎜ = = ⎟ ⎝ Vcn ⎠ Ro ⎝ Vcn ⎟⎠ Ro
⎛ V ⎞ tanh ⎜ ⎝ Vcn ⎟⎠ V Vcn
(8.35)
Vc = εcL = 0.38 V for a 1 μm is obtained from experimental values [11] of εc = 0.38 V/μm. The saturation current is Isat/W = 565 μA/μm. With low Vc = 0.38 V value for a 1 μm resistor, the current is closer to its saturation limit for any reasonable voltage applied. The last expression in Equation 8.35 clearly shows a current–voltage relation that goes beyond Ohm’s law. The factor is 1 in the long-channel limit (L → ∞), validating Ohm’s law. In the short-channel ⎛ V ⎞ V limit (L → 0) the factor tanh ⎜ approaches (V/Vcn)−1 giving the saturation current. ⎝ Vcn ⎟⎠ Vcn Such approximations in the extreme cases are useful for students to understand the transition as the voltage exceeds its critical value. Equation 8.35 thus expresses Ohm’s law being ⎛ V ⎞ V modulated by the factor tanh ⎜ . ⎝ Vcn ⎟⎠ Vcn The experimental results given in [11] conform well to the empirical expression for a micro resistor:
In =
V ⎧ 1 V ⎪ R = on ⎨ Ron ⎡ ⎛ V ⎞ γ n ⎤ 1/γ n ⎪ I sat = nv qvsat Ac ⎩ ⎢1 + ⎜ ⎟ ⎥ ⎢⎣ ⎝ Vcn ⎠ ⎥⎦
V < Vcn
(8.36)
V >> Vcn
with Vcn( p ) =
vsatn( p ) L μ on( p )
(8.37)
It is commonly believed that γ = 2 for electrons and γ = 1 for holes. However, for a 5 μm InGaAs resistive channel, it has been clearly shown that γ = 2.8. Three sheet resistors of the same material (InGaAs) but different geometries (W1/L1 = 50 μm/5 μm , W2 /L2 = 100 μm/10 μm , W3 /L3 = 100 μm/5 μm) are considered to identify the current response to an applied voltage in the Ohmic and saturation regimes. Figure 8.19 shows the current–voltage characteristics of these three resistors as predicted by the theoretical Equation 8.35. The extracted values [11], shown by markers, are in
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60
Current I (mA)
50 40 30 W1/L1 = 50 μm/5 μm W2/L2 = 100 μm/10 μm W3/L3 = 100 μm/5 μm I1 Exp I2 Exp I3 Exp
20 10 0
0
1
2
3
4
5 Volts (V)
6
7
8
9
10
FIGURE 8.19 Theoretical and experimental I–V characteristics of microresistors (W1/L1 = 50 μm/5 μm, W2/L2 = 100 μm/10 μm, W3/L 3 = 100 μm/5 μm).
excellent agreement with the theory of Equation 8.35. The experimental values of the three resistors are indirectly obtained from the I/W versus ε = V/L graph. It is apparent from Figure 8.19 that the current response of each of the three resistors is Ohmic (linear) at low applied voltages (V < Vc), and then manifests sub-linearity as the applied voltage goes beyond the critical voltage, resulting in eventual saturation. The saturated current I = nsqvsatW is proportional to the width of the resistor and depends on the doping density and the saturation velocity, but not on the length of the resistor or on scattering-limited mobility, as many earlier works have conjectured. A higher mobility does not necessarily lead to a higher saturation velocity. The resistors 1 and 2 have the same W/L or Ohmic value. The slope of these two resistors (and hence the current response) is the same as in the linear region. However, the current response diverges with larger width having a larger saturation current. The Ohmic resistance of resistor 1 is two times that of resistor 3, having the same width W. The slope of the I–V characteristic of resistor 1 in the Ohmic regime is half that of resistor 3. However, the saturation current is equal in both resistors. The approach of the current to its saturation value is faster for the shorter resistor as the voltage across the length is increased. The critical voltage depends only on the length, and not on the width, of the resistor. As I–V characteristics given by the theoretical relation of Equation 8.35 become nonlinear, the resistance R = V/I and the signal resistance r = dV/dI as a function of voltage V and the current I are given by
Rn( p ) =
V I n( p )
= Ron( p )
⎛ I ⎞ ⎛V⎞ tanh −1 ⎜ ⎟ ⎜⎝ V ⎟⎠ ⎝ I satn( p ) ⎠ c = Ron( p ) I ⎛V⎞ tanh ⎜ ⎟ I satn( p ) ⎝ Vc ⎠
(8.38)
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Device and Circuit Modeling of Nano-CMOS
rn( p ) =
dV ⎛V⎞ = Ron( p ) cosh 2 ⎜ ⎟ = Ron( p ) ⎝ Vc ⎠ dI n( p )
1 ⎛ I ⎞ 1− ⎜ ⎟ ⎝ I satn( p ) ⎠
(8.39)
2
Similar expressions [11] from the empirical Equation 8.37 are obtained as
Rn( p ) =
rn( p )
V I n( p )
⎡ ⎛ V ⎞ γ n( p ) ⎤ = Ron( p ) ⎢1 + ⎜ ⎟ ⎥ ⎥⎦ ⎢⎣ ⎝ Vc ⎠
⎡ ⎛ V ⎞ γ n( p ) ⎤ dV = = Ron( p ) ⎢1 + ⎜ ⎟ ⎥ dI n( p ) ⎥⎦ ⎢⎣ ⎝ Vc ⎠
1/ γ n ( p )
= Ron( p )
1 γ n( p ) ⎤ ⎡ ⎛ I ⎞ ⎥ ⎢1 − ⎜ n( p ) ⎟ ⎥ ⎢ ⎝ I satn( p ) ⎠ ⎦ ⎣
1+ 1/ γ n ( p )
= Ron( p )
(8.40)
1/ γ n ( p )
γ
1 γ n( p ) ⎤ ⎡ ⎛ I ⎞ ⎥ ⎢1 − ⎜ n( p ) ⎟ ⎥ ⎢ ⎝ I satn( p ) ⎠ ⎦ ⎣
1+ 1/γ n ( p )
=
+1
Rn( p ) n( p ) γ Ron( p ) n( p )
(8.41)
Figure 8.20 shows this resistance surge effect as a function of voltage as predicted by the theoretical Equations 8.38 and 8.39. The corresponding values resulting from the empirical relations, Equations 8.40 and 8.41 with γ = 2.8, are also shown. The resistance is close to its Ohmic value when V < Vc, but rises dramatically beyond Vc, with incremental (signal) resistance rising much faster than the direct resistance. Figure 8.19 shows that the γ = 2.8
Resistance (R, r) (Ohms)
104
W1/L1 = 50 µm/5 µm W2/L2 = 100 µm/10 µm W3/L3 = 100 µm/5 µm R1 Exp R2 Exp R3 Exp W1/L1 = 50 µm/5 µm W2/L2 = 100 µm/10 µm W3/L3 = 100 µm/5 µm r1 Exp r2 Exp
103
r3 Exp
102
1
1.5
2
2.5
3 3.5 4 Applied voltage (V ) (V)
4.5
5
5.5
FIGURE 8.20 Direct resistance (R) and indirect resistance (r) as a function of applied voltage from theoretical and experimental relations.
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value extracted in [11] is in excellent agreement with Equation 8.37, giving a solid foundation to the theoretical formulation presented.
8.7 Non-Ohmic Circuit Behavior As current–voltage characteristics become nonlinear, and the resistance is no longer a constant, it is natural that familiar voltage divider and current divider rules may not apply. As stated above, the current–voltage relationship is I=
Vc ⎛V⎞ ⎛V⎞ tanh ⎜ ⎟ = I sat tanh ⎜ ⎟ ⎝ Vc ⎠ ⎝ Vc ⎠ Ro
(8.42)
This version of the I–V characteristics is more compact than any other versions in the literature and hence is used in the analysis below. As the length of a resistor plays the predominant role in transforming I–V and resistive behavior, it is worthwhile to see how a voltage is divided between two resistors having the same Ohmic value (Ro1 = Ro2) but differing dimensions. When connected in series, as in Figure 8.21, an applied voltage V will be divided equally across each resistor, according to the voltage division dictated by Ohm’s law. However, when Equation 8.35 is used in the place of Ohm’s law (I = V/Ro), the voltage V1 across R1 is obtained from ⎛ V − V1 ⎞ ⎛V ⎞ Vc1 tanh ⎜ 1 ⎟ = Vc 2 tanh ⎜ ⎝ Vc 2 ⎟⎠ ⎝ Vc1 ⎠
(8.43)
with Vc1 = 1.9 V for the 5 μm resistor Vc2 = 3.8 V for the 10 μm resistor The output V1,2 across the two resistors is given in Figure 8.22 when the divider circuit is excited by a voltage source of 0–10 V. Also shown are the results expected from Ohm’s law. I
L1 = sL1 λs W1 = sW1 λs T1 = sT1 λs
V
+ –
L2 = sL2 λs
W2 = sW2 λs FIGURE 8.21 Voltage divider circuit with two microresistors (Ro1 = Ro2).
T2 = sT2 λs
R1
R2
+ V1 –
+ V2 –
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Device and Circuit Modeling of Nano-CMOS
8
W1/L1 = 50 µm/5 µm
7
W2/L2 = 100 µm/10 µm Ohmic
V1,2 (V ) (V)
6 5 4 3 2 1 0
0
1
2
3
4
5
6
7
8
9
10
Applied voltage (V ) (V)
FIGURE 8.22 Voltage division in two microresistors with the same Ohmic value (Ro1 = Ro2).
The lower length resistor (L = 5 μm) is more resistive as the voltage across the combination is increased and hence gets more voltage across it. On the other hand, the voltage drop across the 10 μm resistor is less than its Ohmic value. Figure 8.23 shows a current divider circuit where the same two resistors (Ro1 = Ro2 = 67.2 Ω) are considered in a parallel configuration. With a current per unit length of 565 mA/mm, the saturation current for resistor 1 is Isat1 = 28.2 mA and for resistor 2 is Isat2 = 56.5 mA, since the current in the saturation regime is proportional to the width of the resistor. As before, the critical voltages are Vc1 = 1.9 V and Vc2 = 3.8 V. Figure 8.24 shows that the resulting current in each resistor is substantially below its Ohmic value. Only when V < Vc, can the validity of Ohm’s law be assured. As V increases beyond Vc, the maximum current that can be drawn from the current source is 85 mA, as compared to 297.5 mA at V = 10 V as predicted by Ohm’s law. When two parallel channels are conducting, the majority of current will pass through the higher length channel even I I1
+
L1 = sL1 λs
V
+ W =s λ 1 W1 s – T1 = sT1 λs
I2
R1
V1 –
R2
+
L2 = sL2 λs
V2
W2 = sW2 λs
–
T2 = sT2 λs
FIGURE 8.23 Current divider circuit with two microresistors (Ro1 = Ro2).
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300
I1 I2
Current I1,2, I1 + I2 (mA)
250
I 1 + I2
Ohmic
200 150 100 50 0
0
1
2
3
4 5 6 7 Applied voltage (V ) (V)
8
9
10
FIGURE 8.24 Ohmic and non-Ohmic currents in a current divider circuit with two microresistors (Ro1 = Ro2).
though both resistors have the same Ohmic value. Figures 8.22 and 8.24 clearly demonstrate that the circuit laws need close scrutiny when micro/nanoresistors are encountered in VLSI circuits.
8.8 Power Consumption In the VLSI circuit design and signal propagation, the product of power and frequency plays a predominant role. It is, therefore, important to have a limited study on how deviations from the Ohm’s law will modify the power balance in a circuit. The power consumed in a resistor is given by ⎛ V ⎞ tanh ⎜ ⎝ Vcn ⎟⎠ ⎛ V ⎞ V P = VI n = VI satn tanh ⎜ = V ⎝ Vcn ⎟⎠ Ro Vcn 2
(8.44)
In the Ohmic limit (V < Vc), the usual Ohmic expression is obtained: Po =
V2 Ro
(8.45)
In the saturation limit (V >> Vc), Psat = IsatV is a linear function of the voltage. Thus, the power consumption law changes above V = Vc. This transition is clearly visible in Figure 8.25. The solid line shows quadratic behavior for both resistors. However, when current saturation is considered, the resistor of lower length (or lower Vc) consumes less power than the one of
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Device and Circuit Modeling of Nano-CMOS
Power (P) (W)
1.5
W1/L1 = 50 μm/5 μm W2/L2 = 100 μm/10 μm Ohmic
1
0.5
0
0
1
2
3
4 5 6 7 Applied voltage (V ) (V)
8
9
10
FIGURE 8.25 Power as a function of voltage for two resistors with differing Vc but same resistance Ro.
higher length (or higher Vc). Arora’s formalism [1,3,6,12], Equation 8.44, predicts the reduced power consumption in the non-Ohmic domain (V > Vc) with linear dependence on voltage, but correctly predicts the quadratic behavior in the Ohmic domain (V < Vc). Two resistors of the same Ohmic value but different dimensions, when connected in series, do not consume equal power at high voltages. This power consumption behavior is shown in Figure 8.26. The figure also shows that the total power consumed in a series circuit, according to Equation 8.45, is much less than the value predicted by Ohm’s law at high voltages. This is important information for circuit designers, as the power consumption by the devices on a chip and the resulting heat removal are important features of VLSI design. Power consumption multiplied by the frequency is a figure of merit in VLSI design; each component can be traded for optimal design once the framework is implemented (Figure 8.27). 0.8
W1/L1 = 50 µm/5 µm W2/L2 = 100 µm/10 µm Total Ohmic
0.7
Power (P1,2) (W)
0.6 0.5 0.4 0.3 0.2 0.1 0
0
1
2
3
4 5 6 7 Applied voltage (V ) (V)
8
9
10
FIGURE 8.26 Power consumed by two resistors of same Ohmic value but different geometry ratios connected in a series configuration as a function of total applied voltage.
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8 R1 (theoretical) R2 (theoretical) R1 (iteration) R2 (iteration) Ohmic
Voltage (V1,2) (V)
7 6 5 4 3 2 1 0
0
1
2
3
7 4 5 6 Applied voltage (V ) (V)
8
9
10
FIGURE 8.27 Iterative process for obtaining the voltage drop across two resistors connected in series as a function of total applied voltage. The arrows show the convergence of values obtained by iterations to the theoretically obtained values.
8.9 RC Circuit Delay It is worthwhile to examine the transit delay caused by resistive–capacitive elements of the established models in a circuit. Rise/fall time is the duration needed to charge/discharge the load capacitor through a transistor with capacitive gate. Figure 8.28 shows the corresponding equivalent RC circuit for charging process through a PMOS as effective resistor. Firstly, the IDS–VDS characteristics are obtained for PMOS and NMOS. By selecting the curve with maximum gate voltage VGS, we can obtain an empirical, tangent hyperbolic equation in this case to fit the IDS–VDS curve of PMOS/NMOS at VGS = −1 V/1 V for the charging/discharging process. Figure 8.29 demonstrates the fitting of empirical equation known as Arora’s theoretical formalism to our model [3,6] given as ⎛ V a /b ⎞ i = I sat tanh ⎜ ⎝ Vc ⎟⎠
(8.46)
where a and b are fitting parameters V is the supply voltage Effective resistor R
VDD 1V
v
RP
VGS
Vout
0V
i
c
CL
0
FIGURE 8.28 Equivalent RC circuit from the PMOS charging process.
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Device and Circuit Modeling of Nano-CMOS
100
Model Empirical
ID (μA)
80 60 40 20 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VD (V)
FIGURE 8.29 Fitting curve of our model with the empirical equation.
The total supply voltage for the RC circuit in Figure 8.28 equals the resistance voltage and the capacitor voltage: V = vR + vCAP
⎡ ⎛ i ⎞⎤ = ⎢Vc tanh −1 ⎜ ⎝ I sat ⎟⎠ ⎥⎦ ⎣
b/ a
+
q C
(8.47)
where vR is obtained from Equation 8.46 by replacement of V = vR for a given resistor. Equation 8.47 with respect to time t is differentiated to obtain the rise time b⎡ −1 ⎛ i ⎞ ⎤ ⎥ ⎢Vc tanh ⎜⎝ a⎣ I sat ⎟⎠ ⎦
b/a − 1
1
di 1 =− dt RoC ⎛ i ⎞ i 2 1 −⎜ ⎝ I sat ⎟⎠ 2
(8.48)
where Ro = VC/Isat and RoC = τo is the RC time constant. To solve Equation 8.48, integration after separation of variables yields
∫
b⎡ −1 ⎛ i ⎞ ⎤ ⎥ ⎢Vc tanh ⎜⎝ a⎣ I sat ⎟⎠ ⎦
b/ a − 1
1 di t = − + ln k i ⎞⎛ i ⎞ i ⎛ τo ⎜⎝ 1 − I ⎟⎠ ⎜⎝ 1 + I ⎟⎠ sat sat
(8.49)
where k is a constant of integration. The left-hand side (LHS) numerator in Equation 8.49 is simplified to a cubic polynomial equation:
∫
b⎡ −1 ⎛ i ⎞ ⎤ ⎥ ⎢Vc tanh ⎜⎝ a⎣ I sat ⎟⎠ ⎦
b/ a − 1
1 di 1 = Wi 3 + Xi 2 + Yi + Z di i ⎞⎛ i ⎞ i i ⎞⎛ i ⎞ ⎛ ⎛ − + + − 1 1 1 1 ⎜⎝ ⎜⎝ I sat ⎟⎠ ⎜⎝ I sat ⎟⎠ I sat ⎟⎠ ⎜⎝ I sat ⎟⎠
∫
(8.50)
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The right-hand side (RHS) of Equation 8.50 is rearranged using the partial fractions yielding Wi 3 + Xi 2 + Yi +Z B A C = + + +D i i i ⎞⎛ i ⎞ ⎛ i ⎜⎝ 1 − I ⎟⎠ ⎜⎝ 1 + I ⎟⎠ i 1 − I sat 1 + I sat sat sat
(8.51)
and the y-coordinate is redefined as b⎡ ⎛ i ⎞⎤ y = ⎢Vc tanh −1 ⎜ ⎝ I sat ⎟⎠ ⎥⎦ a⎣
b/ a − 1
(8.52)
This is plotted in MATLAB to get a curve fitting cubic expression as depicted in Figure 8.30. Coefficients W, X, Y, Z, A, B, C, and D can be computed from the cubic polynomial. With acquired coefficients A, B, C, and D, Equation 8.49 transforms to A
∫ 1−
i I sat
+
B 1+
i
+
I sat
C t + D di = − + ln k i τo
(8.53)
and ultimately reduces to (I sat + i)Isat B iC 2.718Di = ke − t/τo (I sat − i)Isat A
(8.54)
3 Real equation Polynomial fit
2.5
y
2 1.5 1 0.5 FIGURE 8.30 Approximation for real equation (LHS of Equation 8.50) and polynomial equation (RHS of Equation 8.50).
0
0
0.2
0.4
i (A)
0.6
0.8
1
×10–4
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Device and Circuit Modeling of Nano-CMOS
⎛ V a /b ⎞ When t = 0, initial current i(0) = I sat tanh ⎜ . To solve for k, Equation 8.54 is rewritten as ⎝ Vc ⎟⎠ a/b ⎞ ⎞
⎛ ⎛V I sat B C ⎡⎛ ⎛ ⎤⎡ D ⎜ I sat tanh ⎜ a/b ⎞ ⎞ a /b ⎤ ⎜⎝ ⎛ ⎞ ⎛ ⎞ V V ⎝ Vc ⎢⎜ I sat 1 + tanh ⎥ ⎢ I sat tanh ⎜⎝ V ⎟⎠ ⎟ ⎟ ⎜⎝ V ⎟⎠ ⎥ e ⎢⎝ ⎜⎝ ⎥⎣ ⎠⎠ c c ⎦ ⎦ k= ⎣ I sat A ⎛ ⎛ ⎛ V a /b ⎞ ⎞ ⎞ ⎜ I sat ⎜ 1 − tanh ⎜ ⎟ ⎝ Vc ⎟⎠ ⎟⎠ ⎠ ⎝ ⎝
⎟ ⎟⎟ ⎠⎠
(8.55)
To find the current i(t) as a function of time t, Equation 8.54 is rearranged and the fzero function is utilized to calculate the corresponding i that makes the sum of LHS and RHS of Equation 8.56 zero: (I sat + i)Isat B iC 2.718Di − ke − t/τo = 0 (I sat − i)Isat A
(8.56)
The source code is given below. tin1=0e-12:1e-12:12e-12; %for a given time, find i for kVg=1:13; t=tin1(kVg); f=@(i)(((((Isat+i).ˆ(Isat.*B)).*((i).ˆC).*(exp(D.*i)))./((Isat-i)... .ˆ(Isat.*A)))- (k).*exp(-t/t0)); [i]=fzero(f, 87e-06); %initial value vector Iout1(kVg)=i; %saving I r=r+1; end
Figure 8.31 shows the current versus time in an RC circuit derived from Equation 8.56.
1
×10–4
0.8
i (A)
0.6 0.4 0.2 0
0
0.2
0.4
0.6 Time (s)
0.8
1 –10
×10
FIGURE 8.31 The current i(t) response to an RC circuit.
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1.0
VR (V)
0.8 0.6 0.4 0.2 FIGURE 8.32 The resistor voltage in an RC circuit as a response to time.
0.0
0
0.2
0.4
0.6 Time (s)
0.8
0.6
0.8
1 ×10–10
1.0
Vcap (V)
0.8 0.6 0.4 0.2
FIGURE 8.33 The capacitor voltage in an RC circuit as a response to time.
0.0
0
0.2
0.4
Time (s)
1 –10
×10
By applying Equations 8.46 and 8.47, the following resistor and capacitor voltage responses are easily obtained as illustrated in Figures 8.32 and 8.33, respectively. As shown in Figure 8.33, the rise time of our PMOS model is approximately 26 ps with the 2 fF load capacitor. By changing the simulation time step, we are able to track the capacitor voltage response in Figures 8.34 and 8.35. The developed m-file scripts are as follows. figure (121) zero3=10; %time duration of V = 0V one3=10; %time duration of V = 1V Vzero3= zeros(1,zero3); Vone3=ones(1,one3); unit=7; %7 rising and falling curve end3=1.00e-10*unit+0.05e-10*(unit-1)+zero3*0.05e-10*3+... one3*0.05e-10*4; tout3=0:0.05e-10:end3;
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Device and Circuit Modeling of Nano-CMOS
1.2 1.0
Vcap (V)
0.8 0.6 0.4 0.2 0.0 –0.2
0
0.2
0.4
0.6
0.8
1
1.2 ×10–9
Time (t)
FIGURE 8.34 RC waveforms with large timescale.
1.2 1.0
Vcap (V)
0.8 0.6 0.4 0.2 0.0 –0.2
0
0.2
0.4
0.6
0.8
Time (t)
1
1.2
1.4
×10–10
FIGURE 8.35 RC waveforms with smaller timescale.
vout3=[Vone3 1-Vcap2 Vzero3 Vcap2 Vone3 1-Vcap2 Vzero3 Vcap2 Vone3 1-Vcap2 Vzero3 Vcap2 Vone3 1-Vcap2]; h1=plot (tout3, vout3,'b'); figure (122) xi4 = 0:0.001e-10:0.134e-10; %time needed to achieve 0.63V Vcap4 = interp1(xi2,Vcap,xi4,'spline'); zero4=0; %time duration of V = 0V one4=0; %time duration of V = 1V Vzero4= zeros(1,zero4); Vone4=ones(1,one4); unit=10; %10 rising and falling curve tout4=0:0.001e-10:0.134e-10*unit+0.001e-10*(unit-1); vout4=[ Vcap4 0.63-Vcap4 Vcap4 0.63-Vcap4 Vcap4 0.63-... Vcap4 Vcap4 0.63-Vcap4 Vcap4 0.63-Vcap4]; h1=plot ( tout4, vout4,'b');
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8.10 Conclusion and Further Reading MATLAB is a powerful software tool used in device modeling and circuit simulation. A simple nano-MOSFET modeling is described in this chapter to obtain its drain current– voltage characteristics. MATLAB function subroutines can be used to transform IDS–VDS graph to obtain IDS–VGS graph easily to further investigate the performance of an established model. In addition, this chapter also demonstrates a procedure to extract the transit time delay with different simulation timescales. The reader is directed to Refs. [16,17] on microcircuit modeling. It is beyond doubt that MATLAB is a simple yet powerful tool in device simulation after the model is developed. The limitation of multiple intersection points in fzero function can be identified by defining a seed value intelligently or by plotting the function plot and identifying the zero-crossing point of the curve. HSPICE is certainly useful for efficiency and accuracy. However, as new concepts are developed, changing HSPICE may pose problems. In this situation, MATLAB comes handy.
Acknowledgments All figures and calculations in this chapter were performed using MATLAB. The soft copies of the files can be requested by sending an e-mail to the authors: Michael L. P. Tan ([email protected]); Desmond C. Y. Chek ([email protected]); or Vijay K. Arora ([email protected]).
References 1. V. K. Arora, High-field distribution and mobility in semiconductors, Japanese Journal of Applied Physics, 24, 537–545, 1985. 2. A. Rothwarf, A new quantum mechanical channel mobility model for Si MOSFET’s, IEEE Electron Device Letters, EDL-8(10), 499–502, 1987. 3. V. K. Arora, Quantum engineering of nanoelectronic devices: The role of quantum emission in limiting drift velocity and diffusion coefficient, Microelectronics Journal, 31(11–12), 853–859, 2000; A. M. T. Fairus and V. K. Arora, Quantum engineering of nanoelectronic devices: The role of quantum confinement on mobility degradation, Microelectronics Journal, 32, 679–686, 2000. 4. F. Stern, Self-consistent results for n-type Si inversion layers, Physical Review B, 5, 4891– 4899, 1972. 5. V. K. Arora, Failure of Ohm’s law: Its implications on the design of nanoelectronic devices and circuits, in Proceedings of the IEEE International Conference on Microelectronics, May 14–17, 2006, Belgrade, Serbia and Montenegro, pp. 17–24. 6. V. K. Arora and M. B. Das, Effect of electric-field-induced mobility degradation on the velocity distribution in a sub-mum length channel of InGaAs/AlGaAs heterojunction MODFET, Semiconductor Science and Technology, 5(9), 967–973, 1990. 7. V. K. Arora, M. L. P. Tan, I. Saad, and R. Ismail, Ballistic quantum transport in a nanoscale metaloxide-semiconductor field effect transistor, Applied Physics Letters, 91, 103510 (3pp), September 2007.
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8. D. C. Y. Chek, M. L. P. Tan, and V. K. Arora, Nano-CMOS circuit design and performance evaluation by inclusion of ballistic transport processes, in 2009 International Semiconductor Device Research Symposium (ISDRS 2009), College Park, MD, 2009. 9. D. W. Zhang, H. Zhang, Z. P. Yu, and L. L. Tian, A unified charge model comprising both 2D quantum mechanical effects in channels and in poly-silicon gates of MOSFETs, Solid-State Electronics, 49, 1581–1588, October 2005. 10. B. L. Anderson and R. L. Anderson, Fundamentals of Semiconductor Devices, McGraw Hill, New York, 2005. 11. D. R. Greenberg and J. A. del Alamo, Velocity saturation in the extrinsic device: A fundamental limit in HFET’s, IEEE Transactions on Electron Devices, 41, 1334–1339, 1994. 12. M. T. Ahmadi, R. Ismail, and V. K. Arora, The ultimate ballistic drift velocity in a carbon nanotubes, Journal of Nanomaterials, Article ID Number 769250, vol. 2008, 8pp. doi: 10.1155/2008/769250, 2008. 13. M. L. P. Tan, V. K. Arora, I. Saad, M. Taghi Ahmadi, and R. Ismail, The drain velocity overshoot in an 80 nm metal-oxide-semiconductor field-effect transistor, Journal of Applied Physics, 105, 074503, 2009. 14. V. K. Arora, Drift-diffusion and Einstein relation for electrons in silicon subjected to a high electric field, Applied Physics Letters, 80(20), 3763–3765, 2002. 15. V. K. Arora, Theory of scattering-limited and ballistic mobility and saturation velocity in lowdimensional nanostructures, Current Nanoscience (CNANO), 5, 227–231, 2009. 16. M. L. P. Tan, I. Saad, R. Ismail, and V. K. Arora, Enhancement of nano-RC switching delay due to the resistance blow-up in InGaAs, Nano, 2, 233–237, 2007. 17. T. Saxena, D. C. Y. Chek, M. L. P. Tan, and V. K. Arora, Microcircuit modeling and simulation beyond Ohm’s law, IEEE Transactions on Education, 54, 34–40, 2011.
9 Computational and Experimental Approaches to Cellular and Subcellular Tracking at the Nanoscale Zeinab Al-Rekabi, Dominique Tremblay, Kristina Haase, Richard L. Leask, and Andrew E. Pelling CONTENTS 9.1 Introduction........................................................................................................................ 333 9.2 Nanoscale Structures in the Cell and Cellular Nanomechanics ................................ 335 9.2.1 Cell Membrane ....................................................................................................... 335 9.2.2 Cytoskeleton ........................................................................................................... 336 9.2.2.1 Actin Filaments ....................................................................................... 336 9.2.2.2 Microtubules............................................................................................ 337 9.2.2.3 Intermediate Filaments .......................................................................... 337 9.2.3 Focal Adhesions ..................................................................................................... 337 9.3 Mechanically Tunable Polymer Substrates for Cell Culture......................................... 338 9.3.1 Gelatin (GXG) Substrates ...................................................................................... 338 9.3.2 Polydimethylsiloxane Substrates ......................................................................... 339 9.3.3 Strategies for Biofunctionalization...................................................................... 339 9.4 Experimental and Theoretical Consideration.................................................................340 9.4.1 Atomic Force Microscope and Nanomechanical Characterization................340 9.4.2 Force Curve and the Hertz Model....................................................................... 341 9.5 Simultaneous Atomic Force and Optical Microscopy..................................................343 9.5.1 Simultaneous AFM and Optical Microscopy ....................................................343 9.5.2 Optical Tracking of Subcellular Organelles in Response to External Forces .............................................................................................. 343 9.5.2.1 MATLAB® Code ......................................................................................344 9.5.3 Traction Force Microscopy, Optical Registration Algorithm, and Particle Tracking ............................................................................................ 350 9.5.3.1 Bead Displacement within the Cell Contact Area.............................. 352 9.6 Conclusion .......................................................................................................................... 357 Acknowledgments ...................................................................................................................... 357 References...................................................................................................................................... 358
9.1 Introduction Mechanical responses of the cell are essential for numerous cellular processes such as cell differentiation, cell migration, and gene expression. Cells are constantly subjected to mechanical stresses, for example, blood flow on the surface of endothelial cells (Shyy and Chien 1997), expansion and contraction produced by cardiac myocytes, and shear 333
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forces due to fluid flow (Dewey et al. 1981; Gimbrone et al. 1981; Galbraith et al. 1998). These mechanical stresses are hypothesized to have profound effects on cellular growth, motility, differentiation, and adhesion. An important goal of cellular nanomechanics is to understand the nanomechanical responses of cells to chemical and physical stimuli and how to use these responses to control cellular behavior. Experimental techniques to study cell nanomechanics have included micropipette aspiration (Hochmuth 2000), cell poking (Petersen et al. 1982), atomic force microscopy (AFM) (A-Hassan et al. 1998), optical tweezers (Sleep et al. 1999), and optical stretching (Guck et al. 2005). These techniques provide quantitative information about the cell by measuring nanomechanical properties directly and investigating the response of cells to applied forces and stresses, through live-cell microscope imaging of cytoarchitectural deformation. Quantifying structural remodeling from microscopic imagery is therefore an important goal that can be achieved with several computational analysis techniques, which will be outlined in this chapter. Cells are also sensitive to mechanical properties of the microenvironment in which they find themselves. Specifically it has been demonstrated that the stiffness, or Young’s Modulus, of the substrate on which cells are growing can dictate the stem cell fate (Engler et al. 2006), the formation of new muscle fibers (Engler et al. 2004; Discher et al. 2005; McDaniel et al. 2007; Zhang et al. 2009), the progression of apoptosis (Wang et al. 2000; Pelling et al. 2009) and cell adherence, migration, and proliferation (Dembo and Wang 1999; Lo et al. 2000; Bershadsky et al. 2003). Many of these cellular responses to the mechanical microenvironment are driven by the ability of cells to pull and push on their substrate. These cellular forces are known as cellular traction forces (CTFs) and are driven in large part by the underlying architecture of the cell. The feedback between the stiffness of the substrate and the magnitude of the forces the cell exert have profound impacts on cellular behavior (Lo et al. 2000; Balaban et al. 2001; Beningo et al. 2001; Munevar et al. 2001; Cai and Sheetz 2009). Therefore, measuring and quantifying CTF have important biomedical relevance. In a technique known as traction force microscopy (TFM) (Dembo and Wang 1999; Lo et al. 2000; Munevar et al. 2001; Beningo et al. 2006), cells are cultured on flexible polymer substrates in which nanoscale fluorescent beads have been embedded as fiduciary markers. As cells exert “pulling” and “pushing” forces on the flexible substrate, the marker beads move proportionately. Computational image routines that are capable of tracking fiduciary marker displacements allow one to determine the magnitude of the exerted forces (Dembo and Wang 1999). Computational cellular image analysis routines are methods that quantify objects, distances, concentrations, velocities of cells, and subcellular structures (Eils and Athale 2003). Many microscopic technologies provide us with vast cytoarchitectural information that needs to be quantitatively analyzed through computational image processing tools in order to extract fruitful information. This can include studies on protein and vesicle kinetics (Singh et al. 2005) and biochemical signaling networks that, in turn, may provide future insight into the mechanochemical transduction pathways in cells (Maintz and Viergever 1998). Numerous post-processing techniques in tracking and measuring movement have been developed such as optical flow registration algorithms (Dembo and Wang 1999; Lo et al. 2000; Munevar et al. 2001), particle tracking (Massiera et al. 2007; Jin et al. 2008; Silberberg et al. 2008), and other image processing techniques utilized in the post-processing step. Generally, these methods involve the registration of static digital images, which can be processed by ImageJ (http://rsbweb.nih.gov/ij/) or other software and dynamic image registration, which involve a temporal dimension. The latter contains fruitful information of dynamical change over time and scripts or macros in image processing software as
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MATLAB® or others are composed for efficient analysis of these dynamics. Some specific examples in cellular image processing with MATLAB are described in this chapter. The process of live-cell imaging begins with registering a digital presentation of the studied structure (nucleus, focal adhesions [FAs], cytoskeleton, mitochondria, etc.) usually by employing an array of fluorescent-labeling strategies. Microscopic acquisition is soon followed by computer-based image manipulation schemes such as thresholding, enhancement, segmentation, filtering, masking, or contouring. Some of the major image analysis techniques include visualization, tracking, and statistical analysis. The image processing toolboxes in MATLAB and custom-built software or scripts can be used to quantify the static and dynamic images in order to quantify the structural dynamics within the living cell. In successive sections, this chapter describes important architectural structures in the cell, experimental approaches to studying cell mechanics, and the image analysis routines, which are employed to analyze the data from these experiments. Finally, the chapter provides examples of computational algorithms and post-processing techniques used to understand the dynamical behavior of living cells.
9.2 Nanoscale Structures in the Cell and Cellular Nanomechanics 9.2.1 Cell Membrane The cell as we know it today is an intricate assemblage of diverse parts, all working in harmony and sustaining a considerable degree of internal order. Moreover, as a functional unit, it is separately maintained from its external environment by a boundary. As a prerequisite to the evolution of life, this boundary was in fact formed for preserving the integrity of the cell and essentially protecting it from physical forces and chemical changes in the microenvironment (Singer and Nicolson 1972; Lodish and Berk 2008). The properties of the cell membrane depend on both the composition of the molecules it is constructed from and the nature of the molecules that the cell contains. The two major organic types of molecules that constitute the cell are either water-soluble (hydrophilic) or water-insoluble (hydrophobic) molecules. However, many proteins found in the cell are those that possess both hydrophilic and hydrophobic chemical groups (i.e., amphipathic molecules). The most popular of the amphipaths are the phospholipids. They are elongated molecules about 2–3 nm long and contain one hydrophilic end and one hydrophobic end (Warren 1987). They are insoluble in water; nevertheless, due to their distinct structure, they can easily self-assemble to hide their hydrophobic groups and form a phospholipid bilayer, where two monolayers are in contact along their hydrophobic ends and only the hydrophilic sides are exposed to the aqueous surroundings (see Figure 9.1). This structure is the major component of the cell plasma membrane and is found embedded with many proteins that are required for transport across the membrane, signaling, and recognition. The plasma membrane is a semi-permeable structure, which controls the movement of substances in and out of the cells through either active or passive transport. Moreover, it contains proteins and lipids, which are involved in a variety of cellular processes such as cell adhesion, ion channel conductance, and cell signaling, all crucial in the transduction mechanism (Warren 1987). In addition, the membrane also serves as the attachment point for the cytoskeleton.
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Moreover, tertiary structures can be formed including fiber bundles (stress fibers) or a threedimensional lattice–lattice-like structure formed from the actin filaments with the involvement of various actin-binding proteins such as Arp2/3, fimbrin, and α-actinin (Burridge and Chrzanowska-Wodnicka 1996). Actin is believed to be one of the primary structural components of most cells. The actin cytoskeleton also responds actively to external forces and is crucial during the migration process in the formation of the leading-edge lamellipodium protrusions. 9.2.2.2 Microtubules They are the second major ingredient of the filamentous cytoskeleton. They primarily consist of α- and β-tubulin monomers, which polymerize into filaments. Both filament monomers have a molecular weight of 55 kDa and they arrange themselves into a small hallow cylinder during polymerization. The outer diameter of the cylindrical tube is about 25 nm, and interestingly microtubules have the same effective Young’s Modulus as actin (1–3 Pa); in addition these, filaments have a persistence length of about 6 μm (Gittes et al. 1993). During mitosis, microtubules form mitotic spindles, which are an essential structure for the separation of chromosomes into daughter cells (Snyder and Mcintosh 1975). 9.2.2.3 Intermediate Filaments These filaments are part of a subfamily of proteins containing more than 50 different members and have an average diameter of ∼10 nm. The common structure they share is the central α-helical domain, which consists of over 300 residues that form an entangled coil. The dimers assemble themselves into a staggered array forming tetramers that connect end-to-end forming protofilaments. These in turn organize into ropelike structures, where each contains eight protofilaments with an average persistence length of about 1 μm (Mucke et al. 2004). Intermediate filaments are relatively stable, and they are involved in providing tensile strength for the cell. In addition, they may be involved in specialized cell–cell junctions (Herrmann et al. 2007). For example, lamins, one of the various types of intermediate filaments form filamentous support inside the inner nuclear membrane; therefore, they are vital to the reassembly of the nuclear envelope after cell division (Georgatos and Blobel 1987; Herrmann et al. 2007; see Figure 9.2). 9.2.3 Focal Adhesions The structures linking cells and the extracellular matrix (ECM) are FAs (Burridge et al. 1988). FAs consist of several cytoplasmic proteins such as talin, vinculin, zyxin, and others (Burridge and Chrzanowska-Wodnicka 1996) that act to chemically and mechanically couple integrins to the actin cytoskeleton that are linked to the ECM. FAs are usually formed minutes after cultured cells come into contact with an ECM-coated surface (Nobes and Hall 1995). These complexes are generally small and located at the boundary of a spreading cell or at the leading edge of a migrating cell (Nobes and Hall 1995; Kiosses et al. 2001). The FAs are able to mechanically transmit forces (cell traction forces) from the cytoskeleton to the ECM and vice versa. It has been shown that larger FAs will transmit larger forces to the ECM (Ridley and Hall 1992; Balaban et al. 2001) as the application of an external force causes FAs to increase in size, remodel, and strengthen their adhesion to the substrate (Felsenfeld et al. 1999; Riveline et al. 2001; Galbraith et al. 2002).
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surface to change its affinity with the ligand or use various protein coatings that have an affinity with both the cells and the treated or non-treated surfaces. PDMS is relatively chemically inert, and the –CH3 group exposed on its surface makes the elastomere very hydrophobic. Its hydrophobic properties significantly reduce or totally inhibit cell adhesion (Cunningham et al. 2002). Fortunately, it is possible to convert the hydrophobic surface into a hydrophilic surface using an oxygen plasma treatment (Wang et al. 2009). Such treatment introduces silanol groups (Si-OH) on the surface from the oxidation of existing methyl groups (Si-CH3) (Ferguson et al. 1993). The most common protein coatings on PDMS are ECM proteins such as fibronectin and collagen. In addition to enhancing cell adhesion, they will mimic the physiological substrate found in vivo than a bare silanol PDMS surface alone. Moreover, it is possible to change surface charge by coating with charged molecules like poly-d-lysine and lecithin and vary the wettability of the surface (Wang et al. 2009).
9.4 Experimental and Theoretical Consideration 9.4.1 Atomic Force Microscope and Nanomechanical Characterization AFM is a technique that scan surface with a small probe to generate topographical images at high resolution (0.1–10 nm) (Binning et al. 1986). Initially, this tool was mainly used to image and characterize nonconducting surfaces and investigate binding interactions between molecules tethered to the surface and the small scanning probe. The AFM has now become a very useful tool in the biological sciences. With the ability to operate in liquid as well as in air, the AFM can be used in cell culture conditions and provide mechanical and physical information about single and/or a collection of cells. Typically, the probes that are used to scan surfaces have the shape of a cone, a halfsphere, or a pyramid. The probe is attached at the tip to a flexible cantilever, which is mounted at a piezoelectric crystal with the ability to move in three dimensions. The tip can be approached to the sample and scanned vertically and horizontally the surface. A laser is reflected off the tip of the cantilever and as the cantilever deflects over the topography of the surface, the position of the reflected laser beam varies on a photodiode. If the spring constant (kc ) of the cantilever is known, the AFM can be used to monitor forces applied to a cell by the tip or generated by the cell acting on the cantilever. Various techniques have been employed to measure the spring constant of the cantilever; the thermal noise method (Hutter and Bechhoefer 1993), measuring the resonance frequency (Sader et al. 1999), or using the intrinsic material properties of the cantilever (Cleveland et al. 1993). Determining the spring constant of the cantilever allows the force acting on it to be easily calculated (refer to Section VII.B). This technique provides information on cellprobe adhesion (Florin et al. 1994) or cell stiffness (Radmacher et al. 1995) using an indentation model, which reflects mainly the mechanical properties of the cytoskeleton (refer to Section 9.4.2). The AFM has been employed in many studies of cell nanomechanics such as detecting the mechanical variations in lamellopodium as a function of the distance from the edge (Laurent et al. 2005). Martens and Radmacher (2008) investigated the mechanical properties of fibroblast cells after adding the myosin inhibitor blebbistatin and the Rho-kinase inhibitor Y-27632 to uncover the effect of myosin to cell stiffness. It has also been used in
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mechanosensitive experiments where applying stain to osteoblasts initiated intracellular calcium response (Charras and Horton 2002). In addition to applying forces, the AFM can also be used to measure forces as it has been used to measure protrusive forces at the edge of migrating cells (Prass et al. 2006; Radmacher 2007) and the contraction of the cardiomyocytes (Domke et al. 1999; Pelling et al. 2004). 9.4.2 Force Curve and the Hertz Model In an AFM indentation experiment, a so-called force curve can be measured to determine the material properties of a sample. In this measurement, the tip is brought in and out of contact with the sample and the deflection of the cantilever is recorded (Radmacher et al. 1996). Hence, the deflection d is related to the loading force (F) by the spring constant (kc) of the cantilever. Moreover, Hooke’s law can be used because the cantilever deflection is linear for small deflections: F = kc d
(9.1)
When performing experiments, the deflection may not be exclusively zero (when out of contact) because there may be stresses in the cantilever, which may deform it even if no external load exists and hence an offset d0 must be subtracted from all deflections (Radmacher et al. 1995; Radmacher et al. 1996), and thus Equation 9.1 becomes F = kc (d − d0 )
(9.2)
On very flexible substrates, the tip indents the sample to some extent due to the application of the loading force and hence the indentation (δ) is given by the difference between the sample height and the deflection of the cantilever d: δ = z−d
(9.3)
Once again, the offsets have to considered for both the height and the deflection, and Equation 9.3 becomes δ = ( z − z0 ) − (d − d0 )
(9.4)
Here d0 is the deflection offset z0 is the z position at the point of contact between the tip and the sample We can apply the Hertzian model, which describes the indentation of an infinitely extended sample by a simplistically shaped tip. Moreover, the two most common tips used in the AFM have conical and parabolic shapes. In the case of both indenters, the loading force can be calculated from the material properties of the sample (Young’s modulus, E and Poisson ratio, v), the opening angle of the tip, α, and the indentation: Fconical =
2 E δ2 tan(α) π (1 − υ2 )
(9.5)
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Fparabolic =
4 E δ 3/ 2 R 3 (1 − υ2 )
(9.6)
where R is the radius on a spherical indenter. The Poisson ratio ν can have values ranging between 0 and 0.5. Generally, a value of 0.5 is given to incompressible samples and 0.3 for soft materials as cells. Moreover, Equations 9.5 and 9.6 assume that the material is homogenous, isotropic, and infinitely passive (Radmacher et al. 1995; Carl and Schillers 2008). When using the Hertz model to calculate the Young’s modulus, E, of very soft samples, one of the most challenging issues is determining the position of the contact point and identifying the portion of the curve, which is well described by the Hertz model. In reality, the Hertz model is based on assumptions, which are not necessarily true for a living cell (Radmacher et al. 1995; Carl and Schillers 2008). For example, assuming that the cell acts as a viscoelastic, isotropic, and infinitely thick material is not realistic. Therefore, it is not surprising that only the shallow indentation portion (typically <500 nm) of the force– displacement curve is typically in agreement with the Hertz model. Determining the appropriate Hertz model can be verified by plotting the natural logarithm of the force (F) versus the natural logarithm of the indentation (δ) and calculating the value of the slope of the linear portion, which will have a slope approaching the values 3/2 or 2. With a value of 2 and 3/2, the appropriate model to use is the conical and parabolic model, respectively. This verification will confirm the model to use even if each model is associated to an indenter shape. This verification becomes important as a conical indenter may become contaminated or damaged during handling and behave as a parabolic indenter. Once the appropriate model is chosen, a rearrangement of the equation of the models will plot a linear relationship between F2/3 or F1/2 and δ in function of the appropriate model (Carl and Schillers 2008): ⎛4 ⎞ E Parabolic : F 2/3 = ⎜ R⎟ ⎝ 3 (1 − υ2 ) ⎠ ⎛2 ⎞ E Conical: F 1/2 = ⎜ tan(α )⎟ ⎝ π (1 − υ2 ) ⎠
2/3
δ
(9.7)
δ
(9.8)
1/2
The value of the slope can be calculated graphically and the Young’s modulus value, E, can be easily extracted from each model. MATLAB can be used to obtain the elasticity of the sample from the force–displacement curves measured with the AFM. Generally the data are provided as a text file, and it can be easily loaded into MATLAB. However, one may still be required to determine the contact point (z0; Equation 9.4), which is where the AFM touches the surface of the material. There have been several approaches in determining this contact point (Burnham et al. 1993; Domke and Radmacher 1998; Heinz and Hoh 1999). Considering z0, all data points prior to it are ignored since that is the region where the tip is not in contact with material. All points beyond the contact point are considered up to a user-specified indentation, and the data are fitted with the appropriate Hertz model to determine E.
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9.5 Simultaneous Atomic Force and Optical Microscopy 9.5.1 Simultaneous AFM and Optical Microscopy High-resolution AFM can be performed simultaneously with optical techniques such as fluorescence, confocal, and phase-contrast microscopy. The combined techniques provide complementary information about the sample, which establishes the basis for a better understanding of cellular processes and functions under physiological conditions (Haupt et al. 2006). Integrating AFM and other optical techniques can provide a more comprehensive study of biological systems. For example, a simultaneous AFM and optical microscope consists of an AFM being positioned onto an inverted microscope and simply adding a CCD camera to the side port of the microscope and linking it to a PC. Therefore, in this configuration, it is possible to perform simultaneous AFM measurements on fixed or live cells, under physiological conditions with light, DIC, phase contract, epifluorescence, confocal, or total internal reflection fluorescence (TIRF) microscopy (Haupt et al. 2006). For example, in simultaneous AFM and TFM, the AFM is mounted unto an epi-fluorescence microscope, and simultaneous AFM measurements and sequential fluorescence and phase-contrast images are recorded and registered for a span of 120 s on live cells, which are plated on flexible substrates with fluorescently embedded beads in GXG substrates. In addition, one can combine AFM and confocal microscopy in tracking filamentous actin, microtubules, intermediate filaments, vinculin, or zyxin when working with cells transfected plasmids encoding these proteins linked to green fluorescent proteins (GFPs) or red fluorescent proteins (RFPs). Such experiments allow researchers to track the filamentous movements of actin or the complicated entangled polymer-like microtubules when mechanically perturbating the cell. Determine local regions in the cell, where FAs grow, assemble, or remodel when poking the nucleus; and, hence, by measuring the intensity field of those bright regions (RFP-zyxin or GFP-vinculin), we may understand the mechanotransduction pathways involved in transducing mechanical stimulation into a biochemical signal. 9.5.2 Optical Tracking of Subcellular Organelles in Response to External Forces Single-point tracking (SPT) has been successfully used for different biological applications, including tracking the motion of molecules, proteins, and viruses and investigating intracellular organelles over time (Saxton and Jacobson 1997). Moreover, feature-point tracking is an SPT algorithm that provides an efficient automated two-dimensional way of tracking the trajectories of particles in motion (Sbalzarini and Koumoutsakos 2005). In this method, the center of the particle is identified and tracked by detecting the positions of particles in a digital video sequence that generates particle trajectories over time. Quantitative data can be extracted from the particle trajectories, which contain useful parametric information such as the velocity of the particle, the direction of motion, and the nature of the motion (random Brownian motion or directed motion) (Saxton and Jacobson 1997). Therefore, this method is advantageous because no assumptions are made regarding the smoothness of the trajectories and the algorithm accepts different modes of motion for single trajectories. The above is useful for many biological applications where the type of motion is not known or where the motion is random and changes abruptly.
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Optical flow registration is used to determine the overall trajectory of motion from one image to another in the form of a displacement field. This field is defined throughout the image and is modeled as a time-dependent viscous fluid flow. Force vectors are derived in order to improve the image similarity between two images and the response of the fluid to the forces is obtained by solving the Navier–Stokes equation for a compressible viscous fluid. In this method, flow registration maximizes a measure of the image similarity so that a perfect registration results in two structurally identical images, with the relative motion between the image features encoded in the displacement field. The Navier–Stokes equation determines the velocity field of a flow with an associated viscosity and applied force. Moreover, the velocity field is determined from the gradient of the image-similarity measure and is a function of the actual displacement field. The image similarity is obtained from the sequence of the identity correlation coefficient (Freeborough and Fox 1998). Using SPT and optical flow registration approaches, Silberberg et al. (2008) focused on the motion of fluorescently tagged mitochondria in live cells. NIH-3T3 fibroblast cells were stained with MitoTracker Red, used to specifically target live mitochondria and fluoresces once it is in contact with the mitochondrial membrane, which proves to have a high signalto-noise ratio. The displacements of individual mitochondria were tracked over time. The team quantified the mitochondrial displacements by tracking sequential image frames with the cell at rest (first and second frame) and when perturbed (second and third frame) with the AFM tip. They found that ∼80% of the cells displayed an increase in mitochondrial displacement and ∼20% showed a decrease in displacement. After the cells were pushed with 10 nN from the AFM tip, the displacement increased from 114 ± 6 to 160 ± 10 nm; therefore, this implied that locally applied forces over the nucleus induced statistically significant rearrangement of mitochondria at the cell edges, increasing by 39.7% following nano-indentation. In order to investigate the cytoskeleton’s role in transmitting the forces, Silberberg et al. (2008) used cytochalasin D (CytD) and nocodazole to disrupt the actin and microtubule networks, respectively. They found no statistical significant when the pre- and postperturbation displacements with wither drugs. This suggested that the application of force after mitochondrial displacements depended on both the actin and microtubule networks. 9.5.2.1 MATLAB® Code In this section, we present an example of a MATLAB script for multiple particle tracking. In the past, various techniques were used for this type of tracking such as crosscorrelation of subsequent images (Gelles et al. 1988), two-dimensional Gaussian function fitting (Anderson et al. 1992), and brightness-weighed centroid (Ghosh and Webb 1994). Here, we present an example of particle tracking using the centroid method. This script was first written for marker tracking painted using a tissue dye on the endothelium of aortic tissues (Tremblay et al. 2010) (see Figure 9.4). This was employed to track the position of the markers and compute the tissue strain field during biaxial tensile testing. This script has been optimized for background with a uniform intensity profile and for tracking four markers or particles at the same time. It can be easily modified to compute centroid position of more than four markers or simply one single marker. The input file is a four-dimensional matrix from time-lapse image acquisition using the MATLAB acquisition toolbox. It corresponds to the variable “data(:, :, :, :).” The first two indexes refer to the pixel intensity for a given x–y position in the frame. The last index refers to the frame number of the acquisition.
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(A)
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(B)
FIGURE 9.4 Square aortic tissues taken from a pig ascending aorta. The tissue was biaxially stretched with hook-shaped surgical needles. Markers were painted on the endothelium to compute strain field. Boxes are the regions in which the centroid method is applied using MATLAB. Boxes were relocated from frame to frame. (A) Tissue sample at a zero-stress state. (B) Tissue sample under ∼25% green strain.
% R_x and R_y are the centroid position from frame to frame; directory and name are the location % and the name of the 4D file that contains the frames with the pixels intensity. function [R_x,R_y] = get_markers_position(data) data_size = size(data); number_frames = data_size(4); % To get initial distance between markers; reference frame% ref_frame=data(:,:,:,1); % Display the ref_frame% imshow(ref_frame) % Define 4 boxes around the 4 markers or beads % for i=1:4 buffer = getrect; for j=1:4 rect_marker(i,j) = buffer(j); end end % Close the figure in which the ref_frame was displayed% close % Put box positions in for i=1:4 xmin_rect_marker(i) xmax_rect_marker(i) ymin_rect_marker(i) ymax_rect_marker(i) end
variables % = = = =
int16(rect_marker(i,1)); int16(rect_marker(i,1) + rect_marker(i,3)); int16(rect_marker(i,2)); int16(rect_marker(i,2) + rect_marker(i,4));
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% At first, we first calculate the position of the four markers on the reference frame % Rename reference frame frame_binary = ref_frame; % Draw the four boxes that were defined by the function getrect; see draw_rectangle function below drawing = frame_binary; for i=1:4 drawing = draw_rectangle(drawing,xmin_rect_marker(i),xmax_rect_marker(i),… ymin_rect_marker(i),ymax_rect_marker(i)); end % Showing the ref_frame with the four boxes just drawn imshow(drawing); % Get background intensity inside the boxes % for i=1:4 backg(i) = get_background(frame_binary,xmin_rect_marker(i),… xmax_rect_marker(i),ymin_rect_marker(i),ymax_rect_marker(i)); end % Calculate centroid position for the 4 markers for marker_number=1:4 %initialization of various variables sum_x = 0; sum_y = 0; total_mass = 0; % Scanning of the area defined by the box % for k=xmin_rect_marker(marker_number):xmax_rect_marker(marker_number) for l=ymin_rect_marker(marker_number):ymax_rect_marker(marker_number) %Summation for the caculation of the barycenter of the marker intensity = abs(double(frame_binary(l,k)) - backg(marker_number)); sum_x = sum_x + double(k)*intensityˆ4; sum_y = sum_y + double(l)*intensityˆ4; total_mass = total_mass + intensityˆ4; end end %Calculation of the barycenter of the four markers for ref_frame Bar_x = sum_x/total_mass; Bar_y = sum_y/total_mass; R_x(marker_number,1) = Bar_x; R_y(marker_number,1) = Bar_y; drawing = draw_barycenter(drawing,Bar_x,Bar_y); end
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% Showing the ref_frame with the four boxes and the centroids imshow(drawing); pause; % For the next frames, we will use the position of the markers on the % previous frame to do the positioning of the box in which the scanning will occurs % For all frames except the first one which has been done already for i=2:number_frames %this variable specifies which frame is processed% frame_number=i; %let's define a box of x pixels per sides; box_size = 50; % this variable do a correction on the position of the box % considering the speed of the markers calculated from the two previous frames for marker_number=1:4 if i==2 corr_x(i) = 0; corr_y(i) = 0; else c orr_x(i) = R_x(marker_number,frame_number-1)-R_x(marker_ number,frame_number-2); c orr_y(i) = R_y(marker_number,frame_number-1)-R_y(marker_ number,frame_number-2); end end % Put the new box position in variables for marker_number=1:4 x min_rect_marker(marker_number) = int16(R_x(marker_number,frame_ number-1))-(box_size/2)+corr_x(i); x max_rect_marker(marker_number) = int16(R_x(marker_number,frame_ number-1))+(box_size/2)+corr_x(i); y min_rect_marker(marker_number) = int16(R_y(marker_number,frame_ number-1))-(box_size/2)+corr_y(i); y max_rect_marker(marker_number) = int16(R_y(marker_number,frame_ number-1))+(box_size/2)+corr_y(i); end frame_binary = data(:,:,:,frame_number); % Draw the four boxes that were defined by the function getrect; see draw_rectangle function below drawing = frame_binary; for i=1:4 d rawing = d raw_rectangle(drawing,xmin_rect_marker(i),… xmax_rect_marker(i),ymin_rect_marker(i),… ymax_rect_marker(i)); end
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% Showing the ref_frame with the four boxes just drawn imshow(drawing); pause; % Get background intensity inside the boxes % for i=1:4 b ackg(i) = g et_background(frame_binary,xmin_rect_marker(i),… xmax_rect_marker(i),ymin_rect_marker(i),… ymax_rect_marker(i)); end % Calculate centroid position for the 4 markers for marker_number=1:4 sum_x = 0; sum_y = 0; total_mass = 0; % Scanning of the area defined by the box % for k=xmin_rect_marker(marker_number):xmax_rect_marker(marker_number) f or l=ymin_rect_marker(marker_number):ymax_rect_marker(marker_ number) %Summation for the caculation of the barycenter of the marker i ntensity = abs(double(frame_binary(l,k)) - backg(marker_ number)); sum_x = sum_x + double(k)*intensityˆ4; sum_y = sum_y + double(l)*intensityˆ4; total_mass = total_mass + intensityˆ4; end end %Calculation of the barycenter of the four markers Bar_x = sum_x/total_mass; Bar_y = sum_y/total_mass; R_x(marker_number,frame_number) = Bar_x; R_y(marker_number,frame_number) = Bar_y; drawing = draw_barycenter(drawing,Bar_x,Bar_y); end % Showing the ref_frame with the four boxes and the centroids imshow(drawing); pause; end
Computational and Experimental Approaches at the Nanoscale
========================================================== function image = draw_rectangle(image,xmin,xmax,ymin,ymax) for i=xmin+1:xmax-1 image(ymin,i) = 0; end for i=ymin+1:ymax-1 image(i,xmax) = 0; end for i=xmin+1:xmax-1 image(ymax,i) = 0; end for i=ymin+1:ymax-1 image(i,xmin) = 0; end ========================================================== function backg = get_background(image,xmin,xmax,ymin,ymax) counter = 0; sum = 0; for i=xmin+1:xmax-1 sum = sum + double(image(ymin,i)); counter = counter + 1; end for i=ymin+1:ymax-1 sum = sum + double(image(i,xmax)); counter = counter + 1; end for i=xmin+1:xmax-1 sum = sum + double(image(ymax,i)); counter = counter + 1; end for i=ymin+1:ymax-1 sum = sum + double(image(i,xmin)); counter = counter + 1; end backg = sum/counter; =========================================== function image = draw_barycenter(image,x,y) % draw in black contour of the circle for i=int16(y-1):int16(y+1) image(i,int16(x-3)) = 0;
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image(i,int16(x+3)) = 0; end for i=int16(x-1):int16(x+1) image(int16(y-3),i) = 0; image(int16(y+3),i) = 0; end image(int16(y-2),int16(x-2)) image(int16(y-2),int16(x+2)) image(int16(y+2),int16(x-2)) image(int16(y+2),int16(x+2))
= = = =
0; 0; 0; 0;
% draw in white the inside of the circle for i=int16(x-2):int16(x+2) for j=int16(y-1):int16(y+1) image(j,i)=1; end end for i=int16(x-1):int16(x+1) image(int16(y-2),i) = 1; image(int16(y+2),i) = 1; end % draw a cross in black image(int16(y-1),int16(x)) image(int16(y),int16(x+1)) image(int16(y+1),int16(x)) image(int16(y),int16(x-1))
= = = =
0; 0; 0; 0;
9.5.3 Traction Force Microscopy, Optical Registration Algorithm, and Particle Tracking Cells generate CTFs as a result of adhesion to their surrounding environments. Furthermore, cells are constantly subjected to mechanical stresses and most importantly they respond to such mechanical variations in their MME by altering their motile behaviors, cellular growth, and adhesion (Discher et al. 2005; Buxboim et al. 2010). For example, cells cultured on flexible substrates with a rigidity gradient respond by crawling toward the region of greater stiffness and altering their behaviors in accordance with their MME. Adhesion of the cells to flexible substrates triggers the onset of transmembrane receptors (Burridge and ChrzanowskaWodnicka 1996), which in turn triggers the contraction of myosin. Therefore, substrate deformation (Engler et al. 2004) is due in large to actomyosin contractility and FA dynamics of cells cultured on flexible substrates (Matzke et al. 2001; Bershadsky et al. 2003). One of the most popular methods of calculating a CTF is using TFM (Dembo and Wang 1999; Lo et al. 2000; Munevar et al. 2001; Beningo and Wang 2002; Beningo et al. 2006), a technique that involves the tracking of marker fluorescent nanoparticles (200nm in diameter) embedded in the flexible elastic substrates as described above (Pelham and Wang 1997; Dembo and Wang 1999). The steps followed to obtain the CTFs generated by the cells are as follows: (1) Cells are cultured on the flexible substrates at low density (∼100 cells/mm2) on 35mm plates and are allowed to adhere and spread. (2) CTFs are generated by the cells that are transmitted to the flexible substrates causing the small movement of the fiduciary fluorescent beads. (3) A cell-of-interest is selected and a digital image of the field is recorded. This image consists of simultaneous phase contrast and epi-fluorescence. (4) (Optional) The cell is removed from the
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Parameters related to the characteristics of the three images (unstressed, stressed, and phase images): 1. Number of pixels in the x and y dimensions 2. Measured pixel size in meters 3. Bits/pixel Finally the optical section is optimized depending on certain important functions such as the number of bead displacement vectors successfully tracked by the similarity method using the cross-correlation function (integer), beads that have sufficient contrast to be reliably tracked (integer), radius of the initial correlation box (integer, pixel units), radius of initial search box (pixel units) used by the similarity method for tracking the beads, and finally, the x-component of the registration error vector (pixel unit). The first function operates by defining a box of pixels centered on the bead’s unstressed location. The tracking is performed by searching the “strained box,” which has centers within a defined search radius of the unstressed box center. The centre of the strained box where the intensity pattern closely resembles that of the unstressed box is taken as an initial estimate of the strained position of the ith particle. The similarity method is used to compare any two boxes, and, hence, we use the crosscorrelation function to compute that (for further explanations refer to Micah Dembo’s technical Manual on LIBTRC). 9.5.3.1 Bead Displacement within the Cell Contact Area The raw displacement data for the movement of the beads can be registered and analyzed with optical flow registration algorithms and LIBTRC libraries. To obtain the raw magnitude and direction of the marker bead displacements only within a cell contact area, we calculated a WN for each bead position in MATLAB (see Figure 9.6). In a complex plane, the WN of a closed curve (i.e., cell contour, C, which outlines the cell contact area) can be expressed in terms of the complex coordinates; therefore the WN is WN =
1 2πi
dz
∫ z
(9.9)
C
where z is the bead location integrated over a closed cell contour, C. If the closed contour represents the contour of the cell and we have (x, y) beads covering a pixel X pixel window, then integrating over the closed contour of that distance in the complex plane will result in two conditions: the marker being inside the closed cell contour or outside (WN = 1 or 0, respectively). In MATLAB, a code was written to employ the WN as a filter allowing one to only consider fiduciary markers within the cell contour or contact area. 9.5.3.1.1 MATLAB Code The raw displacement data contain the (x, y) coordinates of the bead and in the data we also have the (δx, δy), which corresponds to the change in the initial (x, y) bead position. Moreover, these initial bead locations will be positive, negative or both, and so we can separate these values in accordance with the geometric quadrant system.
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Creating a new matrix with the following columns, (x, y, magnitude, angle) for each quadrant creating a new list for each quadrant containing the initial x and y beads, magnitude and direction of the displaced beads; List1= [NN(:,1) NN(:,2) mag_NN(:,1) thetaNN_true(:,1)]; List2= [PP(:,1) PP(:,2) mag_PP(:,1) thetaPP_true(:,1)]; List3= [NP(:,1) NP(:,2) mag_NP(:,1) thetaNP_true(:,1)]; List4= [PN(:,1) PN(:,2) mag_PN(:,1) thetaPN_true(:,1)]; List_final= [List2; List3; List1; List4];
If I were to plot this, I will get beads everywhere, but I am not interested in all beads only those within the cell contour; hence, using the WN I can obtain just the beads of interest. The analog to Equation 9.9 is the MATLAB function inpolygon. function [in, on] = inpolygon(x,y,xv,yv) %INPOLYGON True for points inside or on a polygonal region. % IN = INPOLYGON(X,Y,XV,YV) returns a matrix IN the size of X and Y. % IN(p,q) = 1 if the point (X(p,q), Y(p,q)) is either strictly inside or % on the edge of the polygonal region whose vertices are specified by the % vectors XV and YV; otherwise IN(p,q) = 0. % % [IN ON] = INPOLYGON(X,Y,XV,YV) returns a second matrix, ON, which is % the size of X and Y. ON(p,q) = 1 if the point (X(p,q), Y(p,q)) is on % the edge of the polygonal region; otherwise ON(p,q) = 0. % % Example: % xv = rand(6,1); yv = rand(6,1); % xv = [xv; xv(1)]; yv = [yv; yv(1)]; % x = rand(1000,1); y = rand(1000,1); % in = inpolygon(x,y,xv,yv); % plot(xv,yv,x(in),y(in),'.r',x(∼in),y(∼in),'.b') % % Class support for inputs X,Y,XV,YV: % float: double, single % Copyright 1984–2004 The MathWorks, Inc. % Revision: 1.14.4.3 Date: 2004/03/02 21:47:43 % If (xv,yv) is not closed, close it. xv = xv(:); yv = yv(:); Nv = length(xv); if ((xv(1) ∼= xv(Nv)) ‖ (yv(1) ∼= yv(Nv))) xv = [xv; xv(1)]; yv = [yv; yv(1)]; Nv = Nv + 1; end inputSize = size(x); x = x(:).'; y = y(:).';
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mask = (x >= min(xv)) & (x <= max(xv)) & (y>=min(yv)) & (y<=max(yv)); if ∼any(mask) in = false(inputSize); on = in; return end inbounds = find(mask); x = x(mask); y = y(mask); % Choose block_length to keep memory usage of vec_inpolygon around % 10 Megabytes. block_length = 1e5; M = numel(x); if M*Nv < block_length if nargout > 1 [in on] = vec_inpolygon(Nv,x,y,xv,yv); else in = vec_inpolygon(Nv,x,y,xv,yv); end else % Process at most N elements at a time N = ceil(block_length/Nv); in = false(1,M); if nargout > 1 on = false(1,M); end n1 = 0; n2 = 0; while n2 < M, n1 = n2+1; n2 = n1+N; if n2 > M, n2 = M; end if nargout > 1 [in(n1:n2) on(n1:n2)] = vec_inpolygon(Nv,x(n1:n2),y(n1:n2),xv,yv); else in(n1:n2) = vec_inpolygon(Nv,x(n1:n2),y(n1:n2),xv,yv); end end end if nargout > 1 onmask = mask; onmask(inbounds(∼on)) = 0; on = reshape(onmask, inputSize); end mask(inbounds(∼in)) = 0; % Reshape output matrix. in = reshape(mask, inputSize);
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%––––––––––––––––––––––––––––––– function [in, on] = vec_inpolygon(Nv,x,y,xv,yv) % vectorize the computation. % Translate the vertices so that the test points are % at the origin. Np = length(x); x = x(ones(Nv,1),:); y = y(ones(Nv,1),:); xv = xv(:,ones(1,Np)) - x; yv = yv(:,ones(1,Np)) - y; % Compute the quadrant number for the vertices relative % to the test points. posX = xv > 0; posY = yv > 0; negX = ∼posX; negY = ∼posY; quad = (negX & posY) + 2*(negX & negY) +… 3*(posX & negY); % Compute the sign() of the cross product and dot product % of adjacent vertices. m = 1:Nv-1; mp1 = 2:Nv; signCrossProduct = sign(xv(m,:).* yv(mp1,:)… - xv(mp1,:).* yv(m,:)); dotProduct = xv(m,:).* xv(mp1,:) + yv(m,:).* yv(mp1,:); % Compute the vertex quadrant changes for each test point. diffQuad = diff(quad); % Fix up the quadrant differences. Replace 3 by −1 and −3 by 1. % Any quadrant difference with an absolute value of 2 should have % the same sign as the cross product. idx = (abs(diffQuad) == 3); diffQuad(idx) = -diffQuad(idx)/3; idx = (abs(diffQuad) == 2); diffQuad(idx) = 2*signCrossProduct(idx); % Find the inside points. in = (sum(diffQuad) ∼= 0); % Find the points on the polygon. If the cross product is 0 and % the dot product is nonpositive anywhere, then the corresponding % point must be on the contour. on = any( (signCrossProduct == 0) & (dotProduct <= 0)); in = in | on; So for the above case it will be something like the following after combining List1, List3 and List4 together. % Make sure that List2 contains the initial N × M matrix before separating them into quadrants;
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% the above will be a binary system (0 outside contour and 1 inside contour); [IN]=inpolygon(List_final(:,1), List_final(:,2), c_x, c_y); % Make four columns since we are working with (x, y, magnitude, angle); [IN_new]= [IN IN IN IN]; % convert these indices (0, 1) to the appropriate (x, y, magnitude, angle) matrix; C=List_final(IN_new);
9.6 Conclusion Image processing in the study of cell biology and cell mechanics is generally motivated by observing changes in the dynamics of subcellular features that may play critical roles in the onset of numerous cellular processes. Computational image processing techniques are generally used in treating problems involving dynamical structural changes in the cell. Generally, the computational measurements and calculations are performed after processing the acquired and registered data in order to understand and quantify what is occurring inside the cell. Mathematical and computational software such as MATLAB is one of the many options of available tools that contain important image processing and mathematical toolboxes, which are valuable in the post-processing steps of image segmentation and quantification. Live-cell imaging research is expanding rapidly, and this field provides valuable insight into understanding major mechanochemical pathways that are fundamental to many cellular processes in the cell as outlined here. For instance, tracking the motion of organelles or proteins over time can provide constructive information on how such constituents or molecules respond to variations in substrate stiffness, external force, or inhibition of certain functionalities of important proteins. Therefore, such precious data may possibly contribute to the fields of drug therapy and delivery and regenerative medicine in the future. Here, we reviewed the major cellular structures that are involved in the integrity of the cell and the important protein complex that plays a crucial role in force transmission due to actomyosin contractility in the cell and demonstrated how the cell “senses” its mechanical microenvironment as the stiffness of the substrates or external stimulation by an AFM tip. By using polymer hydrogel substrates together with combined live microscopy, structural regions-of-interest in the cell can be investigated and quantified in order to expand on our modest understanding of the complex pathways involved in the cell. In addition, using the available computational methods such as particle tracking, optical flow registration algorithms, image segmentation, thresholding, masking etc., we can facilitate our understating of how cells respond to environmental cues, respond to variations in stiffness, and transmit mechanical forces.
Acknowledgments The authors are grateful for generous support from the Ottawa NSERC Create Training Program in Quantitative Biomedicine (to ZAR), an NSERC Discovery Grant, an NSERC Discovery Accelerator Supplement Award, and support from the Canada Research Chairs program (to AEP).
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10 Computational Simulations of Nanoindentation and Nanoscratch Cheng-Da Wu, Te-Hua Fang, and Jen-Fin Lin CONTENTS 10.1 Introduction........................................................................................................................ 363 10.2 Physical Model and Assumption of Boundary Condition ..........................................364 10.3 Nanoindentation and Microcontact Analyses............................................................... 366 10.3.1 Effect of Indentation Loads .................................................................................. 366 10.3.2 Effect of Loading Rate........................................................................................... 368 10.3.3 Effect of Thermal Softness.................................................................................... 370 10.3.4 Comparisons of MD and Experimental Results ............................................... 371 10.3.5 Nanoscratch and Nanoindentation Analysis of a Diamond Asperity on a Diamond Plate ............................................................................................... 373 10.3.6 Effect of Indentation Interferences ...................................................................... 374 10.3.7 Effect of Interference on Sliding Friction ........................................................... 376 10.3.8 Experiment and Microcontact Model of Fractal Analysis ............................... 377 10.3.9 Results of Combination of Statistical Analysis and Fractal Theories ............ 379 10.4 Conclusion .......................................................................................................................... 379 References...................................................................................................................................... 380
10.1 Introduction Due to the improvements in manufacturing technology, the thickness of films in semiconductors or micro devices now has reached the nanometer level. However, there is a marked difference between the mechanical characteristics of buck materials and nanofilms. In order to clearly understand these differences, it is essential to investigate the physical properties of thin films. Nanoindentation and nanoscratch are the most frequently used techniques for measuring thin film properties such as Young’s modulus, hardness, and the frictional coefficient [1,2]. However, in order to obtain precisely measured film properties, especially for very thin films on the order of less than 1 nm thick, significant redesign and improvement of nanoindentation measurement systems, with special emphasis on tips, is required [3]. Current systems have difficulty maintaining controlled and reliable small-depth penetration, which is required for minimizing substrate influences during film measurements [4]. Various aspects of the interaction between tool tip and specimen during the nanoindentation process have been investigated. Previous studies have included the phenomena of tip–specimen adhesion and fracture of the specimen [5,6], nanoscale etching and indentation using carbon nanotube [7], phase transformations [8], fatigue [9], elastic and plastic 363
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contact behavior [10], and the influence of different tool materials on the indentation mechanism [11]. These previous studies provide significant insight into tip–specimen interaction phenomena. In order to obtain precisely measured mechanical and contact properties of films at the nanometer scale, molecular dynamics (MD) is a greatly analyzed tool and has been used on the nanoscratch [12], nanoimprinting contact [13–15], nanoindentation behavior [16], monolayer sliding friction [17,18], and mechanical properties of nanoscale materials [19]. However, the effect of indentation load, temperature, loading, and sliding rate are also important in the nanocontact process. Thus, based on these viewpoints, the main object of this study is to evaluate these effects on the nanoindentation and nanoscratch of hard (diamond) and soft materials including gold and copper. Especially, diamond or diamondlike carbon films have been widely used as protective coatings, for instance, on optical storage disks, optical window, biomedical, and microelectromechanical devices [20]. The MD results compare well with experimental results.
10.2 Physical Model and Assumption of Boundary Condition In most previous studies, the MD simulation method has been used to investigate phenomena occurring during the nanoindentation process [16]. It is proposed that the same method be used for this study. MD simulation of nanoindentation is used to investigate various aspects of the interaction between a rigid diamond tip and a monocrystalline gold and diamond film. The film and the diamond tool are used in these simulations as shown in Figure 10.1. The direction of indentation vertical to the specimen surface is taken as the negative z-axis. In this study, films with 157,000 atoms and diamond tool with 6,290 atoms are used. Periodic boundary conditions (PBC) [21] are used in the transverse directions (x- and y-directions), and the bottom three layers of atoms of the specimen are fixed in space [22]. The force acting on an individual atom is obtained by summing the forces contributed by the surrounding atoms. The tight-binding second-moment approximation [23] and the Loading Unloading Indenter
z x
FIGURE 10.1 MD simulation model for nanoindentation. (From Liu, C.L. et al., Mater. Sci. Eng. A, 452–453, 135, 2007. With permission.)
y
Specimen
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Tersoff [24] and Morse [12] potentials were selected for gold, carbon, and the interaction between the tool and the specimen, respectively. The force on atom i resulting from the interaction of all the other atoms can be derived from the above potential functions ϕ(rij), such that N
Fi = −
∑
∇iφ(rij ) = mi
j 1 ( j≠ i)
d 2ri (t) dt 2
(10.1)
where Fi is the resultant force on atom i mi is the mass of atom i ri is the position of atom i N is the total number of atoms The data (tip velocity, integration time, etc.) are used, and the initial configuration of the sample material is perfect gold and diamond. Initial velocities are assigned from the Maxwell distribution [21], and the magnitudes are adjusted so as to keep the temperature in the system constant according to
Vi
new
⎧ ⎪ N f NkBT0 =⎨ 2 ⎪⎩
⎡ ⎢ ⎢⎣
N
∑ i 1
mi (Viold )2 ⎤ ⎥ 2 ⎥⎦
−1 1/2
⎫ ⎪ ⎬ ⎪⎭
⋅ Viold
(10.2)
where Vi is the velocity of atom i T0 is a specified temperature kB is Boltzmann’s constant (=1.381 × 10−23 JK−1) Nf is the freedom of the system The initial displacement and velocity are values determined independently, and the time integration of motion is performed by Gear’s fifth predictor–corrector method [21]. For the simulation, the specimen is initially assumed to have a well-defined atomic surface and the time step is set to 1 fs. The atomic array model of the surface is constructed for a specific temperature at equilibrium and the velocities of atoms of the specimen at this state are satisfied with a Maxwell velocity distribution. Young’s modulus is calculated using the composition response modulus, Er, which takes into account the combined elastic effects of indention tip and film [16]: Er =
1 2
π dP Ac dh
where dP is the slope of the beginning of the unloading curve dh Ac is the projected contact area at the maximum load
(10.3)
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The final Young’s modulus of the specimen, Es, is obtained from the expression 1 1 − v 2s 1 − vi2 = + Er Es Ei
(10.4)
where Ei is the elastic modulus of the indenter vs and vi are the Poisson’s ratios of the specimen and indenter, respectively A key point when determining an elastic modulus from an indention experiment is how to ascertain the projected contact area, Ac. Oliver and Pharr [1] developed an improved technique for the measurement of indentation impressions. They used data directly drawn from the indentation curve and correlated the projected contact area to the contact depth, hc, as follows: Ac = f ( hc )
(10.5)
where hc may be expressed as hc = hmax − 0.72
Pmax Smax
(10.6)
where hmax is the maximum depth Pmax is the maximum load Smax is the slope of the unloading curve at the maximum load The material properties used in this study’s calculations are Ei = 1140 GPa and vi = 0.07 for an ideally Berkovich diamond indenter [16]. The hardness of a material is defined as its resistance to local plastic deformation. Thus, the hardness, H, is determined from the maximum indentation load, Pmax, divided by the projected contact area, that is, H=
Pmax Ac
(10.7)
10.3 Nanoindentation and Microcontact Analyses 10.3.1 Effect of Indentation Loads In order to investigate the effect of loading on the nanoindentation process, the process was simulated using various indentation depths. Figure 10.2a and b show load–displacement curves for diamond and gold films under different indentation loads and at different depths, respectively. As the indentation depth of the diamond tip continues to increase, the load curve starts to go up until it reaches a maximum depth. After reaching the maximum depth, the tip begins to unload and return to its original position. With the diamond tip drawn back, the plastic deformed region undergoes a partial elastic recovery indicating the irreversibility of the plastic behavior during nanoindentation that manifests in the load–displacement curves as a hysteretic form. A discrete yielding event for the diamond is observed at the loading curve. The penetration depth burst generally occurred when
Computational Simulations of Nanoindentation and Nanoscratch
η We
Diamond Gold
0.6
10,000 0.4
1,000
η
We ( 10–20 N-m)
1,00,000
369
0.2
100 80
120
160 200 240 Loading rate (m/s)
280
FIGURE 10.6 Elastic energy and ratio of the elastic energy to the total energy. (From Liu, C.L. et al., Mater. Sci. Eng. A, 452– 453, 135, 2007. With permission.)
0 320
The elastic energy We and the ratio of elastic energy to total energy η is shown in Figure 10.6. For a hard material (diamond), the elastic energy decreased slightly as the loading rate increased. It indicated that the diamond tool is used for ultraprecision machining due to its higher stiffness and lower plastic deformation. For a soft material (gold), both We and η decreased significantly as the loading rate increased. The effect of loading rates on Young’s modulus and the hardness is shown in Figure 10.7. Young’s modulus and the hardness of the diamond film under different indentation rates are approximately 1376–1862 and 90–107 GPa, respectively. Young’s modulus and the hardness of the gold film under different indentation rates are approximately 66–129 and 5–7 GPa, respectively. The energy of the impact of the indenter on a soft film surface is greater at the higher loading rates, which causes the Young’s modulus and the hardness to increase up to a critical level. The critical velocity occurs at the indentation rate of around 200 m/s, as shown
H
E
1,000 Diamond Gold 100
1,000
100
10 80
10
120
160 200 240 Loading rate (m/s)
280
1 320
Hardness (GPa)
Young's modulus (GPa)
10,000
FIGURE 10.7 Young’s modulus and the hardness of diamond and gold. (From Liu, C.L. et al., Mater. Sci. Eng. A, 452–453, 135, 2007. With permission.)
Computational Simulations of Nanoindentation and Nanoscratch
1,00,000
η We
371
0.8 Diamond Gold 0.6
0.4
η
We (10–20 N-m)
10,000
1,000 0.2
100
300
400
500
600
FIGURE 10.9 Elastic energy and ratio of the elastic energy to the total energy under different temperatures. (From Liu, C.L. et al., Mater. Sci. Eng. A, 452–453, 135, 2007. With permission.)
0
Temperature (K) 1,000
1,000
100 H
E
Diamond Gold
100
10
10
300
400
500
600
Hardness (GPa)
Young's modulus (GPa)
10,000
1
Temperature (K)
FIGURE 10.10 Young’s modulus and the hardness of diamond and gold under different temperatures. (From Liu, C.L. et al., Mater. Sci. Eng. A, 452–453, 135, 2007. With permission.)
temperature increase, leading to decrease in hardness. This finds confirmation in the literature with the experimental microscale study of Lebedev et al. [17], which found a decrease of the elastic modulus in submicrocrystalline copper as the temperature increased. 10.3.4 Comparisons of MD and Experimental Results The nanoindentation experiments were performed using Hysitron Triboscope (Hysitron Inc., USA) with a diamond probe. A diamond Berkovich indenter, with a tip radius of 50 nm and a maximum load ranging from 500 to 5000 μN, was used for evaluating Young’s modulus and the hardness of the diamond and gold films. The diamond film and the gold film are deposited on a silicon substrate using chemical vapor deposition and sputtering deposition, respectively. In order to avoid the indentation size effect (ISE), the thicknesses of the
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z x
y
Fixed layer Diamond plane
Diamond asperity Fixed layer FIGURE 10.13 The molecular model of diamond single asperity with a diamond plane. (From Lin, J.F. et al., Curr. Appl. Phys., 10, 266, 2010; Lin, J.F. et al., Comput. Mater. Sci., 40, 480, 2007. With permission.)
A diamond asperity with 44,173 atoms and a diamond plane with 29,584 atoms were used. The plane size was 15.26 nm × 15.26 nm × 0.62 nm. In order to use a relatively small number of particles to experience forces as if they were in a bulk material in the transverse, PBC [21] were used at the diamond plane and the specimen in the x- and y-directions, and the bottom three layers of atoms of the specimen were fixed in space. In experiments, the diamond film was deposited on a silicon (100) substrate using hot filament chemical vapor deposition method. In order to avoid the ISE, the thickness of the diamond film was about 10 μm. AFM (Seiko SPA300HV, Japan) was used to measure the diamond surface. The nanoindentation and nanoscratch experiments were performed using Triboscope (Hysitron, USA) and NanoTest (Micro Materials, UK), respectively. A diamond Berkovich indenter had a radius of about 150 nm. Maximum loads of 55 and 100 μN were used for evaluating indented contact and frictional behavior of the diamond films, respectively. 10.3.6 Effect of Indentation Interferences Figure 10.14a through c show the indentation profiles at interferences of 0.2, 1.0, and 1.7 nm. A deformed area in the diamond asperity was observed under indentation by the diamond plane. The formation of this area was due to unrecoverable plastic deformation. As expected, the higher the contact interference, the larger the deformed area. At maximum indentation depth, it was observed that the diamond atoms’ adhesion beneath the diamond plane and the atomic order beneath the plane differed considerably from its original pattern. The slip direction of diamond was observed along the (110)-direction. The lattice transformation varied according to the crystallographic orientation of the contact surface and was directly linked to the slip system of diamond. Figure 10.15 shows the contact force–interference curves of the diamond asperity at loading velocities of 30, 50, and 100m/s. It can be found that as the interference increased, the contact force increased. Also, a lower loading velocity caused a steeper contact force slope, that is, a lower contact force. The slope of the contact force–interference curves could be defined as the contact stiffness. The lower stiffness at interferences of 0–1.2nm was 5 × 103 N/m. The higher stiffness at interferences of 1.2–1.8nm was 25 × 103 N/m. This is due to the fact that larger amount of dislocations caused larger deformed area of the single asperity top.
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50,000
Experimental results in the nanoindentation tests of diamond film 1. Loading velocity = 38 nm/s 2. Loading velocity = 25 nm/s
Contact force (μN)
40,000
30,000
20,000 FIGURE 10.19 The relationship of indent depth and contact force obtained by nanoindentation experiment at loading velocities of 25 and 38 nm/s. (From Lin, J.F. et al., Curr. Appl. Phys., 10, 266, 2010. With permission.)
10,000 0
40
80 120 Depth (nm)
160
200
were approximately 90 ∼ 100 and 1002 ∼ 1100 GPa, respectively. As the indentation depth of the diamond probe continued to increase, the load curve started to go up until it reached a maximum depth. After reaching the maximum depth, the tip began to unload and returned to its original position. With the diamond tip drawn back, the plastic deformed region underwent a partial elastic recovery indicating the irreversibility of the plastic behavior during nanoindentation that manifested in the contact force–depth curves as a hysteretic form. The higher loading velocity caused a higher hardness and lower hysteretic work. Figure 10.20 shows the experimental dynamic frictional coefficient–sliding distance of nanoscratch at a sliding velocity of 166 nm/s. The dynamic frictional coefficient ranged 2
Dynamic frictional coefficient
1.6
1.2
0.8
0.4 FIGURE 10.20 The experimental dynamic frictional coefficient of nanoscratching at a sliding velocity of 166 nm/s. (From Lin, J.F. et al., Curr. Appl. Phys., 10, 266, 2010. With permission.)
0
0
2
4 6 Sliding distance (μm)
8
10
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nanoindentation and sliding processes. Even for a nanopenetration depth (<2 nm), the simulation results show that the films deform plastically and give a good description. Both Young’s modulus and the hardness decrease as the indentation loads increase. But, for the indentation rates simulation part, Young’s modulus and the hardness decrease at a critical velocity of 200 m/s. In addition, the results also indicate that the indentation rates, which are the main reason to domain the deformation mechanism; the thermal softens during indentation. The simulation results indicate that both Young’s modulus and the hardness become smaller as the temperature increases. Hence, with the aid of MD simulations expanded by statistical and fractal analysis, the microcontact and sliding processes of diamond films and diamond plate during indentation and the mechanism responsible for them are clearly shown.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
Oliver, W. C. and Pharr, G. M. 1992. J. Mater. Res. 7:1564. Fang, T. H., Chang, W. J., and Lin, C. M. 2005. Microelectron. Eng. 77:389. Bhushan, B. and Koinkar, V. N. 1994. Appl. Phys. Lett. 64:1653. Fang, T. H. and Chang, W. J. 2002. Microelectron. Eng. 65:231. Landman, U. W., Luedtke, D. N., Burnham, A., and Colton, R. J. 1990. Science 248:454. Li, X. and Bhushan, B. 1998. Thin Solid Films 315:214. Dzegilenko, F. N., Srivastava, D., and Saini, S. 1999. Nanotechnology 10:253. Cheong, W. C. D. and Zhang, L. C. 2000. Nanotechnology 11:173. Li, X. and Bhushan, B. 2003. Surf. Coat. Technol. 163–164:521. Fang, T. H. and Chang, W. J. 2004. Microelectron. J. 35:595. Isono, Y. and Tanaka, T. 1999. JSME Int. J. Ser. A Mech. Mater. Eng. 42:158. Fang, T. H. and Weng, C. I. 2000. Nanotechnology 11:148. Hsu, Q. C., Wu, C. D., and Fang, T. H. 2004. Jpn. J. Appl. Phys. 43:7665. Hsu, Q. C., Wu, C. D., and Fang, T. H. 2005. Comput. Mat. Sci. 34:314. Fang, T. H., Wu, C. D., and Chang, W. J. 2007. Appl. Surf. Sci. 253:6963. Liu, C. L., Fang, T. H., and Lin, J. F. 2007. Mater. Sci. Eng. A 452–453:135. Wu, C. D., Lin, J. F., and Fang, T. H. 2006. Comput. Mat. Sci. 39:808. Wu, C. D., Lin, J. F., Fang, T. H., Lin, H. Y., and Chang, S. H. 2008. Appl. Phys. A 91:459. Chang, W. J. and Fang, T. H. 2003. J. Phy. Chem. Solids 64:1279. Fang, T. H. and Chang, W. J. 2006. Appl. Surf. Sci. 252:6243. Haile, J. M. 1992. Molecular Dynamics Simulation: Elementary Methods. New York: Wiley. Fang, T. H., Weng, C. I., and Chang, J. G. 2002. Surf. Sci. 501:138. Liu, H. B., Chen, K. Y., Gong, Y. D., An, G. Y., and Hu, Z. Q. 1997. Philos. Mag. A 75:1067. Tersoff, J. 1989. Phys. Rev. B 39:5566. Chung, J. C. and Lin, J. F. 2006. ASME J. Appl. Mech. 73:143. Lin, J. F., Fang, T. H., Wu, C. D., and Houng, K. H. 2010. Curr. Appl. Phys. 10:266. Lin, J. F., Fang, T. H., Wu, C. D., and Houng, K. H. 2007. Comput. Mater. Sci. 40:480.
11 Modeling of Reversible Protein Conjugation on Nanoscale Surface Kazushige Yokoyama CONTENTS 11.1 Introduction........................................................................................................................ 381 11.2 Experimental: Sample Handling and Method ..............................................................384 11.3 Spectral Description on Protein Conjugations .............................................................. 385 11.4 Protein Conformation in Reversible Self-Assembly ..................................................... 391 11.5 Temperature Dependence of Conjugation of Amyloid Beta Protein on the Surfaces of Gold Colloidal Nanoparticles .......................................................... 396 11.5.1 15 nm Gold Colloids Coated with Aβ1–40 ............................................................. 398 11.5.2 20 nm Gold Colloids Coated with Aβ1–40............................................................. 399 11.5.3 30 nm Gold Colloids Coated with Aβ1–40............................................................. 401 11.5.4 40 nm Gold Colloids Coated with Aβ1–40............................................................. 402 11.6 Suggested Model for Nanoscale Temperature-Dependent Conjugation ................... 402 11.7 Conclusion ..........................................................................................................................405 Acknowledgments ...................................................................................................................... 406 References...................................................................................................................................... 406
11.1 Introduction As nanotechnology continues to cultivate revolutionary materials, it has been continuously and greatly contributing to the biomedical field. As one of the great advantages, nanoscale materials are used to describe function of biomolecules placed in vivo or in vitro. Specifically, nanoscale-size colloids play key roles in monitoring biological functions under interfacial conditions to represent environments such as human membranes or cells. For example, the conjugation of colloidal gold with bio-specific macromolecules has been extensively studied and has revealed unique optical and electronic properties [1–13]. Zare and coworkers demonstrated that conformational change of Cytochrome-c can be observed on the surface of the gold colloidal nanoparticles, and proved that protein-coated gold nanoparticles can be used to investigate the conformational change of proteins at a solid–liquid interface [14]. We have applied this approach to investigate the conjugation of amyloid β protein (Aβ) over gold nanocolloid in conjunction with a mechanism of fibrillogenesis, which is associated with Alzheimer’s disease [15]. Pathologically, a key hallmark of the neuritic and cerebrovascular amyloid in Alzheimer’s disease is the formation of insoluble fibrillar deposits of the amyloid beta peptide (Aβ) as both diffuse and senile amyloid plaque that invades the brain’s seat of memory and cognition (brain parenchyma and vasculature) before spreading to other areas [16–19]. 381
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Hydrophilic tail (coil) Hydrophobic tail (rod) Aqueous-extracellular domain Transmembrane domain 1 5 10 15 20 25 30 35 40 NH2–DAEFRHDSGYEVHHQKLVFFAEDVGSNKGA I I GLMVGGVV–COOH – – ++ – – ++ + –– + Aβ1–40 monomer
1–
dr
hy
( 16
)
ilic
h op
0 c) –4 bi 17 pho o dr hy
(
FIGURE 11.1 Entire sequences of Aβ1–40 studied in this work. The charges (+ or −) at neutral pHs are given beneath the sequences. Hydrophobic residues are given in bold letters. A coil-rod sketch of Aβ1–40 is also given.
The 42- and 40-residue, Aβ1–42 and Aβ1–40 are capable of assembling into 60–100 Å diameter β-sheet fibrils that exhibit the characteristic cross-β x-ray fiber diffraction pattern [20,21]. Highly hydrophobic Aβ1–42 is implicated in amyloid fibril nucleation [22], while the more soluble Aβ1–40 is the main form circulating in normal plasma and cerebrospinal fluid [23,24] (see Figure 11.1 for the sequences of Aβ1–40). The fibrillogenesis is considered to be nucleation dependent and progresses as a polymerization process originating from a nucleus unit formed by certain numbers of monomeric Aβ as shown in Figure 11.2 [25,26]. This involves the initial formation of a seeding aggregate that establishes the amyloid fibril lattice [27] followed by the elongation of the fibril by the sequential addition of subunits.
Aβ monomer
Step 1
Monomers FIGURE 11.2 Model of fibrillogenesis. Fibrillogen esis is considered to be a nucleationdependent polymerization process. In Step 1, monomeric Aβ forms nuclei from which protofibrils emanate (Step 2). These protofibrils give rise to full-length fibers (Step 3).
Protofibril
Nucleus Step 2
Step 3
Protofibril
Fibers
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Absorbance
An initial stage of fibrillogenesis involving the soluble Aβ complex (Step 1 in Figure 11.2) is regarded as a pathologically important step and a key onset for the subsequent aggregation of the protein [28–30]. As these precursors of the fibril are thought to be non-fibrillar soluble oligomer forms of Aβ, they possess neurotoxic properties and play a role in the molecular pathogenesis of Alzheimer’s disease [28]. Although still relatively little is known about the structure of this soluble oligomer Aβ, a selective detection of the oligomer form of Aβ aggregate will be an important step in conducting a study of determining its neurotoxicity and clarifying its role in fibrillogenesis. A direct detection of the reversible process associated with conformation in an oligomer has not yet been fully achieved. The conformation found on the membrane surface is a critical stage for initial fibrillogenesis, so the direct investigation of Aβ placed under the interfacial environment must realistically project the conformation associated with intermediates in fibrillogenesis. Generally speaking, the proteins immobilized at an interface are expected to behave differently from their counterparts in bulk solutions [31–34], and Aβ located on human membranes or brain cells must progress through fibrillogenesis in a different way from those that possesses maximum degrees of freedom in solution. Understanding the interactions of the proteins on the surface is challenging but crucial to fully understand the mechanism of fibrillogenesis involving the membrane surface. In order to investigate the fibril precursor at the interfacial environment, we took an approach preparing Aβ on an interfacial boundary on the surface of gold colloidal nanoparticles; the associated structural information of this conjugated Aβ was then investigated as summarized in Figure 11.3. This study aims to create an environment that will reproduce a structure similar to the Aβ located on brain cells or membrane interfacial surfaces. While metal colloidal surfaces do not create the same environment as physiological conditions, our approach enables us to examine Aβ’s structure on a size-controlled interfacial
Step 1
(a) Aβ1–40 monomers
Nucleus
pH 10
500
(c)
pH 4
600 700 Wavelength (nm)
1 μm
Gold colloid
(b)
pH 10
800
pH 4
(d)
pH 10
pH 4
FIGURE 11.3 Schematic chart indicating a background and approach conducted in this study. (a) A targeted reversible process in fibrillogenesis of amyloid beta (Step 1 in Figure 11.1). (b) A corresponding reversible protein assembly, which is aimed to be observed over the gold nanocolloids’ surfaces at each pH. (c) The absorption spectrum Aβ1–40-coated 20 nm gold colloids at pH 10 and pH 4. (d) TEM images of Aβ1–40-coated 20 nm gold colloids at pH 10 and pH 4.
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environment by externally varying the pH. Thus, it enables us to explore nanoscale dimensionality of protein structure at the interfacial environment. The conjugation of Aβ on gold colloid was particularly interesting since the destruction of amyloid fibrils attached to gold nanoparticles was possible by exposure to weak microwave fields without harming healthy cells [35]. If the conjugation space contributes to a specific form of the interfacial structure, the variation in the size must require more or less aggregates on the available adsorption surface. At the same time, the structure associated with the aggregation condition may vary. Also, the role of the thermal condition in the fibrillogenesis taking place at the interface is considered to be significant. The variation of the thermal condition must change the surface potential enough to overcome the physical restriction (particularly size of the gold colloids) preventing the initial reversible process from taking place at room temperature otherwise.
11.2 Experimental: Sample Handling and Method The ultrapure Aβ1–40 (MW: 4329.9 Da), in the form of lyophilized powder (97% by HPLC), was purchased from American Peptide (Sunnyvale, California), and stored at −12°C. The stock solution of 100 μM Aβ1–40 was prepared at approximately 18°C using double-distilled deionized and filtered water. The concentration of Aβ1–40 was determined by UV absorption at 280 nm (absorbance of Tyrosine at 275 nm, ε275 = 1390 cm−1 M−1) [36]. The Grade II chicken egg albumin (ovalbumin, MW: 44.3 kDa) was purchased from Aldrich Co. (St Louise, Missouri) and the stock solution of 10 μM was also prepared using double-distilled deionized and filtered water. Gold colloidal nanoparticles were purchased from Ted Pella Inc. (Redding, California) with the following diameters (and particles mL−1 in O.D. = 0.2 at 528 nm): 5 ± 1.0 nm (1.25 × 1013 particles mL−1), 9.8 ± 1.0 nm (1.4 × 1012 particles mL−1), 15.2 ± 1.5 nm (2.8 × 1011 particles mL−1), 19.7 ± 1.1 nm (1.4 × 1011 particles mL−1), 30.7 ± 1.3 nm (4.0 × 1010 particles mL−1), 40.6 ± 1.1 nm (1.8 × 1010 particles mL−1), 51.5 ± 4 nm (8.2 × 109 particles mL−1), 60 ± 1.0 nm (4.3 × 109 particles mL−1), 80 ± 1.0 nm (2.2 × 109 particles mL−1), and 99.5 ± 1.3 nm (1.6 × 109 particles mL−1). Both Aβ and ovalbumin proteins were dissolved in deionized distilled water in order to prevent the destabilization of colloidal particles, avoiding ionic species of the buffer solutions [37]. In the procedure, any water used was purified to more than 18 M Ω using a Milli-Q water system (Millipore). The ratio of the particles between the protein and the gold colloidal nanoparticle solution was adjusted to a factor of 1000 more than the gold colloids (e.g., 20 nm colloid particles in this experiment were 2.30 nM and the concentration of the protein was 2.30 μM). The gold nanoparticles coated with optimal levels of Aβ1–40 or ovalbumin solution were prepared at various pHs using the following procedures. After Aβ or ovalbumin was mixed with the gold colloid solution, the samples were left for at least 1 h and the pH was then adjusted between 2 and 10. Based on spectroscopic analysis, protein–gold nanoparticle conjugation is completed after 10 min. The pH change from 7 to 2 was completed by drop-wise addition of hydrochloric acid (HCl) and the change between pH 7 and pH 10 was done by addition of sodium hydroxide (NaOH). Since the commercial nanoparticle solutions contained some residual acid, resulting in a buffer against basic conditions, the solution pH was monitored for a long time and additional base was continuously added as required for the range between pH 8 and pH 10. The absorption spectra of bare metal nanoparticles and Aβ or ovalbumin-coated nanoparticles were taken between pH 10 and pH 2. The pH
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value of each sample was directly measured in a cuvette cell while stirring, using a micro pH electrode with an accuracy of ±0.01. The reversibility of the color change of the solution was investigated for all Aβ and ovalbumin solutions between pH 4 and pH 10 by adding an adequate amount of base or acid solution as its absorption spectrum was monitored. The temperature control of the absorption cell was conducted by a feedback loop temperature control system with the Peltier device. The temperature of the solution was monitored by a temperature sensor installed in the cuvette holder and was confirmed by a digital thermometer inserted in a sample cuvette for the temperature range between 5°C and 50°C taking 10°C, 20°C, 30°C, and 40°C as middle data points with an accuracy of ±0.2°C. A step of 1°C was taken when more fine temperature steps were required to clearly observe the transition in spectral features. The pH of the solution was repeatedly altered between pH 4 and pH 10 for 10 cycles, and the corresponding absorption spectrum was monitored at each pH condition. Here, one cycle indicates the transition where pH changes from pH 4 to pH 10 to pH 4. The pH value of each sample was directly measured in a cuvette cell while stirring, using a micro pH electrode. For each operation of acid/base insertion, the pH value was managed to reproduce pH 4 or pH 10 with an accuracy of ±0.1. The original pH of freshly prepared sample was around pH 7. The pH change to pH 4 from pH 7 was completed by drop-wise addition of HCl, and the change to pH 10 from pH 4 was made by addition of NaOH. The utilized acids (HCl) and bases (NaOH) of various concentrations were prepared in a temperature control cell holder with targeted temperature in order to minimize the temperature variation during the pH adjustment. For transmission electron microscopy (TEM) measurement, Formvar-coated grids were used for preparing samples of Aβ1–40-coated gold beads. Formvar was cast onto microscope slides, floated off onto water, and grids were applied to the film. For the protein solution, 8 μL of protein was mixed with 40 μL of gold colloid, and 1 μL of solution was pipetted onto the surface of the grid. After 2 min to allow the sample to bind to the grid, excess solution was removed from the grid with filter paper. Samples were examined with a Morgagni model 268 TEM (FEI Co., Hillsboro, Oregon) operated at 80 kV. Images were taken at a nominal magnification of 28,000 or 71,000 on a model XR-40 four megapixel CCD digital camera (AMT) [38].
11.3 Spectral Description on Protein Conjugations The average peak position of the surface plasmon resonance (SPR) band of the regions between 400 and 800 nm was monitored as the pH was repetitively changed between pH 4 and pH 10. We used the identification of operation to alter pH condition by “n,” which varies from 1 to 21 for the process of 10 cycles. Here, n = 1 corresponds to the starting pH condition of solution before acid is inserted and its pH was around pH 7. After n = 1, odd numbers of n (nodd = 3, 5, 7, …, 21) indicate an operation of acid addition leading the pH of solution to be pH 4, whereas even numbers of n (neven = 2, 4, 6, …, 20) show an operation of base addition to increase the pH of solution to be pH 10. The average peak position at given temperature (T) and operation (n), λpeak(n, T), was calculated based on Equation 11.1 λ peak (n, T ) =
∑ a (n, T )λ (n, T ) i
i
i
(11.1)
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where λi(n, T) and ai(n, T) represent the peak position and fraction of the ith component band. In this study, the fraction ai was determined by the fraction of the area (Ai ) of the band to the area of the total sum of the entire bands, for example, a1(n, T) = A1/(A1 + A2) for the case of two bands. Most of the bands observed in our study were analyzed with two components or one component with a large background band with a maximum at 350 ± 50 nm. All the absorption bands were fully explained by the fit with Gaussian profile using the peak-fit module of ORIGIN (Version 7.0) in the range of 400–800 nm. After the background band was excluded, the average peak shift for two components band is given by λpeak(n, T) = a1(n, T)λ1(n, T) + a2(n, T)λ2(n, T). The color change is considered to be caused by electric dipole–dipole interactions and coupling between surface plasmons of neighboring particles inside the aggregates, provided that the interparticle distance inside the aggregate is smaller than the gold colloidal particle diameter. The initial color of the bare gold nanoparticle solutions or Aβ1–40-coated gold particles had a pH around neutral with reddish color; thus, the SPR band is maximum at 528 nm. The color of the solution that changed into bluish color had bimodal components with absorption band maximum around 600 nm as the pH was adjusted to pH 4. These bimodal absorption peaks consisted of a band with maximum around 528 ± 1 nm (λ1(n, T)) indicating the existence of free gold colloidal nanoparticles and a band with a maximum between 590 and 630 nm (λ2(n, T)). The contribution of the λ1(n, T) component was larger at higher pH values (pH > 7), whereas the λ2(n, T) component shifts to red and gains more amplitude at pH 4; this component became more prominent than the λ1(n, T) component at this pH. The absorption spectrum was monitored as a function of pH (Figure 11.4), and the position of the peaks, λpeak, as a function of the pH values were plotted as shown in Figure 11.5. For our analysis, an index showing color change as a function of pH was defined as pHo, which was extracted by an analytical formula characterized by a Boltzmann formula: λ peak ( pH ) =
[λ min − λ max ]
{1 + e
pH − pH o dpH
}
(11.2)
+ λ max
The λmin and λmax stand for the minimum and maximum of the band peak positions, respectively. The pHo is the pH value at which λpeak = (λmin + λmax)/2. The dpH is defined as dpH = (1) (λ max − λ min )/4λ (1) peak , where λ peak is the first derivative of the λpeak (pH) [39,40]. According to the sigmoidal plots shown in Figure 11.5, the color of the 20 nm gold colloid solutions or the Aβ1–40-coated 20 nm gold colloid solutions are clearly distinguishable between pH 4 0.20
pH = 7 ~ 10
pH = 6 pH = 5
Absorbance
pH = 4
FIGURE 11.4 Four representative absorption spectra of gold 20nm nanoparticles and Aβ1–40 solutions between pH 4 and pH 10. (From Yokoyama, K. and D.R. Welchons, Nanotechnology, 18(10), 105101, 2007. With permission.)
0.10
0 500
600
700
Wavelength (nm)
800
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λpeak(n, T = 20°C) (nm)
620 Aβ1–40-coated 20 nm gold
600 580 560
20 nm gold
540
pH 4 pH 10
5
10
15
20
n, Acid or base inductions
(a) 620 Aβ1–40-coated 30 nm gold λpeak(n, T = 20°C) (nm)
600 580
Aβ1–40-coated 20 nm gold
560 Aβ1–40-coated 15 nm gold 540
pH 4 pH 10
5 10 15 n, Acid or base inductions
20
(b) FIGURE 11.6 Comparison of peak shift (λpeak) as a function of pH changes (a) between 20 nm gold colloid (white circles) and Aβ1–40-coated 20 nm gold colloid particles (black circles) at 20°C and (b) between Aβ1–40-coated 15, 20, and 30 nm gold colloid particles at 20°C. The upward and downward arrows indicate an injection of HCl and NaOH, adjusting the pH of the solution to pH 4 and pH 10, respectively. The dashed lines indicate the values predicted by Equation 11.3. The thin line is provided to indicate the trend of the peak shift observed for 20 nm gold colloid particles. (From Yokoyama, K., Gaulin, N.B., Cho, H., and Briglio, N.M., Journal of Physical Chemistry A, 114, 1521, 2010.)
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50 nm
λpeak, peak (nm)
640
30 nm
600
10 nm 560
520 0
4
8 12 16 n, Acid or base inductions
20
FIGURE 11.7 Demonstration of the color change reversibility seen in ovalbumin-coated 10 nm (closed circles), 30 nm (shaded circles), and 50 nm (open circles) gold colloidal particles. The upward and downward arrows indicate an injection of HCl and NaOH, adjusting the pH of the solution to pH 4 and pH 10, respectively. The dashed lines indicate the values predicted by Equation 11.3 and the parameters given in Table 11.1. (From Yokoyama, K., et al., Nano-Technolog., 19, 375101, 2008.)
peak position does not necessarily match with the absorption wavelength corresponding to the color of each solution. For example, if a two component spectra consists of λ1 = 570 nm and λ2 = 700 nm components, the color of the solution becomes red or blue as λ1 or λ2 is a predominant component, respectively. As the number of cycles increased, the peak at pH 10 shifted gradually from 528 to 560 nm. While the ending peak varied depending on the size of the gold colloid, the same general trend was observed for ovalbumin conjugated with gold colloid for all tested sizes. For most of the data, the spectra were collected with a maximum wait time of less than 30 s between pH changes. A wait time of 10 min resulted in no spectral shift. Thus, the condition of solution after each pH was assumed to be stable. To show the dispersed or aggregated form of gold colloids at basic or acidic condition, the TEM images are shown for representative n in Figure 11.8. The TEM image analysis was performed by converting the image to data of pixel coordinate and corresponding color index. We set the threshold in the color index to recognize the group of pixels corresponding to the gold particles and the average size of the gold particles, ratio of the area occupied by the gold particles (occupancy, %), and the number of the gold particles were calculated. We observed a clear correlation between spectroscopic (or colorimetric) indication of the solution and the conformation of the gold colloids observed in TEM. Generally speaking, the gold particles were dispersed for the initial (pH = 7) and basic conditions, and the gold particles aggregated at acidic condition. This phenomenon was quantified by the occupancy rate of the area by the gold particles. For example, the occupancy rate of the Aβ1–40-coated 20 nm gold particles at n = 2 is ∼73% and that for n = 3 is only ∼0.6%. Even at the basic condition, we observed small clusters of gold particles that consist of 10 or 20 gold particles as n increases. However, the cluster size never approached the acidic condition where clusters contained thousands of gold particles. The number of particles that form aggregates decreased as the number of cycles increased. This may imply that the Aβ1–40 underwent irreversible structural changes as the cycle number was raised. Thus, the number of gold particles per aggregate dropped in the low pH 4.0 samples. This presumably also explains the λpeak shifts toward larger wavelengths after several cycles of pH changes.
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TEM 600
λpeak(n, T = 20°C) (nm)
n = 14 1 μm 580
n=8
n = 20 n = 21
n=2
n = 13 n=7
560
n=3
540 n=1
5
10
n
15
20
FIGURE 11.8 Peak shift (λpeak) as a function of pH changes for Aβ1–40-coated 20 nm gold colloid particles at 20°C is shown along with the TEM images of Aβ1–40-coated 20 nm gold colloids: n = 1 (pH 7), n = 2 (pH 4), n = 3 (pH 10), n = 7 (pH 10), n = 8 (pH 4), n = 13 (pH 10), n = 14 (pH 10), n = 20 (pH 4), and n = 21 (pH 10). (From Yokoyama, K., Gaulin, N.B., Cho, H., and Briglio, N.M., Journal of Physical Chemistry A, 114, 1521, 2010.)
In our study, the undulation of wave peak of each acid or base addition operation, λpeak(n, T), is represented by the following analytical formula: CTd
λ peak (n, T ) = ATd + BTd (n − 1)
+ DTd e
(( n −1)ETd ) cos(nπ)
(11.3)
Where n indicates the operation of pH change and varies from 1 to 21. The n = 1 indicates the starting pH, which is around pH 7. Odd numbers of n (n = 3, 5, 7, …) indicate an operation of acid addition, decreasing the pH of solution to pH 4, whereas even numbers of n (n = 2, 4, 6, …) show an operation of base addition to increase the pH of solution to pH 10. The parameters ATd , BTd , CTd , DTd , and ETd were extracted and tabulated in Table 11.1 for each temperature, T (°C), of gold colloids with diameter of d nm. Since the meaning and implication of each parameter was already explained in detail [41], a brief description of the parameters is given here. Equation 11.3 was derived to reproduce the spectral shift and repetition change. In this equation, an initial peak position at neutral pH (i.e., λpeak(1, T)) is given by ATd − DTd , and the parameters BTd and CTd show the average wave peak position shift as pH varies between pH 4 and pH 10. The parameter DTd and ETd imply amplitude and damping factor for the repetitive event, and the cosine function was used to indicate increase and decrease of λpeak(n, T) upon change between pH 4 and pH 10. The values calculated by Equation 11.3 are effective only for each n value, not the values between each n. Thus, the dotted lines shown in Figures 11.6 through 11.8 are given only for the purpose of clarifying the repetitive trend. In Figure 11.9, three cases of simulations of reversible peak shift are demonstrated to indicate the role of parameters B and E. The case (1) shows a reproduction of λpeak(n, T) 20 for Aβ1–40-coated 20 nm gold colloids based on the values given in Table 11.1, where A20 = 20 20 20 20 20 542 nm, B20 = 21 nm, C20 = 0.29, D20 =10 nm, E20 =0.07. In case (2), the parameter B20 and
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TABLE 11.1 d d d d d List of Extracted Parameters at 20°C (A20 , B20 , C20 , D20 , and E20 ) with the Use of Equation 11.3 for Ovalbumin or Aβ1–40-Coated Gold Colloids for the Sizes Where the Reversible Color Change between pH 4 and pH 10 Were Observed
Size of Gold Colloid, d (nm)
d A20 (nm)
d d A20 - D20 (nm)
d B20 (nm)
d C20
d D20 (nm)
d E20
Ovalbumin coated 5 10 20 30 40 50 60 80 100
540.5(8) 540(4) 545(4) 550(6) 561(6) 563(12) 573(4) 599(7) 587(3)
530.5(8) 530(4) 531(4) 531(6) 532(6) 530(12) 546(4) 570(8) 563(3)
1.9(8) 4(3) 14(4) 27(6) 17(6) 18(11) 20(4) 5(6) 4(2)
0.4(1) 0.5(2) 0.42(7) 0.15(5) 0.28(9) 0.4(2) 0.29(5) 0.6(3) 0.7(1)
10.0(4) 10(2) 14(2) 19(3) 29(4) 23(6) 27(2) 29(4) 14(2)
0.003(4) 0.02(2) 0.02(1) 0.08(2) 0.08(2) 0.03(3) 0.025(7) 0.07(2) 0.03(1)
Aβ1–40 coated 20
542(2)
532(2)
21(2)
0.29(3)
10(1)
0.07(2)
600 Case 1 Case 2
λpeak(n) (nm)
580
Case 3 560
540
520
0
5
10 15 20 n, Acid or base inductions
FIGURE 11.9 Simulations of reversible peak shift. Case (1) A reproduction of λpeak(n, T) for Aβ1–40-coated 20 nm gold colloids with the parameters listed on Table 11.1. Case (2) The repetitive feature 20 20 when B20 = E20 = 0 . Case (3) The repeti20 = 1. tive feature when E20
20 damping factor E20 were removed (B↓20↑20 = E↓20↑20 = 0) and a fully repetitive feature was obtained for each cycle. Comparing the cases (1) and (2), the observed repetitive feature in our study can be clearly categorized as “quasi” repetitive color change. In case (3), the 20 = 1 showing almost negligible undulation in peak shift damping factor was set to be E20 between pH 4 and pH 10.
11.4 Protein Conformation in Reversible Self-Assembly The reversible color changes observed for the Aβ1–40-coated 20 nm gold colloid indicated a production of a reversible self-assembly of this protein over the nanocolloidal surfaces. While the colloidal size dependence was suspected to be due to chemicals, such
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as a stabilizer, used to prepare each gold colloidal size, we confirmed with Ted Pella Inc. that the stabilizer used for all gold colloids were the same. All sizes of gold colloids were formed by a Frens-derived citrate reduction method possessing traces of citrate <0.000 01%, tannic acid <0.000 0001%, and potassium carbonate <0.000 000 01%. Thus, the observed size dependence in this study was not determined by the stabilizer of the gold colloids. It is, however, still uncertain which specific portions of the segments of the Aβ1–40 are responsible for this reversibility. Aβ1–40 is a protein categorized as hydrophilic with two α-helical coils, while hydrophobic Aβ1–42 protein is inert to the gold colloid surface and never showed a reversible process. Thus, a hydrophilic and α-helical portion of the protein may be significantly involved in the structure responsible for the reversibility. Since other tested segments of hydrophilic tail (Aβ1–11) or α-helical portion (Aβ12–28) of Aβ did not show any signs of reversibility, hydrophilic or α-helical segments alone are not capable of causing a reversible process. The reversible process occurs only when the three-dimensional network of hydrophilic and α-helical segments are combined and specifically aligned in Aβ1–40 to satisfy a certain structural constraint for the reversibility. The size specificity of 20 nm gold for the reversible process suggested that a reversible self-assembly of Aβ1–40 requires a specific core size. Considering that the reversible process was not observed with adjacent available sizes (i.e., 15 or 30 nm gold colloid), the restriction for the core size is very selective. Since Aβ1–40 monomers conjugate around the gold colloidal particle’s surface, they must form a cage-like structure for any size gold particle as illustrated in Figure 11.10. Thus, the surface curvature or volume space created by 20 nm gold colloid must perfectly match with the conditions for this particular cage structure to support a reversible process. Since the acidic condition (pH 4) is reported to irreversibly denature Aβ1–40 [42,43], the structure responsible for the reversible process must minimize this denaturization. It is speculated that the conformation of the aggregates over 20 nm core size prevents large damage to the α-helix portion of the Aβ1–40. As a step to generalizing a required structural portion for the reversibility, ovalbumin protein was utilized to explore the reversible process and its size dependence. The chicken egg ovalbumin has 385 residues consisting of 11 α-helices and 14 β-chains. It also contains six Cys (cytosine) in which –SH causes strong binding to the gold colloidal surfaces. Since the reversible color change was observed for all tested ovalbumin-coated gold colloids (sizes of 5, 10, 15, 20, 30, 40, 50, 60, 80, and 80 nm), it indicates that there was no specific self-assembly core size under this many amount of α-helices in ovalbumin. However, we observed some size dependences in parameters representing reversible self-assembly. Parameter ATd − DTd represents the λ peak at the starting point (n = 1), when pH is about neutral. While the values of λ(n = 1, T = 20°C) were around 530 nm for gold colloid with sizes Aβ1–40 monomer Cage structure
FIGURE 11.10 Sketches of a formation of a cage-like structure consisted of Aβ1–40 dimers over a 20 nm gold colloid’s surface.
Gold colloid 20 nm
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smaller than 50 nm, the red shift of λ(n = 1, T = 20°C) were observed for gold colloids with sizes 60 nm and above. Parameter BTd is a measure of the average peak wavelength at each induction, which converges at n = ∞. Therefore, a larger value of BTd means that the spectrum converges to a structure stabilized under acidic conditions by exhibiting more red shift at a relatively early induction number (a small value of n). Our result indicated that gold colloid 30 nm has the maximum value in BTd . This may mean that the conformation created around 30 nm gold colloid can be easily denatured by acidic conditions, resulting in more red shift than other conditions. The parameter CTd implies an induction number dependence of parameter BTd . Since there is no size dependence and all values are less than 1.0, parameter BTd is weakly affected by the induction number. The pre-exponential factor DTd represents the half value of the maximum difference of peak wavelength between pH 4 and pH 10 (i.e., the difference between λpeak (pH = 4) and λpeak (pH = 10) is given by 2 × DTd nm). It indicates that the peak difference gets larger as the colloidal size increases, except for 100 nm gold colloid. Therefore, interfacial structural change between basic and acidic conditions must be more drastic as the surface area increases (or surface curvature decreases). The parameter ETd exhibits how fast the oscillation character dies down (i.e., a damping index), and the 5 nm size showed an exceptionally negligible damping index, indicating a possibility of a long-lasting reversible process. For the rest of the sizes, the damping factors were not negligible and the index peaked around 30 and 40 nm of gold colloidal size. The damping index mainly corresponds to a damping factor of the basic condition side (pH = 10), since the acidic condition side (i.e., peak wavelength at pH = 4) reaches an asymptote at an early induction number, whereas the basic condition side (i.e., peak wavelength at pH = 10) kept increasing at each induction. The gradual change of value at the basic condition side implies that the conformation formed at basic condition is less reproduced as the induction number increases. From the observed size dependence, the conformation assembled over the core sizes of 30–40 nm must be more easily denatured. Since ovalbumin did not require any specific size to exhibit a reversible process, it further highlights the specificity of the Aβ1–40 and 20 nm combination. While a specific structure of Aβ1–40 monomers assembled over 20 nm gold is not determined from this study, comparison of 20 nm gold colloid coated with Aβ1–40 and ovalbumin shown in Figure 11.11 can hint some differences in self-assembled conformations between Aβ1–40 and ovalbumin. A remarkable difference, however, is that a parameter indicating a damping factor of amplitudes, ETd , was larger in Aβ1–40-coated gold colloid by a factor of approximately 3.5. This indicates that the reproducibility of the structure over the colloid surface is higher for ovalbumin proteins. It can imply that the segments of ovalbumin that are responsible for the reversible structure are more resistant to the acid. This may also mean that the conformation of Aβ1–40 located on the gold colloid surface was more affected by the surface field (most likely electrostatic interaction) resulting in less freedom for Aβ1–40 to repeat the conformational change. From the fact that the entire peak is shifting toward a longer wavelength from the original starting value, protein may be unfolded (or denatured) by acid, and Aβ1–40 can be said to be denatured more easily by the acid. Our study revealed existence of a reversible structure observed only at an interfacial environment over gold nanocolloidal surfaces. Under no presence of gold colloidal particles, Aβ1–40 was reported to form the β-sheet conformation consisting of the Aβ-oligomers around pH 5.4 [42,43]. This is true for ovalbumin treated by base, and a predominant β-sheet conformation was observed. It was also determined that base-treated ovalbumin exposed hydrophobic segments of the proteins [44]. As reported for Elastin bound to gold nanoparticles, the plausible bond taking place between the gold colloidal surfaces and Aβ proteins can be concluded to be a hydrophobic interaction between –CN– and
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600
λpeak(n) (nm)
580
Ovalbumin-coated 20 nm gold n = 14 Aβ1–40-coated 20 nm gold
n = 20
n=8 n=2
n = 21
n = 13 n=7
560 n=3 540 n=1
pH = 4
5
10
15
20
10 15 n, Acid or base inductions
20
pH = 10 5
FIGURE 11.11 Demonstration of the color change reversibility seen in Aβ1–40-coated 20 nm gold colloidal particles (open circles) and ovalbumin-coated 20 nm gold colloid (closed circles). The upward and downward arrows indicate an injection of HCl and NaOH, adjusting the pH of the solution to pH 4 and pH 10, respectively. The dashed lines indicate the values predicted by Equation 11.3. (From Yokoyama, K., et al., Nano-Technolog., 19, 375101, 2008.)
–NH– groups with the anionic gold colloidal surface [5]. In the case of ovalbumin, the sequence encloses the cytosine including a disulfide (–SH) bond, which is known to have a strong bonding interaction with gold colloidal surfaces. Thus, it is appropriate to assume that much stronger bonding is taking place for the case of ovalbumin and the gold colloid surface by –S–Au bonds through the hydrophilic portion of the cytosines of ovalbumin. Since the hydrophobic portion was considered to be used for bonding to the gold colloidal surfaces, the hydrophilic portion of the protein must be exposed outside of the gold colloidal surfaces and are therefore responsible for interacting with other protein-coated gold colloid. At lower pH, the hydrophilic portion of the protein must be neutralized and aggregated with other protein-coated gold colloids. The aggregation of protein-coated gold colloidal nanoparticles affect surface plasmons, which should result in color changes in the solutions. Based on the fact that both Aβ1–40 and ovalbumin proteins converged into a structure stable under acidic conditions as the reversible structure was repeated, we interpret that the conformation of the hydrophobic portion (i.e., binding portion to the gold surface) remained relatively unaltered. On the other hand, the hydrophilic portion may fold or unfold as the pH changes to lower or higher, respectively. Thus, it is also speculated that the hydrophilic portion is responsible for the reversible process as a function of pH and determines a degree of reversibility (i.e., magnitude of damping index, ETd in Equation 11.3) (see Figure 11.12). To enable Aβ to repetitively converge on a corresponding stable structure under acidic or basic conditions as the reversible structure, the conformation of the section of Aβ1–40 used for binding to the gold surface must remain relatively unaltered. Also, it is reasonable to assume that the segment corresponding to β-pleated sheet formed at pH 4 and α-helical structure formed at pH 10 may be geometrically separated by having the section
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Modeling of Reversible Protein Conjugation on Nanoscale Surface
(b) In solution
(a) Aβ monomer
vo pi 2( –2 17
17 c) 1– phili r do (hy
t) 0 ic) –4 hob 3 2 op r yd h (
pH 10
pH 4
(c) At the interface
Gold surface
pH 10
pH 4
FIGURE 11.12 (a) Sketches of suggested “pivotal structure model,” where sequences Aβ17–22 play as a pivot. (b) In solution, sections of Aβ1–16 and Aβ22–40 unfold around a pivot. (c) At the interface, section of Aβ1–16 contacts with the outer monomer leaving a section of Aβ22–40 on the surface of a particle.
responsible for binding to gold in between. Because the segments 1–17 are assigned as hydrophilic tail (coil) and sequences 18–40 as hydrophobic tail (rod), the former segments are considered to be responsible for the formation of α-coil and the latter component must be responsible for the β-sheet formation. Thus, we speculate that the central segments are used to conjugate the gold colloidal surface as a “pivotal point” and must be interconnected regardless of solution pH (see Figure 11.12). In the conjugation of Aβ1–40 to Congo red (CR) dye, the specific segments of Aβ monomer is claimed to be critical for the binding with CR and they can be estimated as those between sequences 17 and 22 [45]. Additionally, dimers of full-length Aβ indicate that the rate-limiting step involves the formation of a multimeric β-sheet was found to span the central hydrophobic core sequences between 17 and 21 [46]. On the basis of this idea, the mainly hydrophobic segments 17–22 can be considered to be the most plausible section used as a pivotal point as mentioned in this speculation. TEM study revealed that the Aβ1–40-coated gold colloids aggregate at the acidic condition and that they exist as monodispersed particles at the basic condition. The aggregates of the gold colloids at the acidic condition must be connected through the bonding between Aβ1–40 monomers attached on the surface. Since the first reversible process of fibrillogenesis is assumed to be the transition between monomer and oligomer aggregates, the main monomer structure is predominated by α-helices, whereas the β-sheet may be mainly produced in the formation of the aggregates or intermediates. Therefore, the major structural component observed in this reversible process can be simplified as the involvement between β-sheet and α-helices, while the internal bonding between Aβ1–40-coated gold colloids are formed (at pH 4), or disconnected (at pH 10), respectively. The simulation on aggregation of Aβ16–22 peptides in water showed a hydrophilic character of aggregation with decrease in density of hydration water, whereas the peptide surface exposed to water became more hydrophobic with increasing aggregate size [30,47]. Thus, it supports the fact that
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hydrophobic segments are utilized for the aggregation between Aβ monomers. In addition, the binding of Aβ to the anionic lipid membrane was reported to be an entropy-driven reversible transition between random coil ↔ β-sheet and the binding enthalpy was found to always be endothermic. The large positive entropy together with the large negative heat capacity suggests a prominent role of hydrophobic interactions in peptide aggregation [48]. An aggregation of soluble Aβ conducted at polar-nonpolar interfaces was reported to produce the aggregates with β-structure rich conformation, which resembles Aβ protofibrils [49]. Therefore, we conclude that the β-sheet section is aligned outward at the acidic condition and is used for interconnecting gold colloids to each other to form the aggregates.
11.5 Temperature Dependence of Conjugation of Amyloid Beta Protein on the Surfaces of Gold Colloidal Nanoparticles The reversible color change was further examined at various temperature conditions between 5°C and 50°C for Aβ1–40-conjugated 15, 20, 30, and 40 nm gold colloids as shown in Figure 11.13. Prior to this study, only the 20 nm gold colloid conjugated with Aβ1–40 was considered to exhibit the reversible color change. This study intends to understand the role λpeak(n, T ) 640
50°C
600 560 520
30°C
600 560 520 10°C
600 560 520
5°C
600 560 520 (a)
0
10
20
0 (b)
10
20
(c)
0
10
20
0 (d)
10
20
n
FIGURE 11.13 Collections of peak shift, λpeak(n, T), as a function of pH changes (n) for Aβ1–40-coated gold colloid particles of (a) 15 nm, (b) 20 nm, (c) 30 nm, and (d) 40 nm at selected temperatures between 5°C and 50°C. The dashed lines are provided to indicate the feature of the peak shift at a given pH. (From Yokoyama, K., Gaulin, N.B., Cho, H., and Briglio, N.M., Journal of Physical Chemistry A, 114, 1521, 2010.)
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of the thermal condition in the reversible process as well as the corresponding conformation of the Aβ aggregates formed over the gold colloid surface. In principle, the change in thermal condition must impact the surface potential. If the surface potential can possess enough energy, this may overcome the physical size restriction of the gold colloids preventing the reversible process. This implies that the reversible process may be observed in sizes of the gold colloids other than 20 nm diameter. When the reversible process was examined for Aβ1–40-coated gold colloid with the sizes of 15, 20, 30, and 40 nm at the temperature ranges between 5°C and 50°C, all Aβ1–40-coated gold colloids exhibited an undulating feature in λpeak(n, T) as the pH changed between pH 4 and pH 10. However, in most cases the amount of shift at the basic condition is not significant enough to provide reddish color preventing visual observation of repetitive color change between reddish and bluish color. For example, the reversible process was visually observed for Aβ1–40-coated 20 nm gold at all shown temperatures except for the case when temperature is at 5°C. For Aβ1–40-coated 30 nm the amplitudes of λpeak(n) were significantly large enough to provide the reversible color change at 10°C. This was true at 5°C for Aβ1–40-coated 40 nm gold colloids. To clarify the transition point where the reversible change took place, the pH-induced peak shift was investigated with a step of 1°C as shown in Figure 11.14, panels a–c. (In order to avoid congestions, some representative peaks are selected here.) For 20 nm, the peak shift was investigated from 10°C to 5°C for every 1°C, since 5°C is the lowest temperature available in this experiment. The amplitude of peak shift was significantly reduced at 5°C
600
600
600
560
560
520
520
520
7°C
600
600
560
560
520
520
520
600
600
560
560
560
520
520
520 640
600
5°C
600
560 520
0
10
20
n
17°C
560
520
520
(b)
6°C
600
560 0
7°C
640
18°C
6°C
8°C
640
19°C
560
600
(a)
640
20°C
560
600
λpeak(n, T )
8°C
10
20
n
5°C 0
10
20
n
(c)
FIGURE 11.14 Demonstration of color reversibility of (a) Aβ1–40-coated 20 nm gold particles between 5°C and 8°C, (b) Aβ1–40-coated 30 nm gold particles between 17°C and 20°C, and (c) Aβ1–40-coated 40 nm gold particles between 5°C and 8°C. (From Yokoyama, K., Gaulin, N.B., Cho, H., and Briglio, N.M., Journal of Physical Chemistry A, 114, 1521, 2010.)
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TABLE 11.2 List of Extracted Parameters (AT15, BT15 , CT15 , DT15, and ET15 ) with the Use of Equation 11.3 for Spectrum Peak Shift (λpeak) between pH 4 and pH 10 for Aβ1–40-Coated 15 nm Gold Colloids Temperature (T)/°C
AT15 /nm
AT15 - DT15/nm
BT15 /nm
CT15
DT15 /nm
ET15
5 10 20 30 40 50
534(2) 534(1) 532(2) 533(2) 532(2) 531(1)
531(2) 531(1) 529(2) 529(2) 529(2) 528(1)
12(2) 16(1) 23(2) 27(2) 36(2) 44(1)
0.16(4) 0.15(2) 0.14(2) 0.10(2) 0.05(1) 0.006(4)
3.2(9) 3.7(6) 3.3(8) 3.6(9) 3.3(7) 2.2(3)
0.02(2) 0.04(2) 0.05(3) 0.04(3) 0.02(2) 0.002(10)
TABLE 11.3 List of Extracted Parameters (AT20 , BT20, CT20 , DT20, and ET20) with the Use of Equation 11.3 for Spectrum Peak Shift (λpeak) between pH 4 and pH 10 Observed in Aβ1–40-Coated 20 nm Gold Colloids Temperature (T)/°C
AT20/nm
AT20 - DT20/nm
BT20/nm
CT20
DT20 /nm
ET20
5 10 20 30 40 50
534(2) 551(2) 545(6) 550(3) 545(3) 560(4)
530(2) 532(2) 531(6) 535(3) 530(3) 534(4)
46(2) 14(2) 19(6) 29(3) 44(4) 44(4)
0.10(1) 0.21(4) 0.29(7) 0.17(2) 0.13(2) 0.39(6)
4.8(9) 19.6(9) 14(3) 15(1) 15(2) 26(2)
0.02(2) 0.010(4) 0.04(2) 0.04(1) 0.03(1) 0.03(1)
as shown in Figure 11.14a when reversible color change became visually unidentifiable. For 30 nm gold coated with Aβ1–40, the peak shift was investigated between 20°C and 10°C in 1°C increments in order to find the temperature where the reversible color change began. A significant increase in amplitude was found at 18.0°C ± 0.2°C and lower, and reversible color change was visually identified (Figure 11.14b). As for 40 nm gold covered with Aβ1–40, we investigated the peak shift by scanning the temperature between 10°C and 5°C by 1°C step (Figure 11.14c). We found a sign of the reversible color change when temperature was lowered to 6.0°C ± 0.2°C. The observed λpeak(n, T) for each gold colloidal size were analyzed by Equation 11.3 and fitting parameters (ATd , BTd , CTd , DTd , and ETd ) were tabulated in Tables 11.2 through 11.5. Here, we show the details of temperature dependence of each parameter for each size of gold colloid (15, 20, 30, and 40 nm) conjugated with Aβ1–40. 11.5.1 15 nm Gold Colloids Coated with Aβ1–40 The reversible color change was not colorimetrically observed at all temperatures for 15 nm gold colloid conjugated with Aβ1–40. The peak shift, λpeak(n, T), shows the undulating feature with a small amplitude as a function of pH as shown in Figure 11.13a, that is, DT15 < 5.5 nm (Figure 11.15d–i), where DT15 indicates a critical threshold required to support a visual observation of the reversible color change. As the overall position of the peak shift
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TABLE 11.4 List of Extracted Parameters (AT30, BT30, CT30, DT30, and ET30) with the Use of Equation 11.3 for Spectrum Peak Shift (λpeak) between pH 4 and pH 10 for Aβ1–40-Coated 30 nm Gold Colloids Temperature (T)/°C
AT30 /nm
AT30 - DT30/nm
BT30/nm
CT30
DT30 /nm
ET30
5 10 20 30 40 50
540(2) 544(2) 533(1) 533(2) 535(3) 536(3)
530(2) 530(2) 531(1) 531(2) 531(3) 531(3)
7(2) 11(2) 27(1) 27(2) 55(4) 55(3)
0.40(6) 0.28(4) 0.16(1) 0.05(1) 0.08(1) 0.09(1)
9.8(8) 13.2(8) 2.7(4) 2.8(8) 4(2) 5(1)
−0.008(6) 0.013(6) 0.03(1) 0.04(3) 0.06(5) 0.04(3)
TABLE 11.5 List of Extracted Parameters (AT40, BT40, CT40, DT40, and ET40) with the Use of Equation 11.3 for Spectrum Peak Shift (λpeak) between pH 4 and pH 10 for Aβ1–40-Coated 40 nm Gold Colloids Temperature (T)/°C
AT40/nm
AT40 - DT40/nm
BT40/nm
CT40
DT40/nm
ET40
5 10 20 30 40 50
537(6) 534(2) 533(2) 533(1) 534(1) 534(2)
529(6) 531(2) 531(2) 531(1) 531(1) 531(2)
23(6) 58(3) 64(2) 73(1) 73(1) 74(3)
0.36(7) 0.09(1) 0.07(1) 0.051(3) 0.032(4) 0.045(7)
8(2) 3(1) 3(1) 2.7(5) 3.5(5) 3(1)
−0.04(2) 0.04(3) 0.02(2) 0.08(2) 0.02(1) 0.05(4)
shows in the figure, the color of the solution was bluish (or dark purple) regardless of pH except for that at the initial condition (n = 1). All parameters of Equation 11.3 extracted for Aβ1–40-coated 15 nm gold colloids exhibited a monotonic, or continuous, trend as shown in Figure 11.15a-i, b-i, c-i, d-i, and e-i. The parameters AT15 , DT15, and ET15 showed almost constant values under all tested temperature ranges, whereas the parameters BT15 and CT15 showed slight monotonic increases or decreases as the temperature increased. The trend observed for Aβ1–40-coated 15 nm gold colloids was used as the standard feature for identifying the case where no visual confirmation of the reversibility was made. 11.5.2 20 nm Gold Colloids Coated with Aβ1–40 The reversible color change was confirmed for the entire temperature range visually except for that at 5°C, and it was confirmed by the fact that D520 did not go over the threshold DTd = 5.5 nm. The DTd values had a local maximum at 10°C and an abrupt increase at 50°C among the tested temperature range. The high value of DT20 at high temperature is an especially unique feature found only for 20 nm gold colloid conjugated with Aβ1–40. The trend seen in parameter CT20 somewhat correlates with parameters AT20 and DT20. As seen in Figure 11.15e-ii, all ET20 values stayed above zero, indicating that the amplitude of the undulating feature damped down as n increased.
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d
AT
570 560 550 540 530
(i) Aβ1–40-15 nm gold
570 560 550 540 530
(iii) Aβ1–40-30 nm gold
570 560 550 540 530
(ii) Aβ1–40-20 nm gold
570 560 550 540 530
(iv) Aβ1–40-40 nm gold
(a)
10
20 30 40 50 Temperature (°C)
10
20 30 40 50 Temperature (°C)
d
BT 100 80 60 40 20
(i) Aβ1–40-15 nm gold
100 80 60 40 20
(ii) Aβ1–40-20 nm gold
(b)
100 80 60 40
(iii) Aβ1–40-30 nm gold
20 100 80 60 40 20
10 20 30 40 50 Temperature (°C)
(iv) Aβ1–40-40 nm gold
10
20 30 40 50 Temperature (°C)
CTd 1.0 0.8 0.6 0.4 0.2
(i) Aβ1–40-15 nm gold
1.0 0.8 0.6 0.4 0.2
(iii) Aβ1–40-30 nm gold
1.0 0.8 0.6 0.4 0.2
(ii) Aβ1–40-20 nm gold
1.0 0.8 0.6 0.4 0.2
(iv) Aβ1–40-40 nm gold
(c)
10
20 30 40 50 Temperature (°C)
10
20 30 40 50 Temperature (°C)
FIGURE 11.15 (a) Plotting of the parameter ATd in Equation 11.3 as a function of gold colloidal sizes for Aβ1–40 -conjugated (i) 15 nm, (ii) 20 nm, (iii) 30 nm, and (iv) 40 nm gold colloid particles. (b) Plotting of the parameter BTd in Equation 11.3 as a function of gold colloidal sizes for Aβ1–40-conjugated (i) 15 nm, (ii) 20 nm, (iii) 30 nm, and (iv) 40 nm gold colloid particles. (c) Plotting of the parameter CTd in Equation 11.3 as a function of gold colloidal sizes for Aβ1–40conjugated (i) 15 nm, (ii) 20 nm, (iii) 30 nm, and (iv) 40 nm gold colloid particles.
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d
2.5 2.0 1.5 1.0 5 2.5 2.0 1.5 1.0 5 (d)
(i) Aβ1–40-15 nm gold
DT 2.5
2.0 1.5 1.0 5 2.5
(ii) Aβ1–40-20 nm gold 10
2.0 1.5 1.0 5
20 30 40 50 Temperature (°C)
(iii) Aβ1–40-30 nm gold
(iv) Aβ1–40-40 nm gold
10
20 30 40 50 Temperature (°C)
ETd 0.1 0.05 0 –0.05 –0.1 0.1 0.05 0 –0.05 –0.1 (e)
(i) Aβ1–40-15 nm gold
0.1 0.05 0 –0.05 –0.1 (iii) Aβ 1–40-30 nm gold
(ii) Aβ1–40-20 nm gold
0.1 0.05 0 –0.05 –0.1 (iv) Aβ 1–40-40 nm gold
10
20 30 40 50 Temperature (°C)
10
20 30 40 50 Temperature (°C)
FIGURE 11.15 (continued) (d) Plot of the parameter DTd in Equation 11.3 as a function of gold colloidal sizes for Aβ1–40-conjugated (i) 15 nm, (ii) 20 nm, (iii) 30 nm, and (iv) 40 nm gold colloid particles. The horizontal dotted line at 5.5 nm indicates the threshold and is the minimum amount required to support visual observation of the reversible color change. (e) Plot of the parameter ETd in Equation 11.3 as a function of gold colloidal sizes for Aβ1–40 conjugated (i) 15 nm, (ii) 20nm, (iii) 30nm, and (iv) 40nm gold colloid particles. (From Yokoyama, K., Gaulin, N.B., Cho, H., and Briglio, N.M., Journal of Physical Chemistry A, 114, 1521, 2010.)
11.5.3 30 nm Gold Colloids Coated with Aβ1–40 When Aβ1–40 was coated over 30 nm gold colloid, the reversible color change was visually observed at 18°C ± 0.02°C and lower. Although the feature at 18°C possessed a high damping mode, a clear reversible change with relatively high amplitude and lower damping mode was observed from 20°C to 17°C as shown in Figure 11.14b. The induction of the visually observable reversible change was clearly confirmed in the plot of parameter DT30 (Figure 11.15d-iii) as a stepwise increase at 18°C. Since all other parameters indicated the noncontinuous spot at 18°C as shown in Figure 11.15a-iii, b-iii, c-iii, and e-iii, 18°C is a critical temperature for Aβ1–40-coated 30 nm gold colloid to cause a significant change in the structure of Aβ1–40 constructed over the surface. The parameters AT30 and CT30 showed almost the same trend as that seen in parameter DT30 , where a stepwise increase began at 18°C. Especially, the relatively higher values of CT30 at T < 20°C reflect the non-undulating feature of the average peak shift as a function of n. This is highly contributed by the
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feature seen at smaller n, where the peak shift shows very small changes between neven and nodd. Quite interestingly, the damping factor represented in parameter ET30 showed a cusp at both 18°C and 19°C, whereas the overall trend indicated a smooth decrease as temperature decreased. The high damping factor at these temperatures can be easily confirmed from the features seen in the peak shift plot shown in Figure 11.14b, where amplitude of λpeak(n, T = 18°C and 19°C) decreased drastically as n increased. Since the damping factor E530 is negative, the amplitude was slightly larger at 5°C. 11.5.4 40 nm Gold Colloids Coated with Aβ1–40 The reversible color change was visually observed at 6°C and lower where a clear stepwise increase in parameter D640 was seen as shown in Figure 11.15d-iv. However, λpeak(n = 1–3, T = 5°C and 6°C) exhibited a peculiar feature; an initial peak shift at n = 2 and 3 did not indicate a clear trend of the reversible change as shown in Figure 11.14c. This was reflected in CT40 at T = 5°C and 6°C (Figure 11.15c-iv) as non-monotonic change in peak shift between neven and nodd. The parameters BT40 , DT40, and ET40 indicated the noncontinuous spot around 6°C as shown in Figure 11.15b-iv, d-iv, and e-iv. The parameters CT40 highly correlated with parameter DT40 , where a stepwise change seen at 6°C. The negative ET40ʺ 8 indicates that the amplitude was slightly larger at these temperatures as n increased. Among listed parameters, the parameter DTd is critical; it indicates whether the reversible process can be recognized visually or not. The half amount of peak shift between λpeak(nodd, T) and λpeak(neven, T) is represented by parameter DTd . Our observation concludes that the visual confirmation of the reversible process was possible when parameter DTd was over 5.5nm, which was indicated in Figure 11.15d with dashed lines in order to clarify that the reversible change was observed at the temperatures when DTd resides above this threshold.
11.6 Suggested Model for Nanoscale Temperature-Dependent Conjugation We discovered that the gold colloidal surface with diameters of 30 and 40 nm exhibit the reversible process when the temperature was lowered to 18°C ± 0.2°C and 6°C ± 0.2°C, respectively. Although 30 and 40 nm are commercially available sizes close to 20 nm, and these sizes can be speculated to support the reversible change, it is puzzling that no sign of the reversibility was found in 15 nm gold colloids coated with Aβ1–40. This reveals that 20 nm is the smallest commercially available diameter that supported the reversible structure of the Aβ1–40. It is still a possibility that the threshold size for a reversible color change is between 15 and 20 nm. The present observation would confirm an involvement of a particular structure or conformation by Aβ monomers that may be stable at a particular temperature as seen in Aβ1–40-conjugated 20, 30, or 40 nm gold colloids. It is conceivable that the formation of the intermediates can be regarded as the transition state with relatively higher thermal energy than those of reactants or products, and the reversible process would be supported at relatively higher temperatures but not at the relatively lower temperatures as observed in our study. For example, the nucleation constant on an Aβ monomer is known to follow the Arrhenius law with an activation energy, EA, of 311.2 kJ/mol at pH 7.4 under the temperatures ranging from ~30°C to ~45°C [50]. Fibril elongation rate constant was found to increase by two orders of magnitude by following the Arrhenius law with
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403
EA of 96.2 kJ/mol with 0.1 M HCl or 62.8 kJ/mol at pH 3.1 over the temperature range from 4°C to 40°C [51,52]. These activation energies imply significant conformational change in the binding of Aβ monomers to fibril ends; they may be either a conformational change of peptides or oligomers involved in binding to protofibrils or to a local reorganization of each aggregate [51]. Based on this, protofibrils are considered to be formed via a single no cooperative elongation mechanism and is well described as a linear colloidal aggregation due to diffusion and coalescence of growing aggregates. However, it should be emphasized that the above studies were conducted in the Aβ1–40 solution and not under the interfacial environment. On the basis of the fact that the intermediates over the 30 or 40 nm colloidal surfaces were stably prepared at the lower temperatures, the kinetics of Aβ1–40 nucleation over the nanocolloidal surface must be driven by completely different mechanisms than those for Aβ1–40 monomers dispersed in solution. By adopting the concept of homogeneous nucleation theory, it is possible to explain why the reversible color change was observed as the temperature was lowered for 30 and 40 nm gold colloids conjugated with Aβ1–40. Homogeneous nucleation theory hypothesizes that aggregation obeys the same rule as the formation of dew drops from supersaturated water vapor [53], and the Gibbs energy change, ΔG*, with nucleation radius, R* is given by ΔG* =
4πσR*2 16πσ 3 υ2 = 3 3(kT ln S)2
(11.4)
where k is the Boltzmann constant T is the temperature σ is the surface tension of the liquid υ is the molecular volume S is the supersaturation ratio (S) C/Csat, where C is solute concentration and Csat is saturation concentration For the range of radius focused in this study (i.e., nanometers range), the ΔG* required to form an aggregate from the solution phase is positive, thus, the aggregates can be formed only when S > 1. Equation 11.4 also reveals that the ΔG* becomes more positive to support more nucleation as the temperature decreases; this is consistent with the fact that the reversible color change was supported at the lower temperature. Thus, at the lower temperatures, ΔG* must be larger enough to compensate for the size restriction prohibiting the formation of the nucleation at 30 and 40 nm gold colloidal surfaces. On the other hand, the reversible process was observed at even higher temperatures, such as 50°C, for Aβ1–40-coated 20 nm gold colloids. We interpret this to mean that the gold colloid size of 20 nm possesses the interfacial potential that can support the intermediates of the folded form at wider temperature ranges. At room temperature, we speculated that the cage structure consisted of Aβ1–40 aggregates was formed around the 20 nm gold colloid, and this cage structure must be most stabilized with an inner diameter of 20 nm. If cage structure is the major conformation for the nucleation, physical restriction (such as core size of the nucleation) is considered to support the stable cage-formed aggregates under a wide temperature range. This assumption is consistent with the speculation that 20 nm gold colloid supports a reversible color change at wider temperature range, if the critical diameter of the core for nucleation is 20 nm for Aβ1–40 aggregates.
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Although the assumption of the construction of cage structure by Aβ1–40 aggregates is not popularly investigated by any groups, our findings of size dependence in reversible color change strongly suggests that these particular sizes (20, 30, and 40 nm) are associated with the size of stable Aβ1–40 aggregates. Garai et al. defined the nucleation radius to lie between 5 and 50 nm utilizing homogeneous nucleation theory [54]. This is consistent with our studies indicating that the Aβ1–40 aggregates are stabilized with 20 nm core diameter at room temperature and with 30 or 40 nm core diameter at lower temperatures. They also calculated the surface tension of the aggregate of the Aβ/water interface to be more than 4.8 mJ/m 2 at room temperature, and the surface energy barrier to nucleation to be more than 1.93 × 10−19 J with the number of monomers in the nucleus to be more than 29. The structure involved in the reversible color change may correspond to the structural change that exists in the reversible step of fibrillogenesis where a cluster of monomeric units form a nucleotide oligomer (see Figure 11.3b). Our present study provided that the structural change enabling a reversible process; this must be attributed to a three-dimensional size-dependent or surface curvature-dependent hydrophilic potential network only possible at a particular nanosize and a particular temperature range. Plausible structures specific to a certain temperature may be involved in our present observation and can be speculated from the reported structure of Aβ1–40, although they are in dispersed form in solution but not placed at the interfacial condition. For example, molecular dynamics calculation suggest that the shape of the free energy landscape for folding of the Aβ12–36 in a 2,2,2-trifluoroethanol (TFE)/water solution to be strongly dependent on temperature. The overall shape was funnel-like with the bottom of the funnel coinciding exactly with the NMR structure above 325 K, and the landscape became increasingly rugged with the emergence of new conformational clusters connected by low free-energy pathways below 325 K [55]. The Aβ1–40 in an aqueous solution (pH ~4.6) at 15°C is known to have substantially unfolded and/or unusually folded conformation characteristic of an Aβ monomer or dimer containing a rapid reversible coil, while it is a β-strand at 37°C [56]. The dimer is reported to be the predominant species at pH 7.0 as well as at physiological concentrations for Aβ, and the β-sheet formation reaches a maximum around pH 5.4 [42,57,58]. However, the above-mentioned structures and corresponding temperature of the oligomer still provide small implications to the temperatures observed in our study in which the reversible color change began to be observed. More appropriate temperature correspondence and plausible structure of Aβ-oligomers constricted at the nanocolloidal surface can be estimated from a replica exchange molecular dynamics simulation performed with Aβ10–35 [59]. This simulation illustrated several possible oligomers (dimers, trimers, and tetramers) and their characteristic temperature dependence. For example, essentially all strands were calculated to exist as a dimeric form at <10°C up to <77°C, and dimer concentration maximizes at <47°C. The trimeric conformers are in equilibrium with dimeric and monomeric structures from <10°C with the concentration maximum at 17°C; random unfolded structures of monomers dominate above <77°C. Here, the temperature for trimer maximum has a good correspondence with our observation of the reversible color change on 30 nm gold colloids coated with Aβ1–40 that began at 18°C as temperature decreased. Thus, the oligomer form involved at the 30 nm gold colloidal surface may possess mainly the trimeric form (see Figure 11.16). Although a strong implication was not concluded, the oligomer form involved in 40 nm gold colloidal surface may also be in trimeric form based on the fact that the trimeric form is more preferable at the lower temperature in the simulation. We speculate that the dimeric oligomer
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20°C
15°C
Trimer-form
FIGURE 11.16 Schematic sketch showing the formation of the trimer-based cage structure at the lower temperature over the surface of 30 nm gold colloidal particles.
form was involved at the 20 nm colloidal surface, since the dimeric form was supported for wider temperature range in the above-mentioned simulation; thus, it supports our observation of the reversible color change on 20 nm gold colloid coated with Aβ1–40 at the temperature range between 6°C and 50°C. (We assume that each dimeric form composes a cage-like structure around 20 nm gold.) The simulation implies that the monomer form was supported at temperatures lower than 283 K (~10°C) and the monomer was predominantly formed over the dimeric form at 5°C, prohibiting us from observing the reversible color change at this temperature.
11.7 Conclusion In conclusion, evidence of the reversible conjugation of Aβ1–40 and ovalbumin over gold nanocolloid surfaces was obtained. While quasi-reversible color change between pH 4 and pH 10 took place only with Aβ1–40-coated 20 nm gold colloids, all tested sizes of gold colloidal nanoparticles showed a quasi-reversible color change for ovalbumin-coated gold colloids. The reversibility of a structure was speculated to be determined by how acidic conditions denature a hydrophilic portion of the protein. Generally, the ovalbumin-coated 20 nm gold colloid exhibited a higher degree of reversibility than Aβ1–40-coated 20 nm gold colloid. The reversibility was found to be minimized at ovalbumin-coated 30 or 40 nm gold colloid at 20°C. However, Aβ1–40-coated 30 and 40 nm colloids exhibited the reversible color change as temperature was lowered to 18°C ± 0.2°C and 6°C ± 0.2°C, respectively. For the Aβ1–40-coated 20 nm gold colloid nanoparticles exhibited a reversible color change under temperatures ranging from 6°C to 50°C though, the color change was stopped at 5.0°C ± 0.2°C. This specific and unique size and temperature dependence in reversible color change must originate from specific surface potential unique to each nanosize. The reversible color change may correspond to a reversible step reported as initiation of the fibrillogenesis in Alzheimer’s disease. Our study suggests that fibrillogenesis could significantly be enhanced depending upon the nanoscale surface curvature.
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Acknowledgments This work is supported by the National Science Foundation under Grant NSF-NER 0508240. A generous contribution from Geneseo Foundation at an initial stage of this project is greatly acknowledged. The following individuals also should be recognized for their contribution to the project described here: N. B. Gaulin, N. M. Briglio, H. Cho, D. Sri Hartati, S. M. W. Tsang, J. E. MacCormac, and D. R. Welchons. With kind permission from American Chemical Society and Institute of Physics (UK), the contents of the following articles are displayed in this document: “Temperature Dependence of Conjugation of Amyloid Beta Peptide on the Gold Colloidal Nanoparticles,” Journal of Physical Chemistry A, 114, 2010, 1521–1528 by K. Yokoyama, N. B. Gaulin, H. Cho, and N. M. Briglio. “Nano-scale Size Dependence in the Conjugation of Amyloid Beta and Ovalbumin Proteins on the Surface of Gold Colloidal Particles,” Nano-Technolog, 19, 2008, 375101– 375108 by K. Yokoyama, N. M. Briglio, D. Sri Hartati, W. S. M. Tsang, J. E. MacCormac, and D. R. Welchons.
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12 Computational Technology in Nanomedicine Viroj Wiwanitkit CONTENTS 12.1 Introduction........................................................................................................................ 411 12.2 Overview of Computational Approach in Nanomedicine .......................................... 412 12.3 Computational Nanomedicine with MATLAB® ........................................................... 413 12.3.1 Summary of Important Reports on Application of MATLAB in Nanomedicine.................................................................................................... 414 12.4 Computational Tools with MATLAB Programming Environment for Management of Biomedical Data .............................................................................. 415 12.4.1 Usefulness of MATLAB in Developing Other Computational Tools............. 416 12.4.2 Important Computational Tools with MATLAB Programming Environment for Management of Biomedical Data .......................................... 416 12.5 Common Applications of MATLAB in General Nanomedicine Practice.................. 418 12.5.1 Examples of Nanomedicine Researches Based on MATLAB Application.... 419 12.6 Opportunities and Challenges of Computational Technology in Nanomedicine... 424 12.7 Conclusion ..........................................................................................................................425 References......................................................................................................................................425
12.1 Introduction The connection between biological and physical science is the new wave in science at present. Joining and bridging between two different branches of science leads to the emergence of new merit branch of science. Nanoscience is a result of such integration and seems to be the newest science of the present century. There are several branches of nanoscience that can be applied in the present day. At present, nanomedicine is an important branch of nanotechnology dealing with medical aspect. This specific branch of nanoscience is a very useful novelty in medicine that becomes a hope in diagnosis and treatment of several diseases. The application of nanomedicine in an actual practice or real life can be in many dimensions and facets. The application might be in vivo, in vitro, or in silico measures [1]. The in silico approach is the presently focused measure of application. Indeed, in silico technique is a very helpful and widely used technique at present. This technique makes use of computational simulation to answer the study or the research question. The blooming of in silico technique is due to the recent launch of “omics” sciences in bio-chemoinformatics [2]. It is accepted that the application of computational technology is an important factor in drifting the development of nanomedicine researches. At present, several computer programs
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can be useful in nanomedicine research. However, an important and easy-to-use program is MATLAB®. MATLAB can help medical scientists answer the complex question. Simulation of the complex biological processes can be easily performed based on MATLAB. In this brief, specific chapter, the author discusses how MATLAB can be useful in nanomedicine. Also, a summary of important reports on the application of MATLAB in nanomedicine is given in this chapter.
12.2 Overview of Computational Approach in Nanomedicine As earlier noted, nanomedicine is the newest approach in medicine. It deals with the “nano” scale in medicine. Indeed, the “nano” is not a surprising thing in medicine. Several things in medicine are “nano.” For example, a virus is very small and has the size in nanolevel. The action of drug, in bio-pharmacological process, also occurs at a very small, “nano” environment. Hence, the nanomedicine is not actually a new thing but a new look into medical phenomenon. At present, many techniques are available for study in nanomedicine. The production of new medical diagnostic tools via nanoengineering, the synthesis of new medically active molecules by nanopharmacology and nanochemistry, etc. are good examples of activity in nanomedicine. However, sometimes, it is very hard to study something in medicine. For example, the dynamics of biological process in the nanolevel is very hard to study and observe. Hence, there must be some new approaches to get required knowledge. The use of in silico technique might be a good answer [3,4]. In silico technique helps medical scientists in several works. Based on the concept of bioinformatics, the genomics, proteomic, interactomics, and expression can be all studied and explained via in silico approach. Hence, it is no doubt that the computational approach is useful in medicine as well as in nanomedicine. This approach has a significant role in dealing with the already-mentioned hard-to-approach phenomena in nanomedicine. Generally, the specific subbranch of nanoscience is developed for the area of “nano” and “informatics” and called “nanoinformatics [5].” In a recent referencing publication, De La Iglesia et al. said, “Nanoinformatics aims to create a bridge between Nanomedicine and Information Technology applying computational method to manage the information created in the nanomedical domain [5].” The applications in nanomedicine is the present exact usefulness of nanoinformatics. The translation nanoinformatics is set to the aim [6,7]. As a core concept, Chiesa et al. proposed that “Nanoinformatics could expand previous experiences in Biomedical Informatics with new features required to study different scientific biological and physical characteristics at a different level of complexity [6].” The ACTION-Grid project under funding by the European Commission is the best example on attempt to set “medical nanoinformatics [6].” As a conclusion, the computational approach helps (a) understanding the hard to trace the existed process or phenomenon in nanolevel and (b) predicting the possible process or phenomenon in nanolevel (Table 12.1). Also, the computational approach can help reduce the required time and attempts to solve a problem in nanomedicine. (For example, it is easy and fast to trace mutation-prone positions in a nanobiomolecule. The computational approach can effectively replace the in vivo and in vitro molecular–based medical laboratories in searching for such mutation-prone positions.)
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TABLE 12.1 Application of Computational Approach in Nanomedicine Applications
Details
1. Clarification
This application is to clarify on the existed problematic process or phenomenon in medicine. The examples are clarification of the interaction process at nanolevel and clarification of the nanostructure of a medical molecule (drug, enzyme, antibody, etc.) This application is to prediction on the new thing in medicine. The examples are prediction of the complex resulted from interaction of molecules at nanolevel (such as new drugs versus pathogen) and the change due to interaction of molecules at nanolevel (such as energy change due to new proposed biochemical reaction)
2. Prediction
Sources: Haddish-Berhane, N. et al., Int. J. Nanomed., 2(3), 315, 2007; Saliner, A.G. et al., IDrugs, 11(10), 728, 2008.
12.3 Computational Nanomedicine with MATLAB ® Although there are several computer programs for nanomedicine research at present, an easyto-use program is MATLAB. In an easy word, MATLAB, shortened from Matrix Laboratory, is a high-level language for technical calculation (numeric and graphic parameters) and construction of mathematical modeling. For sure, those advantages of MATLAB can be applied in medicine (Table 12.2). In general, the use of MATLAB in medicine can be easily classified into two levels, simple and advanced usage. The first level is the simple usage. This is the use for simple activity such as basic calculation or graphical drawing. Although other computational programs can act on these aspects, the application of MATLAB in this level can help fasten the process. In addition, some difficult graphical drawing (such as threedimensional plot) or mathematical calculation (such as solving for differential equation or integration) cannot be easily performed by other computational programs. The second level is the advanced usage. This is the actual advantage of MATLAB. The simulating on the signal and wavelet can be derived via using MATLAB. This cannot be easily derived by other computational programs. It should be noted that many things or processes in medicine have their signal or wavelet in nature (or it can be said that they have kinetic appearance); hence, it is no doubt that MATLAB can help clarify their existed process and predict their interactions with others. In addition, MATLAB is also applied as a basic program for developing other more integrated programs. For example, Kinlsq, a program for fitting kinetics data with numerically integrated rate equations, is a newly program developed based on MATLAB, which is used for the analysis of medical biochemical reaction [8]. TABLE 12.2 Application of MATLAB® in Medicine Applications 1. Graphical manipulation
2. Mathematical calculation manipulation 3. Functional manipulation
Details This application is to deal with graphical information. This can help create two- or three-dimensional basic graphs (line, circle, pie, etc.) and plots (contour, mesh, surface, etc.) in medicine This application is to deal with numeric information. The mathematical calculation can help perform medical statistical work. Estimation and testing of hypothesis can be performed This application is to further use functional query (signal, wavelet, etc.). This is the basic principle of simulation in medicine
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Focusing on nanomedicine, the same use of MATLAB can be seen. Since the things or processes in nanomedicine are also things and processes in medicine, it is no doubt of the effectiveness of application of MATLAB. The simple use on graphical and mathematical calculation manipulations can be similar to use in general medicine. The advanced use for clarification or prediction on kinetic things or processes in nanomedicine helps bring great progression to the field. Some difficult problems in the past can be easily solved based on application of MATLAB. More details on examples of important reports on this topic can be seen in the next heading.
12.3.1 Summary of Important Reports on Application of MATLAB in Nanomedicine As earlier noted, MATLAB can have a great advantage in nanomedicine. Hence, it is no doubt that MATLAB is presently widely used in nanomedicine [9–12]. Here, the author summarizes on important reports on application of MATLAB in nanomedicine (Table 12.3). 1. Application for clarification The good example of application of MATLAB is the usage in nanopharmacology. Clarification of the system of drug deliver can be possible. In the present day, the novel drug delivery system is usually based on the nanolevel flow of drug via micropumping system. To study the efficacy of the new micropumping system, the clarification on the flow pattern thorough the pump is needed. Recently, Yih et al. use MATLAB for modeling and characterization of a nanoliter drug delivery MEMS (micro-electro-mechanical system) micropump with circular-based bossed membrane [9]. The flow rate via inlet and outlet of micropump was studied via MATLAB [9]. The reason for using MATLAB in this work is due to numerous non-linear parameters in the studied system that is difficult for calculation [9]. With the use of MATLAB, the required multiple calculations can be successfully solved (Table 12.4). TABLE 12.3 Some Important Reports on MATLAB in Nanomedicine Authors Yih et al. [9]
Larsson et al. [10]
Quintás et al. [11]
Sartori et al. [12]
Details An application of MATLAB for study on fluidics system via micropumping system for drug delivery [9]. This is a good example of using MATLAB for clarification on processes in nanolevel [9] Larsson et al. proposed a new four-stepped method for prediction of molecular structure of protein in biochemistry [10]. The MATLAB application-based technique is used in this work [10] Quintás et al. reported on online Fourier transform infrared spectrometric detection in gradient capillary liquid chromatography using nanoliter-flow cells [11]. The MATLABbased algorithm was used for adjustment of changing background absorption due to gradient elution in this spectrometry technique [11] Sartori et al. reported on the newly developed correlative microscopy which could help bridge the gap between fluorescence light microscopy and cryo-electron tomography [12]. MATLAB-based algorithm was used for transferring graphical parameters from cryo-light microscope to electron microscope [12]
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TABLE 12.4 Properties of MATLAB That Are Useful for Using as Core for Developing of Other Tools Properties 1. Basic properties
2. Graphic user interfaces
3. Connectivity with other programs
Details The basic properties of MATLAB can be helpful in clarification and prediction. These properties are still existed in the newly developed MATLAB-based tool The computer users can input the data into the computational system via keyboard, mouse, trackball, drawing pad, etc. and keeps those data for further usage MATLAB can be easily connected with other programs, such as C language, FORTAN language, etc., and construct a specific Mex-file for further usage
2. Application for prediction The good example of application of MATLAB is the usage in nanobiochemistry. Prediction of the structure of protein and its interaction can be possible. Indeed, the major techniques for prediction on nanoprotein are usually computationbased techniques at present. The reason is that the molecule of protein is usually small and can be at nanolevel sometimes. It is difficult to examine the structure or actually study the interaction by in vivo or in vitro techniques. Recently, Larsson et al. reported on a new method for detection of nano-second internal motion and determination of overall tumbling times independent of the time scale of internal motion in proteins from NMR relaxation data [10]. Time scale and amplitude of internal motion within biomolecule were simulated via MATLAB-based protocol in this new proposed method [10]. Nevertheless, it should address that MATLAB is helpful for the present application in nanogenomics [13,14]. Solving the problem in genomics can be probable. The good example is the prediction of nanogenomic structure (such as part of DNA, RNA, and synthetic XNA).
12.4 Computational Tools with MATLAB Programming Environment for Management of Biomedical Data It is accepted that manipulation of data in science can be possible via the use of advanced computational program. This is also applicable to the biomedical data. Basically, biomedical data can be seen in various forms such as numerical and graphical parameters. Any medical sciences can generate and give biomedical data that are valuable for further processing or manipulation. The specific data generated in nanomedical science will be hereby called “nano” biomedical data. Several nanomedical data can be seen in nanomedicine. The examples are the data on the nanostructure of the pathogen, drug, and cellular content. Designing of new diagnostic tool and therapeutic agents based on these “nano” biomedical data can be more specific than previous classical practice. At present, manipulations of these biomedical data can be performed via several computational tools. Several new computational-based tools can be presently available for processing of “nano” biomedical data. The selection of proper tool for proper work has to be done. Thus, it is necessary to understand the property and details of the tool before actually using it. In this chapter, the author briefly summarizes a
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report on important available tools for manipulating “nano” biomedical data. Those computational tools with MATLAB programming environment for management of biomedical data are specifically focused in this chapter. 12.4.1 Usefulness of MATLAB in Developing Other Computational Tools It is no doubt that MATLAB is useful in the manipulation of data, graphical data processing, numeric calculation, and simulation of signal data. Of interest, MATALAB can be a core program for further development of other MATLAB-based tools for further generalization of usage. The reason for this practice is (a) MATLAB is an easy-to-use and implement program, (b) MATLAB can be connected to other programs with a simple connection process, and (c) MATLAB also allows graphic user interfaces. Focusing on the present implementation of MATLAB as an important part of other newly developed tools, it can be as (a) a core computer program and (b) an algorithm specifically to a step of a new tool (which is not required to be a computational program). Using MATLAB to solve a problem or to simulate a solution is effective and efficient, even more so than actual hardware implementation. The MATLAB-based tool is also mentioned as a cost-effective measure, especially for getting rid of noise in processing [15]. Although MATLAB is an expensive program, for certain problems like a VLSI circuit signal simulation, manufacturing a test VLSI chip is more expensive. Indeed, cost for implementation of MATLAB-based technique is approximately 100 U.S. dollar per implementation [16]. Comparing between reduced cost due to the use of newly proposed filtering approach and added cost due to the implementation of the new technique should be clearly studied. Additionally, for solving complex problems, for example, simulating a tsunami effects, or global warming, or nuclear waste decomposition model, there is another problem: “MATLAB is very slow.” Because the problem is so difficult, it might have a better chance to write a new simulation software in a GRID computing environment. Finally, an increased chance of errors due to application of filtering procedure is possible. To solve this issue, a fuzzy vector median-based surface smoothing technique should be added to prevent possible errors [17]. Conclusively, although MATLAB-based tool has several advantages, some disadvantages can still be expected. 12.4.2 Important Computational Tools with MATLAB Programming Environment for Management of Biomedical Data There are several computational tools with MATLAB programming environment for management of biomedical data. Some important ones, which are specific to “nano” biomedical data, are hereby summarized and presented. 1. Lucid Draw [18]: Lucid Draw is a new computational tool developed by He et al. Lucid Draw is a visual analysis tool, which is fast, easy to use, and highly flexible [18]. MATLAB is used as a core program for graphic data manipulation [18]. He et al. said that Lucid Draw was “an auxiliary network analysis tool capable of visualizing complex biological networks in near real-time with controllable layout styles and drawing details [18].” For sure, Lucid Draw can be applicable in visualization of complex biological networks, which are common, in nanoscience. 2. DOSY Toolbox [19]: DOSY Toolbox is a free program for processing PFG NMR diffusion data [19]. Fourier transformation and phasing are used in processing [19].
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For these mathematical calculation manipulation, DOSY Toolbox makes use of MATLAB as a core computational program [19]. Since NMR data are also important data in nanoscience, DOSY Toolbox can be applied. POLYAN [20]: POLYAN is a computer program developed based on MATLAB [20]. It has a point-and-click graphical user interface that can bring easy and efficient polyparametric analysis of cardio-respiratory variability signals [20]. This might be an application in nanophysiological study of cardio-respiratory system. MBEToolbox [21]: MBEToolbox is a high-performance language, developed on MATLAB, for technical computing, integrating computation, visualization, and programming for sequence data analysis in molecular biology and evolution [21]. Cai et al. who developed this tool proposed that MBEToolbox was “an extensible, functional framework for users with more specialized requirements to explore and analyze aligned nucleotide or protein sequences from an evolutionary perspective [21].” This tool is applicable in nanoscience since the main aim of this tool is for manipulating in molecular biology. HEART [22]: HEART is a MATLAB-based program for beat-to-beat analysis of cardiovascular parameters from high-fidelity pressure and flow waveforms [22]. This might be application in nanophysiological study of cardiovascular system. HeiDATAProVIT [23,24]: HeiDATAProVIT is a computational tool based on MATLAB [23]. HeiDATAProVIT has a database for biomedical datasets and an extensible pool of evaluation and visualization [23]. Schablowski et al. concluded that HeiDATAProVIT was “well suited both for data processing in clinical routine and for evaluation of measurement data in any medical research project [23].” Spektr [25]: Spektr is a computational tool for x-ray spectral analysis and imaging system optimization [25]. Spektr makes use of MATLAB for modeling of beam filtration [25]. Siewerdsen et al. who launched this tool proposed that Spektr could “allow convenient calculation of x-ray spectra, selection of elemental and compound filters, and calculation of beam quality characteristics, such as half-value layer, mR/mAs, and fluence per unit exposure [25].” For sure, since x-ray spectral analysis is useful in study in nanoscience, Spektr is a very useful program. Multiscale Hy3S [26]: Multiscale Hy3S is a simulation program for simulating the stochastic dynamics of networks of biochemical reactions [26]. It is useful for investigating the dynamics of realistic biological systems [26]. Multiscale Hy3S makes use of graphic user interface property via MATLAB [26]. OMPC [27]: OMPC is a computational tool that is designed as a platform that uses syntax adaptation and emulation to allow transparent import of existing MATLAB functions into Python programs [27]. This tool is useful in neuroinformatics [27]. CGcgh [28]: CGcgh is a MATLAB-based tool for molecular karyotyping using DNA microarray-based comparative genomic hybridization [28]. Lee et al. proposed that CGcgh could effectively manipulate cDNA microarray, spotted oligonucleotide microarrays, and Affymetrix Human Mapping Arrays [28]. ISFOLD [29]: ISFOLD is a MATLAB-based tool that helps examine patterns of nucleotide substitutions from sequence alignments or mutation experiments and identify plausible bp interactions of RNA [29]. Hence, it is useful for clarification and prediction in nanostructure of RNA.
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12.5 Common Applications of MATLAB in General Nanomedicine Practice 1. Situation 1: Using MATLAB to determine flow pattern in flow cytometry This is an example of using MATLAB for clarification purpose. Indeed, the flow cytometry is a medical device that can be helpful in counting of small particles, especially for cells (usually red blood cell and white blood cell). It can also be applied for counting nano-eukaryotes (1, 10, and or μg/L). Injection of fluid containing cell into the flow track in the flow cytometry analyzer is the first step. Basically, the flow cytometry makes use of sensor for detecting injected flow cell and the primary result is as a pictorial finding. The primary file is hereby called FCS (flow cytometry standard) [30]. The graphic user interfaces of MATLAB can help interpret the graphical data derived from detection by sensor of flow 104 103 102 101
Files in FCS format is listed
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Cell A 125 × 103/dL
Further exporting to Excel sheet
Cell B 75 × 103/dL (A)
Brief steps
(B)
Cell C 840 × 103/dL
FIGURE 12.1 Protocol for MATLAB ® manipulation on FCS data via graphic user interfaces within flow cytometry analyzer. (A) Brief steps. (B) Data appearance. *The upper figure is the file in FCS format that is directly derived from flow cytometry analyzer. The blurred dots in this figure represent the analyzed cells in the automated system. The middle figure shows example of a line plot derived from graphic user interface. The lower figure is the final result appearing in Excel sheet. Grouping of cells from overall analyzed cells can be successfully done.
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cytometry. The final data will be in the numeral pattern and used in clinical practice. The reference protocol is listed in Figure 12.1. As a summary, the result first appears in graphical form known as flow cytometry scattergram, then it will be transformed into lineplot known as flow cytometry histogram, and finalized as numerical parameter that is ready for clinical usage. 2. Situation 2: Using MATLAB to determine EEG pattern This is another example of using MATLAB for clarification purpose. Basically, EEG is a measurement of electrophysiological activity of the brain [31]. This is useful for monitoring and diagnosis of brain disease, especially for epilepsy. The primary data are signal data, analog type. MATLAB can help in transformation of analog data into digital data. The process for transformation can follow standard steps as a. b. c. d.
Analog data as input Sampling Analog to digital converter Digital data as output
This is the basic principle for construction of brain machine interface which is presently available in clinical usage [31]. 12.5.1 Examples of Nanomedicine Researches Based on MATLAB Application 1. Using MATLAB program as a solution to the research question Example 1. A study to determine the impulse response of insulin hormone on target organ. Basically, the pancreas is an important exocrine gland in human beings that has a main function in digestion and regulating glucose homeostatis. Normally, an important hormone from pancreas, insulin, exists in nanolevel in blood circulation and is very difficult to be determined via the blood examination in medical laboratory. Similar to any hormone in human beings, the secretion of insulin is defined as an impulse pattern. The response to the impulse parathyroid hormone is the blood glucose level. The studying on the impulse response as change in blood glucose level can be useful in tracing the change in nanolevel of insulin in vivo. This knowledge can be further used for improvement of present treatment of diabetes mellitus. To get this data, we might set an in vivo study at first to get the set of numeric data (glucose level) and use these data for further writing of the impulse function of the system. The derived impulse function can be further manipulated by MATLAB. It hereby assumes that the impulse function in this specific example is assigned to be T (s) =
s+6 s 3 + 3 s 2 + 2s + 4
Hence, we can solve this function based on MATLAB. In MATLAB Command Window, it can be written as shown in Figure 12.2 and the derived graphical output (amplitude–time graph) can be presented as that in Figure 12.2. As a conclusion, this example represents the usefulness of MATLAB in medicine in the aspect of functional manipulation.
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Impulse–response curve
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Transfer function: s + 6 --------------------sˆ3 + 3 sˆ2 + 2 s + 4 >> >> >> >>
M1 = impulse (sys1); plot(M1) xlabel('t');ylabel('Amplitude'); title('Impulse-response curve');
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>> sys1=tf ([1 6], [1 3 2 4])
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FIGURE 12.2 (A) Impulse function and (B) graphical output in Example 1.
Example 2. A study to analyze transport phenomena of nano-drug carriers in drug delivery system to tumor. Basically, the effect of drug can be derived only if it gets success in molecular action. In the molecular process, drug is considered to be a foreign body or external particle. In order to complete the process, the drug has to pass the transportation process into the cell, which is the actual site that the molecular effect of the drug occurs. Based on the advanced nanotechnology, the new drug carrier particle can be generated by nanotechnology. The nano-drug carrier is accepted as a helpful tool in drug delivery. This is very useful in finding for new drug for treatment of tumor. Similar to any transportation process in human beings, there will be complex transportation steps at the membrane. In order to realize the actual success in transportation, the information of the existed mechanism at tumor’s cell membrane is required. Since it is a dynamic process at a very small level, the use of mathematical modeling technique can be useful. The role of MATLAB at this stage is acceptable. First, it is needed to study the in-flux and out-flux across the tumor’s cell membrane. To get this data, we might set an in vivo study using nanoprobe detector at first to get the set of numeric data (drug level) and use these data for further writing of the transportation dynamic function of the system. It hereby assumes that the impulse function in this specific example is assigned to be EO 2(S + 1) = EI 2S + 1 Hence, we can manipulate this dynamic function based on MATLAB. In MATLAB Command Window, it can be written as shown in Figure 12.3 and the derived pole-zero graphical output can be seen as that in Figure 12.3. According to the pole-zero pattern, pole is equal to −0.5, zero is equal to 1, and gain is equal to 0.5. It can be seen that the present drug transportation system has stability and might be a good system in real clinical usage. As a conclusion, this example represents the usefulness of MATLAB in medicine in the aspect of functional manipulation.
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1 0.8
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>> [z,p,k]=tf2zp([1 1],[2 1]) z = −1 p = −0.5000 k = 0.5000 >> zplane(z,p) In MATLAB command window
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FIGURE 12.3 (A) Dynamic transportation function and (B) graphical output in Example 2.
Example 3. A study to measure the microtubule length in spermatozoa from infertile men. The role of microtubule in cellular movement is acceptable in molecular medicine. The motility of the cell in human beings has to base on the microtubule. An important motile cell in human beings is the spermatozoa. The microtubule plays an important role in spermatozoa movement. In case of male infertility, the problem of motility of spermatozoa movement can be expected. To study the microtubule length can be the way to provide better knowledge on male infertility. The steady-state length distribution of microtubule can be measured using basic ultramicroscopic technique. An analytical expression for the length distribution by transformation of the length data into a single ordinary differential equation can be further solved numerically. The role of MATLAB at this stage is acceptable. It hereby assumes that the differential equation function in this specific example is assigned to be d2x dx +4 + 4 x = sin(t) dt dt Hence, we can manipulate this differential function based on MATLAB. In MATLAB Command Window, it can be written as shown in Figure 12.4 and the derived graphical output can be seen as that in Figure 12.4. As a conclusion, this example represents the usefulness of MATLAB in medicine in the aspect of mathematical calculation manipulation. Example 4. A study to reveal the movement of the treponema bacteria. Treponema bacteria are a group of spirochete bacteria. Many members of treponema bacteria can cause important diseases for human beings. Syphilis, a sexually transmitted disease, is a well-known sexually transmitted disease that is caused by treponema bacteria. The motility of the pathogenic bacteria in human beings is an interesting focus when one studies on the pathogenesis. To observe the movement of the causative pathogen, Treponema pallidum can be done based on dark-field microscopic observation. The derived track of the pathogen movement at different time can be recorded and further manipulated based on MATLAB.
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>> X=dsolve('D2x+4*Dx+4*x=sin(t)','x(0)=2','Dx(0)=1') X = 54/25*exp(−2*t)+26/5*exp(−2*t)*t-4/25*cos(t)+3/25*sin(t) >> ezplot(X) (A)
In MATLAB command window
×105
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0 –1 –2 –3 –4 –5 –6 (B)
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FIGURE 12.4 (A) Differential equation function and (B) graphical output in Example 3. *The primary assumption is x(0) = 2 and dy/dx(0) = 1. According to this assumption, the length of microtubule at the assigned stage is equal to 0.04 (54e−2t + 130te−2t + 3sin(t) − 4cos(t)).
In MATLAB Command Window, the code for the track of pathogen movement can be written as shown in Figure 12.5 and the derived graphical output can be seen as that in Figure 12.5. Based on this example, it can be shown that the movement of this bacteria is a helical-like pattern confirming the name of this group of bacteria, spirochete. The tracking of the movement of the spirochete can be useful in realization of the pathogenesis of the syphilis, which is a sexually transmitted disease that is caused by T. pallidum. As a conclusion, this example represents the usefulness of MATLAB in medicine in the aspect of graphical manipulation. 2. Using MATLAB-based tools as a solution to the research question Example 5. A study to predict the secondary structure of new emerging H1N1 influenza virus. In year 2009, the new emerging infectious disease was firstly detected in Mexico and further spread around the world. The mentioned disease is the swine flu or novel influenza, which is caused by a new variant H1N1 influenza virus infection. In the early phase, the virus can be isolated and sequenced. Based on the primary derived sequence, the data can be further processed and used for prediction of secondary structure of its nucleic acid sequence which can be further useful for new drug search. Here, the author gives an example on prediction of secondary structure of hemagglutinin protein of novel H1N1 influenza virus. The sequence of this protein, called as FASTA
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>> >> >> >> >> >> >> >>
t=linspace(0,10*pi); plot3(cos(t),sin(t),t); xlabel('cos(t)'); ylabel('sin(t)'); zlabel('t'); text(0,0,0,'Start'); grid on title('Track of pathogenic Treponema pallidum');
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In MATLAB command window Track of pathogenic Treponema pallidum
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FIGURE 12.5 (A) Computational code and (B) graphical output showing the track of pathogen movement in Example 4.
format, is firstly searched from database PubMed (http://www.pubmed.com) and further processed by TMHMM online server, an online referencing MATLAB-based tool (http://www.cbs.dtu.dk/krogh/TMHMM). The primary sequence in FASTA format is shown in Figure 12.6. The predicted structure shows that amino acid numbers 1–529 are outside, numbers 530–552 are transmembrane, and numbers 553–575 are inside.
FIGURE 12.6 The primary sequence in FASTA format in Example 5. *FASTA format is the standard code for representing a protein sequence that is mainly used in biology and medicine. It appears as a set of alphabet letters, which is specific code for each protein. Each alphabet is a representative for a specific amino acid.
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Oxidant concentration (cm–3)
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FIGURE 12.7 The results from stimulating on flux change in malignant plasma cell in Example 6.
3.2e +19
3e +19
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This information can be useful for the development of new diagnostic tool and antiviral drug [32]. Example 6. A study to access nanoflux change via a nanomembrane of malignant blood cell. Basically, transmembrane flux is the specific change of ions due to their movements across the plasma membrane. Previously, study of transmembrane flux is not easy and takes very long time. However, based on the new development in nanotechnology, it is probable and feasible to determine the transmembrane flux. Here, the author determines the transmembrane oxidation flux in malignant plasma cell in multiple myeloma compared to the control normal cell. The simulation test to determine the oxidation flux change based on nanomedicine technique based on the available computer program, MOLECULAR SIMULATOR, is used. This technique is acceptable and used in some recent publications [33,34]. Adjustment for critical parameters and conditions in the process, such as oxidant condition, time, initial oxide thickness, temperature, pressure, crystal orientation, as well as an alternative selection between the Deal-Grove’s or Massoud’s model, or a combination of both, can be performed based on this computational tool. The stimulating conditions for concentration of ion in this specific study include wet condition, temperature = 37°C, operated time = 1 min, and oxygen pressure = 0.1 atmosphere according to the normal situation in blood stream. According to the simulating process, the transmembrane oxidant concentration is steady and equal to 3.1e + 19/cm3 at any membrane thickness (Figure 12.7). For control data, no transmembrane oxidant concentration can be detected. The knowledge on the oxidation flux in malignant plasma cell in multiple myeloma can help better understand the pathology and search for new treatment.
12.6 Opportunities and Challenges of Computational Technology in Nanomedicine At present, the concept of convergence of science is introduced. Combination of new sciences aiming at synergistic properties to help deal with scientific problem is accepted. The use of the four sciences, nano-, bio-, info-, and cogno-sciences, leads to the newest
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concept of NBIC [35–37]. Integration among the four sciences leads to many new hydrid sciences such as nanoinformatics. Computational technology, which is a basic fundamental technology in info- and cogno-sciences, can be very helpful in bridging between two sciences. Computational programs have several advantages as tools for management and manipulation of data. There are many opportunities to adapt the computational technology in nanomedicine. Computational technology can help fasten manipulate and predict the data in nanomedicne. Also, computational technology can help analyze and design nanoparticles that are useful in nanomedicine. As earlier noted, medical nanoinformatics is the best example [6]. How to better apply computational technology for corresponding to the increasing demand on data and knowledge management is the big query [5]. Developing new algorithms and methods that can serve various growing demands is actually challenging.
12.7 Conclusion Computational technology can be a great tool in nanomedicine. It can help clarify and answer many problems in nanomedicine. Thus, application of computational technology can help drive nanomedicine researches. At present, there are many computer programs that can be applied. However, MATLAB is an easy-to-use and helpful computational technology that is widely used at present. MATLAB can help medical scientists answer the complex questions in nanomedicine via three main manipulations: graphical manipulation, mathematical calculation manipulation, and functional manipulation. Clarification on the existed scenario and prediction for the new imaginary scenario can be derived from using MATLAB. In addition, there are many available computational tools with MATLAB programming environment for management of biomedical data at present. These new MATLAB-based computational tools also have a lot of advantages in nanomedicine.
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13 Future Directions: Opportunities and Challenges George C. Giakos CONTENTS 13.1 Introduction........................................................................................................................ 429 13.2 Space Surveillance and National Defense...................................................................... 432 13.3 Molecular Nanophotonics, Bionanophotonics, and Nanoparticles............................434 13.4 Nanoelectronics, Nano-Optoelectronics, and Solar Cells ........................................... 439 13.5 Examples of Computational Modeling Using MATLAB® ...........................................440 13.5.1 Computational Example 1 Using MATLAB.......................................................440 13.5.2 Computational Example 2 Using MATLAB.......................................................443 13.6 Conclusion ..........................................................................................................................445 Acknowledgments ......................................................................................................................445 References......................................................................................................................................445
13.1 Introduction The nanotechnology field exhibits tremendous growth. It is anticipated that nanotechnology will have a potential impact over this century on different areas of science and technology with emphasis on space materials and instrumentation, national defense, homeland security, biophotonics and molecular nanophotonics, nanochemistry and material science, electronics/information technology, and environmental and energy technology. Nanotechnology can greatly contribute to the diagnosis and treatment of cancer or tumor at the cellular level with the development of techniques that enable the delivery of analyte probes and therapeutic agents into cells and cellular compartments. Organic and inorganic nanoparticles that interface with biological systems have recently attracted widespread interest in the fields of biology and medicine leading to a new discipline called nanomedicine [1]. Nanoelectronics and nano-optics will assist in the development of efficient stand-off detection techniques for biological and radiological warfare as well as in improving the collection and the distribution of information, yielding to enhanced protection for communication centers, communication lines, and data storage. Nanoscale systems and nanostructured materials cover a wide spectrum of materials from inorganic and organic, amorphous or crystalline nanoparticles over nanocolloid suspensions up to nanostructured carbon compounds such as fullerenes and carbon nanotubes (CNs). The interaction of light with matter integrates major technologies such as photonics, nanotechnology, optical electronics, microfluidics, electrochemistry, leading to the shaping of new research frontiers such as space optics, biophotonics, and biomolecular nanophotonics [2,3]. These emerging frontiers are promising to an array of disciplines and applications such as the design of highly efficient space sensors, with enhanced temporal, spectral, and spatial resolution, sensitivity, and high discrimination potential, for space-based surveillance and 429
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13.2 Space Surveillance and National Defense The unique functional characteristics of nanostructured material, stemming mainly from a large surface-to-volume ratio and quantum effects, can yield numerous potential space applications. Space-based electronic systems such as communications satellites and interplanetary space probes are some of the examples that have become increasingly important during the last few years. Similarly, unmanned aerial vehicles (UAV) and micro UAV and missile defense systems are extremely significant for the national defense. As these systems become smaller, and more sophisticated, the design constraints regarding their ability to carry optical instrumentation, guidance and avionics systems, and radiation shielding to protect against the cosmic radiation becomes more stringent and challenging. Therefore, new lightweight, miniaturized radiation resilient electronics and high-strength materials, as well as materials with programmable intrinsic sensing and compensating properties are needed to meet the demands of new space system designs. Special emphasis will be paid on future directions and challenges regarding the discovery and development of quantum well structures, quantum dot lasers, nanomaterials, metamaterials, and photonic nanocrystals. Diffractive optical elements, optoelectronic transducers, and photonic components can be substantially improved by lateral optoelectronic nanostructures, leading to controllable transmission and reflection optical filters and devices, with high agility, tunability, scalability, and reconfigurability. The impact of nanotechnology on the development of new space components and systems is shown in Figure 13.4. Image formation through detection of the polarization states of light offers distinct advantages for a wide range of detection and classification problems and has been explored by a number of authors, given the intrinsic potential of optical polarimetry to offer highcontrast, high-specificity images under low-light conditions [34–41]. Blending polarimetry with nanotechnology would lead to the development of enhanced resolution, functionality, and high-discrimination potential imaging sensors. Interestingly enough, innovative and efficient detection trends for space-based surveillance and situational awareness missions have been introduced by the Cardimona group at Air Force Research Laboratory (AFRL) [42]. The single-pixel polarimetric optical sensing structures rely on utilizing the interference between multiple diffractions and reflections in a four-layered structure of quantum well infrared photodetectors (QWIPs) with different gratings orientations, which is of particular interest. Since QWIPs are polarization dependent, polarimetric signature information is extracted and measured through different photocurrents, proportional to the incident photons that are read out from each layer, separately. As a result, the four Stokes parameters that describe the polarization state of the incident radiation can be simultaneously and instantaneously described with minimum errors and misregistration effects. A two-layer QWIP structure highlighting the underlying principles of single-pixel polarimetric detection [42] is shown in Figure 13.5. On the other hand, harsh and extreme ambient conditions in space such as intense high-energy cosmic radiation, extreme temperatures, high vacuum, and extreme mechanical and thermal conditions during launch and reentry pose significant technological challenges on the widespread diffusion of nanotechnology by limiting the inherent advantages of certain nanomaterials and nanocomponents. However, there are nanocomponents operating on inherently robust physical principles such as quantumdot-based detectors and sensors, while offering at the same time unique miniaturization capabilities. For instance, there exist a variety of nanocomponents that can be operated on
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optical imagery systems and space instrumentation, in reality it can offer substantial advantages such as cost reduction, energy saving, and reduced form factor. An illustration of the space technology demands for future applications is shown in Figure 13.5. In addition, nanotechnology is expected to play an important role in the protection of space assets and infrastructure such as space satellites, therefore, enhancing the space situational awareness (SSA) and mission protection [44]. Specifically, today’s space infrastructure is largely unprotected and is vulnerable to attack by potential adversaries employing a variety of means, which could damage or disrupt satellite operations, from simple jamming to physical destruction [45]. Satellites orbit the earth in a predictable fashion, due largely to their speed and altitude above the earth’s surface. Depending on the mission, a satellite generally can be placed on low Earth orbits (LEO) for weather and reconnaissance, middle Earth orbits (MEO) for navigation, and geosynchronous orbits (GEO) for communications. Adversaries can use offensive counterpace techniques to degrade U.S. space capabilities. These techniques can rely either on passive means such as denial, deception, and jamming or deploying active techniques such as direct attack using directed energy weapons (lasers, microwaves). Considering the fact that technological advances and proliferation of antisatellite weapons will enable more adversaries to possess the means to attack or interfere with U.S. satellites, the problem becomes very critical [46]. Typical threats are related to the damage of the image sensing equipment from directed energy or to the remote destruction of satellite’s solar cells using high-power laser beams [45]. Therefore, by combining commercial and radiation-hardened nanomaterials, space threats would be mitigated; for instance, radiation-hardened metal-oxide semiconductor (CMOS) circuits [47,48] and carbon-based materials such as CNS are currently being explored as a defense to various space threats. On the other hand, recent developments in nanoelectronics and molecular electronics could lead to the development of reconfigurable and extremely fault-tolerant logic arrays and memory fabrics.
13.3 Molecular Nanophotonics, Bionanophotonics, and Nanoparticles The interaction of biological material with light integrates major technologies such as photonics, nanotechnology, and biotechnology, leading to the shaping of new research frontiers such as biophotonics and biomolecular nanophotonics [2,49–66]. These emerging frontiers are promising to an array of disciplines such as medicine, biology, bioengineering, advanced diagnostic and analytical devices, and instrumentation, including nanoinstrumentation, defense, and homeland security. An illustration is shown in Figure 13.6. For instance, advanced metabolic and functional imaging techniques, operating on multiple physical principles, using high-resolution, high-selectivity nanoimaging techniques, making use of quantum dots, nanoshells, biomarkers, contrast agents, gold nanostructures, PEBBLE (probes encapsulated by biologically localized embedding) nanosensors, can play an important role in the diagnosis and treatment of autoimmune diseases and cancer as well, as well as provide efficient drug-delivery imaging solutions for disease treatment with increased sensitivity and specificity [2,49–55]. Quantum dots are tunable semiconductor nanocrystals that emit light at wavelengths, which depends on the size and composition of the crystals. In other words, they blend semiconductor
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Nanosatellites Space transportation Picosatellites
Cost Commercial off the shelf technologies (COTS)
Space technology demands for future applications
Innovative system concepts
On board autonomy
Increased capabilities Stratospheric platforms, i.e., aerostats and gliders Inspection probes
Improved communication performance Instruments and sensors breakthroughs Innovative components and materials Intelligent space systems operation
Constellations and swarms of miniaturized satellites and probes Gossamer spacecrafts Space elevators FIGURE 13.6 Space technology demands for future applications.
electronics with photonics principles [49]. Quantum dots are made of various materials, such as lead sulfide, cadmium silinate, etc. Typical examples of quantum dots are II–VI semiconductors such as ZnS, CdS, ZnSe, CdSe, or CdTe and their core/shell structures, for example, CdSe/ZnS, CdTe/CdS, or CdS/ZnS. Their wavelength tunability characteristics accompanied with minimal photobleaching and high signal-to-noise ratios are promising factors for a wide range of biological and medical applications [50–53]. Based on their unique characteristics it is expected that tumor-imaging sensitivity will improve 10-fold. Similarly, nanoshells are nanoparticle beads that consist of a silica core coated with a thin gold shell; manipulation of the thickness of the core and the outer shell permits these beads to be designed to absorb and scatter specific wavelengths of light across the visible and NIR spectrum [56]. Aqueous insulin and aqueous alcohol crystals or colloidal macromolecules behave like adaptive reconfigurable photonic crystals and have metamaterial-like characteristics, with tunable defects. In fact, clathrate-like or cage structures, filled with different
Future Directions: Opportunities and Challenges
437
Several nanoparticles such as quantum dots, gold nanoparticles, nanotubes, and nanoshells have been investigated in diverse areas such as in vitro assays, in vitro and in vivo imaging aimed to provide enhanced cancer treatment, and drug-delivery alternatives [62–66]. As a diagnostic tool, these nanoparticles can enhance image contrast between cancerous and healthy tissue, while as a treatment tool, they can either chemically destroy cancer cells through photodynamic therapy (PDT) by the light-activated sensitizer particles or destroy cancer cells through photothermal therapy (PTT) by heating the nanoparticles with an external light energy source, typically an infrared laser. Tumor targeting technologies based on the use of gold nanoparticles are being developed to detect or destroy cancer cells or deliver drugs directly into cancerous tumors [67,68]. In addition, simple, cost-effective, and sensitive diagnostic tests based on the inherent characteristics of gold nanoparticles are being developed for the early detection of prostate and other cancers. Nanoshells can be useful for thermal ablation therapy by exploiting their ability to absorb light, while their ability to scatter light has potential for cancer imaging. In addition, an array of nanodevices, surface plasmon wave sensors, and nanoelectromechanical systems (NEMS), for biochemical, analytical applications, are currently under development; for instance, monolithic integration of silver nanowells in microfluidics promises label-free detections of biochemical reactions in multiplexed aqueous environments utilizing unique surface enhanced Raman scattering (SERS) unlocking at the same time the possibilities for mass production of identical SERS active sites on costeffective polymer substrates [66]. Key examples of emerging nanotechnology platform approaches include surface nanotexturing for MS (mass spectrometry) and RPMAs (reverse phase protein microarrays), the bio-bar code assay, biologically gated nanowire sensors, and nanomechanical devices such as bio-derivatized cantilever arrays [63]. However, in vivo applications of nanoparticles in the biomedical field have still to fulfill expectations. Although there is tremendous growth of application of nanoparticles in material science, with emphasis on nanoelectronics, there are still few commercialized nanoparticle-based agents for cancer diagnosis and treatment. Biocompatibility, toxicity, stability, side effects, and other unchartered territories are some of the issues to be solved before cancer nanotechnology can truly revolutionize the health care industry and applied clinical areas. It is important to understand the mechanisms of cancer cells before applying efficient metabolic imaging and treatment methodologies. A physical algorithm highlighting the properties of cancer cells, in relation to nanotechnology platforms and their dedicated applications, is shown in Figure 13.8. Overall, the development of efficient medical imaging techniques capable of providing physiological and metabolic information for early diagnosis, staging, and treatment and follow-up of cancer is an important area of research [69–72]. Early detection of cancer is desirable as most tumors are detectable only when they reach a certain size and contain millions of cells that may already have metastasized. Currently employed techniques such as medical imaging are insufficiently sensitive and specific to detect most types of earlystage cancers. Moreover, these methods are labor intensive, time consuming, expensive, with limited integration capability. On the other hand, optical imaging per se or in combination with other modalities such as positron emission tomography (PET), single-photon emission tomography (SPECT) magnetic resonance imaging would offer the potential for noninvasive or minimally invasive exploration of molecular targets inside the human body [73,74]. Optical imaging in combination with other modalities and appropriate nanotechnology tools would allow
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exciting discovery where, using optical excitation at wavelengths near the surface plasmon resonance of the gold nanoparticle, photoluminescence enhancement on the order of 20 has been observed at the single-molecule level for heterodimers composed of a core-shell CdSe/ZnS semiconductive quantum dot and a gold nanoparticle of 60 nm diameter separated by a 32 nm long ds DNA [94]. On the other hand, nanomaterial-based energy technologies have great promise for both terrestrial and space applications [95,96]. Enhancement of the solar cell efficiency as well as enhanced short circuit current can be achieved by inserting indium arsenide quantum dots into crystalline III–V semiconductor-based photovoltaics [97].
13.5 Examples of Computational Modeling Using MATLAB® 13.5.1 Computational Example 1 Using MATLAB A computational example aimed to estimate key polarimetric figures-of-merit by estimating the Stokes parameters of backscattered signals from a nanoparticle assembly or aggregation of macromolecules in the far field via Fourier analysis, using a rotating quarter-wave retarder, is offered in the following. In this example, a generator–analyzer experimental arrangement is established and the detected backscattered signal intensities are related to the Stokes parameters through the Mueller matrices of the analyzer optics using the “Fourier analysis with a rotating quarter-wave retarder” method. The Stokes parameters, S′, estimated at the output of the analyzer receiver arm, consisting of a quarter-wavelength retarder and a linear polarizer placed next to the detector, are related to the input to the analyzer receiver arm Stokes parameters, S, of the polarized light directly backscattered from the nanoparticle assembly, via Sʹ = Manalyzer optics ⋅ S
(13.1)
⎛ S0 ⎞ ⎜ S1 ⎟ S=⎜ ⎟ ⎜ S2 ⎟ ⎜⎝ S ⎟⎠ 3
(13.2)
where
and S0, S1, S2, S3 are the Stokes parameters. Step 1: Calculation of the Mueller matrix of the analyzer optics The Mueller matrix of a rotating quarter-wave retarder is ⎛1 ⎜0 M=⎜ ⎜0 ⎜⎝ 0
0 cos 2 2θ sin 2θ cos 2θ sin 2θ
0 sin 2θ cos 2θ sin 2 2θ cos 2 2θ
0 ⎞ sin 2θ ⎟ ⎟ −cos 2θ⎟ 0 ⎟⎠
(13.3)
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Assuming horizontally linearly polarized backscattered laser beam contributions, the Mueller matrix of the horizontal linear polarizer is ⎛1 1 ⎜1 M= ⎜ 2 ⎜0 ⎜⎝ 0
1 1 0 0
0 0 0 0
0⎞ 0⎟ ⎟ 0⎟ 0⎟⎠
(13.4)
The Stokes parameter in the front of the detector, after passing through the retarder–polarizer combination, is ⎛ 1⎞ ⎜ 1⎟ 1 Sʹ = (S0 + S1 …) ⎜ ⎟ 2 ⎜ 0⎟ ⎜⎝ 0⎟⎠
(13.5)
But S′0 = I(θ) is given as I (θ) =
1 (S0 + S1 …) 2
(13.6)
Equation 13.6 is in the form of a truncated Fourier series. By carrying out the Fourier analysis, the four Stokes parameters S0, S1, S2, S3 are calculated and the polarimetric metric is obtained. The polarimetric metrics, namely, the degree of polarization (DOP), degree of linear polarization (DOLP), degree of circular polarization (DOCP), ellipticity, e, and orientation, η, are expressed in terms of Stokes parameters as follows: DOP =
(S12 + S22 + S32 )1/2 S0
(13.7)
(S12 + S22 )1/2 S0
(13.8)
S3 S0
(13.9)
DOLP =
DOCP =
e=
b s3 = a s + s2 + s2 0 1 2
(13.10)
1 ⎡s ⎤ arctan ⎢ 2 ⎥ 2 ⎣ s1 ⎦
(13.11)
η=
where, e and η are the ellipticity and azimuth, respectively.
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MATLAB• Code The following algorithm estimates the Stokes parameters of an assembly of nanomolecules under differed rotation angles: function stokes(I1,targetrotation) %Input a vector of measured intensities (I1) in Vpk-pk at a specific target angle in degrees (targetrotation) %MEASURED INTENSITIES (Vpk) %–––––––––––––––––––––––––––––––––––– I1=I1./2; N=length(I1); %CALCULATING THE FOURIER COEFFICIENTS %–––––––––––––––––––––––––––––––––––– for n=1:N; Bind(n)=(I1(n)*sin(n*(pi/4))); Cind(n)=(I1(n)*cos(n*(pi/2))); Dind(n)=(I1(n)*sin(n*(pi/2))); end A=1/4*sum(I1); B=1/2*sum(Bind); C=1/2*sum(Cind); D=1/2*sum(Dind); %CALCULATING THE STOKES PARAMETERS %––––––––––––––––––––––––––––––––– S0=(A-C); S1=(2*C); S2=(2*D); S3=B; S=[S0 S1 S2 S3]; %The Stokes Vector %CALCULATING the POLARIMETRIC PARAMETERS from the STOKES parameters %–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– DOP=(sqrt(S1ˆ2+S2ˆ2+S3ˆ2)/S0); %Degree of Polarization DOLP=(sqrt(S1ˆ2+S2ˆ2)/S0); %Degree of Linear Polarization DOCP=S3/S0; %Degree of Circular Polarization Ellipticity=(1/2)*asin(S3/S0); %Ellipticity Orientation=(1/2)*atan(S2/S1); %Orientation % Parameters to be displayed fprintf( ... 'Stokes Parameters at Target Rotation of %g degrees.\n', targetrotation) disp('––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––') disp('Intensity Vector (Vpk)') disp(I1) disp('Stokes Vector from Target')
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disp(S) disp ('DOLP at target') disp (DOLP) disp ('DOCP at target') disp (DOCP) disp('Ellipticity at target') disp (Ellipticity) disp('Orientation at target') disp (Orientation) disp ('DOCP/DOLP at target') disp (DOCP/DOLP)
13.5.2 Computational Example 2 Using MATLAB In the following example, a Mueller matrix polarimeter is considered; a coherent light source interrogates a sample, consisting of nanoparticles or a group of molecules, with a series of generator states, gi, and analyzes the light that has interacted with the sample with a set of analyzer states, ai, recording a set of flux measurements arranged in a vector P. The Mueller matrix of the sample is calculated. It is well known that the output Stokes vectors, S′, are related to the input, S, via the Mueller matrix of the sample: ⎡S0ʹ ⎢ Sʹ ⎢ 1 ⎢S2ʹ ⎢ ⎣S3ʹ
⎤ ⎡ m11 ⎥ ⎢m ⎥ = ⎢ 21 ⎥ ⎢ m31 ⎥ ⎢ ⎦ ⎣ m41
m12 m22 m32 m32
m13 m23 m33 m4 3
m14 ⎤ ⎡S0 ⎤ m24 ⎥⎥ ⎢⎢ S1 ⎥⎥ m34 ⎥ ⎢S2 ⎥ ⎥⎢ ⎥ m44 ⎦ ⎣S3 ⎦
(13.12)
A data reduction technique [98] is applied due to its intrinsic potential for optimization studies. To construct a data reduction matrix, we must first know the configurations of the polarization state generator and the polarization state analyzer for each of the Q intensity measurements. These configurations lead to a Q × 16 polarimetric measurement matrix, W, that is utilized to determine the measured flux, Pq, for a given sample Mueller matrix, M, via the equation [98] Pq = Wq ⋅ M
(13.13)
where M is the sample Mueller matrix Wq is the qth row of W that corresponds to a single polarimeter configuration Pq is the measured optical flux at the detector for that configuration Within the physical context, W is a system parameter that takes into account the optical elements that comprise the polarimeter. The Mueller matrix of the sample is then calculated through the relationship M = W −1P
(13.14)
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MATLAB Code Step 1: Calculate the measurement matrix Define elements of an analyzer (A, 16 × 4) and a generator vector (G, 4 × 16) where each row of the analyzer and column of the generator matrix represent a measurement state (qi). A minimum of 16 states (i = 1–16) are required to determine the 16 elements of the target Mueller matrix. For q = 16 % Define an analyzer column vector for all analyzer states (qx1) display('The Analyzer Vector is:') A = [ A2_1; A2_2; A2_3; A2_4; A2_5; A2_6; A2_7; A2_8;... A2_9; A2_10; A2_11; A2_12; A2_13; A2_14; A2_15; A2_16] % Define a generator row vector for all generator states (1xq) display('The Generator Vector is:') G = [ G2_l, G2_2, G2_3, G2_4, G2_5, G2_6, G2_7, G2_8,... G2_9, G2_10, G2_11, G2_12, G2_13, G2_14, G2_15, G2_16] % Calculate W k = 0; for q=1:16 for i=1:4 for j=1:4 k = k+1; W2(q,k) = A2(q,i)*G2(j,q); if k == 16; k = 0; end end end end display('The Measurement Matrix is:') W
Step 2: Calculate the Mueller matrix The measured intensities (P) are related to the measurement vector(W) and the Mueller vector(M) by P=W*M, where 1. Mueller Vector (16x1) w/ M(i,j)=[m11;m12;…m44] 2. M = (Wˆ-1)*(P) 3. Note: If W is not square, the pseudoinverse can be taken. % Solve for M (Mueller Vector) display('The Mueller Vector is:') M = inv(W)*Pdata % Mueller Matrix of the Target display('The Mueller Matrix of the Target is:') MM = [ M(1), M(2), M(3), M(4);… M(5), M(6), M(7), M(8);… M(9), M(10), M(11), M(12);… M(13), M(14), M(15), M(16)]
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13.6 Conclusion In this chapter, the technical challenges of mathematical modeling in the nanotechnology arena were highlighted by introducing future nanoscale system research directions and nanoscience applications in a diverse range of multidisciplinary areas such as space surveillance and national defense, molecular nanophotonics, bionanophotonics and nanoparticles, nanoelectronics, nano-optoelectronics, and solar cells. The goal of this study is to emphasize the significance of the development of new theoretical and computational methodologies using MATLAB that would have the potential of being broadly applicable to a range of challenging problems.
Acknowledgments The author wishes to thank Ms. Stefanie Marotta and Mr. Jeff Petermann, graduate students at the University of Akron, for their valuable assistance.
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Appendix A: Material and Physical Constants
Common Material Constants TABLE A.1 Approximate Conductivity at 20°C Material
Conductivity (S/m)
1. Conductors Silver Copper (standard annealed) Gold Aluminum Tungsten Zinc Brass Iron (pure) Lead Mercury Carbon Water (sea)
6.3 × 107 5.8 × 107 4.5 × 107 3.5 × 107 1.8 × 107 1.7 × 107 1.1 × 107 107 5 × 107 106 3 × 107 4.8
2. Semiconductors Germanium (pure) Silicon (pure)
2.2 4.4 × 10−4
3. Insulators Water (distilled) Earth (dry) Bakelite Paper Glass Porcelain Mica Paraffin Rubber (hard) Quartz (fused) Wax
10−4 10−5 10−10 10−11 10−12 10−12 10−15 10−15 10−15 10−17 10−17
451
452
Appendix A: Material and Physical Constants
TABLE A.2 Approximate Dielectric Constant and Dielectric Strength
Material
Dielectric Constant (or Relative Permittivity), εr (Dimensionless)
Dielectric Strength, E (V/m)
1200 80 8.1 8 7 5–10 6 6 5 5 3.1 2.5–8.0 2.55 2.25 2.2 2.1 1
7.5 × 106 — — — 12 × 106 35 × 106 70 × 106 — 20 × 106 30 × 106 25 × 106 — — — 30 × 106 12 × 106 3 × 106
Barium titanate Water (sea) Water (distilled) Nylon Paper Glass Mica Porcelain Bakelite Quartz (fused) Rubber (hard) Wood Polystyrene Polypropylene Paraffin Petroleum oil Air (1 atm)
TABLE A.3 Relative Permeability Material
Relative Permeability, μr
1. Diamagnetic Bismuth v Mercury Silver Lead Copper Water Hydrogen (STP)
0.999833 0.999968 0.9999736 0.9999831 0.9999906 0.9999912 ≃1.0
2. Paramagnetic Oxygen (STP) Air Aluminum Tungsten Platinum Manganese
0.999998 1.00000037 1.000021 1.00008 1.0003 1.001
3. Ferromagnetic Cobalt Nickel Soft-iron Silicon-iron
250 600 5000 7000
453
Appendix A: Material and Physical Constants
Physical Constants
Best Experimental Value
Approximate Value for Problem Work
Avogadro’s number (/kg mol) Boltzmann constant (J/k) Electron charge (C) Electron mass (kg)
6.0228 × 1026 1.38047 × 10−23 −1.6022 × 10−19 9.1066 × 10−31
6 × 1026 1.38 × 10−23 −1.6 × 10−19 9.1 × 10−31
Permittivity of free space (F/m)
8.854 × 10−12
Permeability of free space (H/m)
4π × 10−7 376.6 2.9979 × 108 1.67248 × 10−27 1.6749 × 10−27 6.6261 × 10−34 9.8066 6.658 × 10−11
120π 3 × 108 1.67 × 10−27 1.67 × 10−27 6.62 × 10−34 9.8 6.66 × 10−11
1.6030 × 10−19 8.3145
1.6 × 10−19 8.3
Quantity
Intrinsic impedance of free space (Ω) Speed of light in vacuum (m/s) Proton mass (kg) Neutron mass (kg) Planck’s constant (J s) Acceleration due to gravity (m/s2) Universal constant of gravitation (m2/kg s2) Electron volt (J) Gas constant (J/mol K)
10 9 36π 12.6 × 10−7
Appendix B: Symbols and Formulas
B.1 Greek Alphabet Uppercase Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω
Lowercase
Name
α β γ δ ε ζ η θ, ϑ ι κ λ μ ν ξ ο π ρ σ τ υ ϕ, φ χ ψ ω
Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega
455
456
Appendix B: Symbols and Formulas
B.2 The International System of Units (SI) Prefixes Power
Prefix
Symbol
Power
Prefix
Symbol
10 10−24 10−21 10−18 10−15 10−12 10−9 10−6 10−3 10−2 10−1
stringo yocto zepto atto femto pico nano micro milli centi deci
— y z a f p n μ m c d
10 101 102 103 106 109 1012 1015 1018 1021 1024
— deka hecto kilo mega giga tera peta exa zetta yotta
— da h k M G T P E Z Y
−35
0
B.3 Trigonometric Identities cot θ =
1 1 1 , sec θ = , csc θ = tan θ cos θ sin θ tan θ =
sin θ cos θ , cot θ = cos θ sin θ
sin 2 θ + cos 2 θ = 1, tan 2 θ + 1 = sec 2 θ, cot 2 θ + 1 = csc 2 θ sin( −θ) = −sin θ, cos( −θ) = cos θ, tan( −θ) = −tan θ csc( −θ) = −csc θ, sec( −θ) = sec θ, cot( −θ) = −cot θ cos(θ1 ± θ2 ) = cos θ1 cos θ2 ∓ sin θ1 sin θ2 sin(θ1 ± θ2 ) = sin θ1 cos θ2 ± cos θ1 sin θ2
tan(θ1 ± θ2 ) =
cos θ1 cos θ2 =
tan θ1 ± tan θ2 1 ∓ tan θ1 tan θ2
1 [cos(θ1 + θ2 ) + cos(θ1 − θ2 )] 2
457
Appendix B: Symbols and Formulas
sin θ1 sin θ2 =
1 [cos(θ1 − θ2 ) − cos(θ1 + θ2 )] 2
sin θ1 cos θ2 =
1 [sin(θ1 + θ2 ) + sin(θ1 − θ2 )] 2
cos θ1 sin θ2 =
1 [sin(θ1 + θ2 ) − sin(θ1 − θ2 )] 2
⎛θ +θ ⎞ ⎛ θ −θ ⎞ sin θ1 + sin θ2 = 2 sin ⎜ 1 2 ⎟ cos ⎜ 1 2 ⎟ 2 ⎝ ⎠ ⎝ 2 ⎠ ⎛θ +θ ⎞ ⎛ θ −θ ⎞ sin θ1 − sin θ2 = 2 cos ⎜ 1 2 ⎟ sin ⎜ 1 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛θ +θ ⎞ ⎛ θ −θ ⎞ cos θ1 + cos θ2 = 2 cos ⎜ 1 2 ⎟ cos ⎜ 1 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛θ +θ ⎞ ⎛ θ −θ ⎞ cos θ1 − cos θ2 = −2 sin ⎜ 1 2 ⎟ sin ⎜ 1 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ a cos θ − b sin θ = a 2 + b 2 cos(θ + φ) ,
⎛ b⎞ where φ = tan −1 ⎜ ⎟ ⎝ a⎠
a sin θ + b cos θ = a 2 + b 2 sin(θ + φ),
⎛ b⎞ where φ = tan −1 ⎜ ⎟ ⎝ a⎠
cos(90 − θ) = sin θ, sin(90 − θ) = cos θ, tan(90 − θ) = cot θ cot(90 − θ) = tan θ, sec(90 − θ) = csc θ, csc(90 − θ) = sec θ cos(θ ± 90 ) = ∓ sin θ, sin(θ ± 90 ) = ± sin θ, tan(θ ± 90 ) = − cot θ cos(θ ± 180 ) = − cos θ, sin(θ ± 180 ) = − sin θ, tan(θ ± 180 ) = tan θ cos 2θ = cos 2 θ − sin 2 θ, cos 2θ = 1 − 2 sin 2 θ, cos 2θ = 2 cos 2 θ − 1 sin 2θ = 2 sin θ cos θ, tan 2θ =
2 tan θ 1 − tan 2 θ
cos 3θ = 4 cos 3 θ − 3 sin θ
458
Appendix B: Symbols and Formulas
sin 3θ = 3 sin θ − 4 sin 3 θ sin
tan
sin θ =
θ 1 − cos θ θ 1 + cos θ =± , cos = ± , 2 2 2 2
θ 1 − cos θ θ sin θ θ 1 − cos θ =± , tan = , tan = 2 1 + cos θ 2 1 + cos θ 2 sin θ
e jθ − e − jθ e jθ + e − jθ e jθ − e − jθ , cos θ = ( j = −1 ), tan θ = jθ − jθ j(e + e ) 2j 2 e ± jθ = cos θ ± j sin θ (Euler’s identity) 1 rad = 57.296 π = 3.1416
B.4 Hyperbolic Functions cosh x =
ex + e−x ex − e−x sinh x , sinh x = , tanh x = 2 2 cosh x
coth x =
1 1 1 , sech x = , csch x = tanh x cosh x sinh x sin jx = j sinh x , cos jx = cosh x sinh jx = j sin x , cosh jx = cos x
sin( x ± jy ) = sin x cosh y ± j cos x sinh y cos( x ± jy ) = cos x cosh y ∓ j sin x sinh y sinh( x ± y ) = sinh x cosh y ± cosh x sinh y cosh( x ± y ) = cosh x cosh y ± sinh x sinh y sinh( x ± jy ) = sinh x cos y ± j cosh x sin y
459
Appendix B: Symbols and Formulas
cosh( x ± jy ) = cosh x cos y ± j sinh x sin y tanh( x ± jy ) =
sin 2 y sinh 2x ±j cosh 2x + cos 2 y cosh 2x + cos 2 y cosh 2 − sinh 2 x = 1 sech 2 + an h 2 x = 1
B.5 Complex Variables A complex number can be written as z = x = jy = r∠θ = re jθ = r(cos θ + j sin θ), where x = Re z = r cos θ, y = Im z = r sin θ ⎛ y⎞ r = z = x 2 + y 2 , θ = tan −1 ⎜ ⎟ ⎝ x⎠ j = −1 ,
1 = − j, j
j 2 = −1
The complex conjugate of z = z* = x − jy = r∠−θ = re−jθ = r(cos θ − j sin θ) (e jθ )n = e jnθ = cos nθ + j sin nθ (de Moivre’ s theorem) If z1 = x1 + jy1 and z2 = x2 + jy2, then only if x1 = x2 and y1 = y2. z1 ± z2 = ( x1 + x2 ) ± j( y1 + y 2 ) z1z2 = ( x1x2 − y1 y 2 ) + j( x1 y 2 + x2 y1 ) = r1r2e j( θ1 + θ2 ) = r1r2∠θ1 + θ2 x y −x y z1 ( x1 + jy1 ) ( x2 − jy 2 ) x1x2 + y1 y 2 r r = = + j 2 21 1 2 2 = 1 e j(θ1 − θ2 ) = 1 ∠θ1 − θ 2 ⋅ z2 ( x2 + jy 2 ) ( x2 − jy 2 ) x2 2 + y 2 2 x2 + y 2 r2 r2 ln(re jθ ) = ln r + ln e jθ = ln r + jθ + j 2mπ (m = integer) z = x + jy = r (e jθ/2 ) = r ∠θ/2
460
Appendix B: Symbols and Formulas
z n = ( x + jy )n = r n e jnθ = r n ∠nθ (n = integer) z1/n = ( x + jy )1/n = r 1/n e jθ/n = r 1/n ∠θ/2 + 2πm/n (m = 0, 1, 2, … , n − 1)
B.6 Table of Derivatives y= c (constant) cxn (n any constant) eax ax (a > 0) ln x (x > 0) c xa
dy = dx 0 cnxn−1 aeax ax ln a 1 x ca x a +1
loga x
log a e x
sin ax cos ax
a cos ax −a sin ax
tan ax
a sec 2 ax =
cot ax
a cos 2 ax a a csc 2 ax = sin 2 ax
sec ax
a sin ax cos 2 ax
csc ax
−a cos ax sin 2 ax
arcsin ax = sin−1 ax arccos ax = cos−1 ax arctan ax = tan−1 ax arccot ax = cot−1 ax sinh ax cosh ax tanh ax sinh−1 ax
a 1 − a2 x 2 −a 1 − a2 x 2 a 1 + a2 x 2 a 1 + a2 x 2 a cosh ax a sinh ax a cosh 2 ax a 1 + a2 x 2
461
Appendix B: Symbols and Formulas
a
cosh−1 ax
2
a x2 − 1
tanh−1 ax
a 1 − a2 x 2
u(x) + v(x)
du dv + dx dx
u(x)v(x)
u
u( x) v( x)
1 v2
1 v( x)
−1 dv v 2 dx
y(v(x))
dy dv dv dx
dv du +v dx dx ⎛ du ⎜⎝ v dx
u
dv ⎞ ⎟ dx ⎠
dy dv du dv du dx
y(v(u(x)))
B.7 Table of Integrals
∫ a dx = ax + c
(c is an arbitrary constant)
∫ x dy = xy − ∫ y dx ∫
x n +1 + c, (n ≠ −1) n+1
x n dx = 1
∫ x dx = ln x + c ∫ ∫
e ax dx =
a x dx =
e ax +c a
ax + c for ( a > 0) ln a
∫ ln x dx = x ln x − x + c
for ( x > 0)
462
Appendix B: Symbols and Formulas
−cos ax +c a
∫ sin ax dx = ∫ cos ax dx =
− ln cos ax +c a
∫ tan ax dx = ∫ cot ax dx = ∫
2a
⎛ x − a⎞ ln ⎜ ⎝ x + a ⎟⎠ 1 dx = + c or x 2 − a2 2a
1 dx = 2 a − x2
∫
1 a +x
2
dx =
∫
+c
⎛ x⎞ tanh −1 ⎜ ⎟ ⎝ a⎠ a
+c
⎛ x + a⎞ ln ⎜ ⎝ x − a ⎟⎠ +c 2a
1
⎛x⎞ dx = sin −1 ⎜ ⎟ + c ⎝a⎠ a −x
∫ 2
⎛ 1 − cos ax ⎞ ln ⎜ ⎝ 1 + cos ax ⎟⎠
⎛ x⎞ tan −1 ⎜ ⎟ ⎝ a⎠ 1 dx = +c x 2 + a2 a
∫
∫
ln sin ax +c a
⎛ 1 − sin ax ⎞ − ln ⎜ ⎝ 1 + sin ax ⎟⎠ sec ax dx = +c 2a
∫ csc ax dx =
∫
sin ax +c a
2
2
⎛ x⎞ sinh −1 ⎜ ⎟ ⎝ a⎠ a 1 2
x −a
2
+ c or ln( x + x 2 + a 2 ) + c
dx = ln( x + x 2 − a 2 ) + c
463
Appendix B: Symbols and Formulas
∫
⎛ x⎞ sec −1 ⎜ ⎟ ⎝ a⎠ dx = +c 2 2 a x x −a 1
∫
( ax − 1)e ax +c a2
∫ x cos ax dx =
cos ax + ax sin ax +c a2
∫ x sin ax dx =
sin ax − ax cos ax +c a2
∫
x ln x dx =
x2 x2 ln x − +c 2 4
∫
e ax ( ax − 1) +c a2
xe ax dx =
e ax cos bx dx =
e ax ( a cos bx + b sin bx) +c a2 + b 2
e ax sin bx dx =
e ax (−b cos bx + a sin bx) +c a2 + b 2
∫ ∫
xe ax dx =
x
sin 2x +c 4
x
sin 2x +c 4
∫ sin x dx = 2 − 2
∫ cos x dx = 2 + 2
∫ tan x dx = tan x − x + c 2
∫ cot x dx = − cot x − x + c 2
∫ sec x dx = tan x + c 2
∫ csc x dx = − cot x + c 2
∫ sec x tan x dx = sec x + c ∫ csc x cot x dx = − csc x + c
464
Appendix B: Symbols and Formulas
B.8 Table of Probability Distributions Probability P(X = x)
1. Discrete Distribution
Expectation (Mean) μ
Variance σ2
Binomial B(n, p)
⎛ n⎞ r n ! p r qn r n r ⎜⎝ r ⎟⎠ p (1 p) = r !(n r )! , r = 0, 1, … , n
np
np(1 − p)
Geometric G(p)
(1 − p)r−1 p
1 p
1− p p2
Poisson p(λ)
λ ne λ n!
λ
λ
r p
r(1 − p) p2
np
np(1 − p)
Pascal (negative binomial) NB(r, p)
Hypergeometric H(N, n, p)
⎛ x 1⎞ r x r ⎜⎝ r 1⎟⎠ p (1 p) , ⎛ Np⎞ ⎛ N Np⎞ ⎜⎝ r ⎟⎠ ⎜⎝ n r ⎟⎠ ⎛ N⎞ ⎜⎝ n ⎟⎠
Density f(x)
2. Continuous Distribution Exponential E(λ)
Uniform U(a, b)
x = r , r + 1,…
λx ⎪⎧λe , x ≥ 0 ⎨ x<0 ⎪⎩ 0,
⎧ 1 , ⎪ ⎨b a ⎪⎩ 0,
Standardized normal N(0, 1)
ϕ( x) =
General normal Gamma Γ(n, λ)
a<x
e
Expectation (Mean) μ
N−n N −1
Variance σ2
1 λ
1 λ2
a+b 2
( b − a )2 12
0
1
1 ⎛ x μ⎞ ϕ⎜ ⎟ σ ⎝ σ ⎠
μ
σ2
λn n 1 x e Γ(n)
n λ
n λ2
p p+q
pq ( p + q)2 ( p + q + 1)
1 ⎛ 1⎞ Γ 1+ ⎟ λ ⎜⎝ β⎠
1 ( A B) λ2 ⎛ 2⎞ A = Γ ⎜1 + ⎟ β⎠ ⎝
2π
λx
Beta β(p, q)
ap ,q x p 1 (1 x)q 1 , 0 ≤ x ≤ 1 Γ ( p + q) ap ,q = , p > 0, q > 0 Γ( p)Γ(q)
Weibull W(λ, β)
λ ββxβ 1e
( λx )β
F( x ) = 1 e
,
x≥0
( λx )β
⎛ 1⎞ B = Γ2 ⎜1 + ⎟ β⎠ ⎝ x2
Rayleigh R(σ)
x 2σ2 e , x≥0 σ2
σ
π 2
⎛ 2σ 2 ⎜ 1 ⎝
π⎞ ⎟ 4⎠
465
Appendix B: Symbols and Formulas
B.9 Summations (Series) 1. Finite element of terms N
∑ n 0 N
∑
N
1 − aN +1 ; a = 1− a
∑ na
n
n=
n 0
n 0 N
N ( N + 1) ; 2
N
∑ n(n + 1) = n 0
n
∑n
2
=
n 0
⎛ 1 − ( N + 1)a N + Na N + 1 ⎞ = a⎜ ⎟⎠ (1 − a)2 ⎝ N ( N + 1)(2N + 1) 6
N ( N + 1)( N + 2) ; 3
N
( a + b )N =
∑ NC a n
N −n n
b , where NCn = NCN − n =
n 0
NPn N! = n! ( N − n)! n !
2. Infinite element of terms ∞
∑
1 , 1− x
xn =
n 0 ∞
∑
∞
(|x|< 1);
n 0 ∞
1
∑n n 0
ex =
2
∑ n 0 ∞
ax =
∑ n 0
k
∂ ⎛ x ⎞ ⎜ ⎟ , (|x|< 1) ; ∂a k ⎝ x − e − a ⎠
∞
( −1)n
∑ n 1
∞
∑ n 0 ∞
∑ n 0
tan x = x +
n 0
( −1)n x 2 n + 1 x 3 x 5 x7 = x− + − + 3! 5! 7 ! (2n + 1)! ( −1)n x 2 n x2 x 4 x6 = 1− + − + (2n)! 2! 4! 6!
∞
tan −1 x =
(±1)n x n x2 x3 = ±x − ± − , (|x|< 1) n 2 3
x 3 2x 5 + + , 3 15
∑ n 0
1
1
1
1
∑ 2n + 1 = 1 − 3 + 5 − 7 + = 4 π
(ln a)n x n (ln a)x (ln a)2 x 2 (ln a)3 x 3 + = 1+ + + 1! 2! n! 3! ∞
cos x =
1 , (|x|< 1) (1 − x)2
1 1 1 xn = 1 + x + x2 + x3 + 1! 2! 3! n!
ln(1 ± x) = − sin x =
=
1 1 1 1 + + + = π2 22 3 2 4 2 6
= 1+
∞
n
n 0
nk x n = lim( −1)k a→ 0
∑ nx
(|x|< 1)
x 3 x 5 x7 ( −1)n x 2 n + 1 = x− + − + , 2n + 1 3 5 7
(|x|< 1)
466
Appendix B: Symbols and Formulas
B.10 Logarithmic Identities log e a = ln a (natural logarithm) log 10 a = log a (common logarithm) log ab = log a + log b log
a = log a − log b b
log a n = n log a
B.11 Exponential Identities ex = 1 + x +
x2 x3 x4 + + + , where e 2.7182 2! 3! 4! e xe y = e x+ y (e x )n = e nx ln e x = x
B.12 Approximations for Small Quantities If |a| << 1, then ln(1 + a) a ea 1 + a sin a a cos a 1 tan a a (1 ± a)n 1 ± na
467
Appendix B: Symbols and Formulas
B.13 Vectors 1. Vector derivatives a. Cartesian coordinates Coordinates
(x, y, z)
Vector
A = Axax + Ayay + Azaz
Gradient
∇A =
Divergence
∇⋅A =
Curl
ax ∂ ∇×A = ∂x Ax
∂A ∂A ∂A ax + ay + az ∂x ∂y ∂z ∂Ax ∂Ay ∂Az + + ∂x ∂y ∂z ay ∂ ∂y Ay
⎛ ∂A =⎜ z ⎝ ∂y Laplacian
∇2A =
az ∂ ∂z Az ⎛ ∂ Ay ∂Az ⎞ ⎟ ay + ⎜ ∂x ⎠ ⎝ ∂x
∂Ay ⎞ ⎛ ∂A ax + ⎜ x ⎝ ∂z ∂z ⎟⎠
∂Ax ⎞ az ∂y ⎟⎠
∂2 A ∂2 A ∂2 A + + ∂x 2 ∂y 2 ∂z 2
b. Cylindrical coordinates (ρ, ϕ, z)
Coordinates Vector
A = Aρaρ + Aϕaϕ + Azaz
Gradient
∇A =
Divergence
∇⋅A =
Curl
aρ 1 ∂ ∇×A = ρ ∂ρ Aρ
∂A ∂A 1 ∂A aρ + aφ + az ρ ∂φ ∂ρ ∂z 1 ∂ 1 ∂Aφ ∂Az (ρAρ ) + + ρ ∂ρ ρ ∂φ ∂z
⎛ 1 ∂Az =⎜ ⎝ ρ ∂φ Laplacian
∇2A =
ρa φ ∂ ∂φ ρAφ
az ∂ ∂z Az
⎛ ∂Aρ ∂Aφ ⎞ aρ + ⎜ ∂z ⎟⎠ ⎝ ∂z
1 ∂ ⎛ ∂A ⎞ 1 ∂ 2 A ∂ 2 A + + 2 ρ ∂z ρ ∂ρ ⎜⎝ ∂ρ ⎟⎠ ρ2 ∂φ 2
1⎛ ∂ ∂Az ⎞ a φ + ⎜ (ρAφ ) ρ ⎝ ∂x ∂ρ ⎟⎠
∂Aρ ⎞ az ∂φ ⎟⎠
468
Appendix B: Symbols and Formulas
c. Spherical coordinates (r, θ, ϕ)
Coordinates Vector
A = Arar + Aθaθ + Aϕaϕ
Gradient
∇A =
Divergence
∇⋅A =
Curl
ar 1 ∂ ∇×A = 2 r sin θ ∂r Ar
∂A 1 ∂A 1 ∂A ar + aθ + aφ ∂r r ∂θ r sin θ ∂φ 1 ∂ 2 1 ∂ 1 ∂Aφ (r Ar ) + ( Aθ sin θ) + r 2 ∂r r sin θ ∂θ r sin θ ∂φ
=
Laplacian
∇2A =
ra θ ∂ ∂θ rAθ
1 ⎛ ∂ ( Aφ sin θ) r sin θ ⎝⎜ ∂θ
(r sin θ)a φ ∂ ∂φ (r sin θ)Aφ ∂Aθ ⎞ 1 ⎛ 1 ∂Ar ar + ⎜ ∂φ ⎠⎟ r ⎝ sin θ ∂φ
⎞ ∂ 1⎛ ∂ (rAφ )⎟ a θ + ⎜ (rAθ ) ∂r r ⎝ ∂r ⎠
∂ ⎛ ∂A ⎞ ∂2 A 1 ∂ ⎛ 2 ∂A ⎞ 1 1 ⎜r ⎟+ ⎜ sin θ ⎟+ ∂θ ⎠ r 2 sin 2 θ ∂φ 2 r 2 ∂r ⎝ ∂r ⎠ r 2 sin θ ∂θ ⎝
2. Vector Identity a. Triple products A ⋅ (B × C) = B ⋅ (C × A) = C ⋅ (A × B) A × (B × C) = B(A ⋅ C) − C(A ⋅ B) b. Product rules ∇( fg ) = f (∇g ) + g(∇f ) ∇(A ⋅ B) = A × (∇ × B) + B × (∇ × A) + (A ⋅∇)B + (B ⋅∇)A ∇ ⋅ ( fA) = f (∇ ⋅ A) + A ⋅ (∇f ) ∇ ⋅ (A × B) = B ⋅ (∇ × A) − A ⋅ (∇ × B) ∇ × ( fA) = f (∇ × A) − A × (∇f ) = ∇ × ( fA) = f (∇ × A) + (∇f ) × A ∇ × (A × B) = (B ⋅∇)A − (A ⋅∇)B + A(∇ ⋅ B) − B(∇ ⋅ A)
∂Ar ⎞ ⎟ aφ ∂θ ⎠
469
Appendix B: Symbols and Formulas
c. Second derivative ∇ ⋅ (∇ × A) = 0 ∇ × (∇f ) = 0 ∇ ⋅ (∇f ) = ∇ 2 f ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇ 2A d. Addition, division, and power rules ∇( f + g ) = ∇f + ∇g ∇ ⋅ (A + B) = ∇ ⋅ A + ∇ ⋅ B ∇ × (A × B) = ∇ × A + ∇ × B ⎛ f ⎞ g(∇f ) − f (∇g ) ∇⎜ ⎟ = g2 ⎝g⎠ ∇f n = nf n −1∇f
(n = integer)
3. Fundamental theorems a. Gradient theorem b
∫ (∇f ) ⋅ dl = f (b) − f (a) a
b. Divergence theorem
∫
∫
(∇ ⋅ A)dv =
∫
(∇ × A) ⋅ ds =
volume
A ⋅ ds
surface
c. Curl (Stokes) theorem
surface
d.
∫ f dl = − ∫
line
e.
∫
f ds =
∫
A × ds = −
surface
f.
∇f × ds
surface
surface
∫
∇f dv
volume
∫
volume
∇ × Adv
∫ A ⋅ dl
line
Appendix C: MATLAB® MATLAB® has become a useful and dominant tool of technical professionals around the world. MATLAB is an abbreviation for Matrix Laboratory. MATLAB is a numerical computation and simulation tool that uses matrices and vectors. Also, MATLAB enables the users to solve wide analytical problems. A copy of MATLAB software can be obtained from The Mathworks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Phone: 508-647-7000 Website: http://www.mathworks.com This brief introduction of MATLAB is presented here to give a general idea about the software. MATLAB computational applications to nanosystems are used to solve practical problems.
C.1 Introduction to Basic of MATLAB ® Throughout this book, we use the latest version of MATLAB at this time, Version 7.11.0 (R2010b). When you double-click on MATLAB icon, it opens as shown in Figure C.1. The Command Window, where the special ≫ prompt appears, is the main area in which the user interacts with MATLAB. To make the Command Window active, you need to click anywhere inside its border. To quit MATLAB, you can select EXIT MATLAB from the File menu, or enter quit or exit at the Command Window prompt. Do not click on the “X” (close box) in the top right corner of the MATLAB window, because it may cause problems with the operating software. Figure C.1 contains four default windows, which are Command Window, Workplace Window, Command History Window, and Current Folder Window. Table C.1 shows a list of the various windows and their purpose in MATLAB. C.1.1 Using MATLAB in Calculations Table C.2 shows the MATLAB common arithmetic operators. The order of operations is first, parentheses ( ), the innermost are executed first for nested parentheses; second, exponentiation (ˆ); third, multiplication (*) and division (/) (they have equal precedence); fourth, addition (+) and subtractions (−).
471
473
Appendix C: MATLAB®
TABLE C.2 MATLAB Common Arithmetic Operators Operators Symbols
Descriptions
+ − *
Addition Subtraction Multiplication
/
a⎞ ⎛ Right division ⎜ means ⎟ ⎝ b⎠
\ ˆ ′ ()
b⎞ ⎛ Left division ⎜ means ⎟ ⎝ a⎠ Exponentiation (raising to a power) Converting to complex conjugate transpose Specify evaluation order
For example, >> a = 7; b = −5; c = 4; >> x = 3*a + cˆ2 − 25 x = 12 >> y = sqrt(x)/5 y = 0.6928
Table C.3 provides a common sample of MATLAB functions. You can obtain more by typing help in the Command Window (≫ help). For example, >> 8+5ˆ(log2(3.25)) ans = 23.4368 >> y=3*cos(pi/6) y = 2.5981 >>z = exp(y+9) z = 1.0889e+005
In addition to operating on mathematical functions, MATLAB allows us to work easily with vectors and matrices. A vector (or one-dimensional array) is a special matrix (or twodimensional array) with one row or one column. Arithmetic operations can apply to matrices, and Table C.4 shows extra common operations can be implemented to matrices.
474
Appendix C: MATLAB®
TABLE C.3 Typical Elementary Math Functions Function
Description
abs (x) acos (x), acosh (x) angle (x) asin (x), asinh (x) atan (x), atanh (x) conj (x) cos (x), cosh (x) cot (x), coth (x) exp (x) fix imag (x) log (x) log2 (x) log10 (x) real (x) sin (x), sinh (x) sqrt (x) tan (x), tanh (x)
Absolute value or complex magnitude of x Inverse cosine and inverse hyperbolic cosine of x (in radians) Phase angle (in radians) of a complex number x Inverse sine and inverse hyperbolic sine of x (in radians) Inverse tangent and inverse hyperbolic tangent of x (in radians) Complex conjugate of x (in radians) Cosine and inverse hyperbolic cosine of x (in radians) Inverse cotangent and inverse hyperbolic cotangent of x (in radians) Exponential of x Round toward zero Imaginary part of a complex number x Natural logarithm of x Natural logarithm of x to base 2 Common logarithms (base 10) of x Real part of a complex number of x Sine and inverse hyperbolic sine of x (in radians) Square root of x Tangent and inverse hyperbolic tangent of x (in radians)
TABLE C.4 Matrix Operations Operations A′ det (A) inv (A) eig (A) diag (A)
Descriptions Transpose of matrix A Determinant of matrix A Inverse of matrix A Eigenvalues of matrix A Diagonal elements of matrix A
A vector can be created by typing the elements inside brackets [ ] from a known list of numbers. For example, >> A = [1 3 5 7 9 11] A = 1 3 5 7 9 11 >> B = [1 3 5; 7 9 11; 2 4 6] B = 1
3
5
7
9
11
2
4
6
Also, a vector can be created with constant spacing by using the command variablename = [a: n: b], where a is the first term of the vector; n is spacing; and b is the last term.
Appendix C: MATLAB®
475
For example, >> x = [1:0.5:5] x = 1 1.5 2 2.5 3 3.5 4 4.5 5
Also, a vector can be created with constant spacing by using the command variablename = linespace (a, b, m), where a is the first element of the vector; b is the last element; and m is the number of elements. For example, >> x=linspace(0,2*pi,4) x = 0 2.0944 4.1888 6.2832
Examples using Table C.4: >> B = [1 3 5; 7 9 11; 2 4 6] B = 1
3
5
7
9
11
2
4
6
>> D=Bˆ2 D = 32
50
68
92
146
200
42
66
90
>> C= A' C = 1 3 5 7 9 11 >> E =[5 10;15 20]; >> inv(E) ans = −0.4000
0.2000
0.3000
−0.1000
>> det(E) ans = −50.0000
476
Appendix C: MATLAB®
TABLE C.5 MATLAB Named Constants Name
Content
Pi
π = 3.14159 …
i or j
Imaginary unit, −1 Floating-point relative precision, 2−52 Smallest floating-point number, 2−1022 Largest floating-point number (2-eps) . 21023 Largest positive integer, 253 − 1 Infinity Not a number Random element Identity matrix An array of 1 s An array of 0 s
Eps realmin realmax bimax Inf or Inf nan or NaN rand eye ones zeros
Special constants can be used in MATLAB. Table C.5 provides special constants used in MATLAB. For example, >> eye(4) ans = 1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
>> ones(3) ans = 1
1
1
1
1
1
1
1
1
>> 1/0 ans = Inf >> 0/0 ans = NaN
477
Appendix C: MATLAB®
TABLE C.6 MATLAB Common Arithmetic Operations on Arrays Operators Symbols on Arrays + − .* ./ .\ .ˆ .′
Descriptions Addition same as matrices Subtraction same as matrices Element-by-element multiplication Element-by-element right division Element-by-element left division Element-by-element power Unconjugated array transpose
Arithmetic operations on arrays are done element by element. Table C.6 provides MATLAB common arithmetic operations on arrays. For example, >> A=[1 2 3; 4 5 6; 7 8 9] A = 1
2
3
4
5
6
7
8
9
>> A.*A ans = 1
4
9
16
25
36
49
64
81
>> C=[1 2;3 4; 5 6]; >> D=[8 10;12 14;16 18]; >> C./D ans = 0.1250
0.2000
0.2500
0.2857
0.3125
0.3333
>> C.ˆ2 ans = 1
4
9
16
25
36
478
Appendix C: MATLAB®
C.2 Plotting MATLAB has nice capability to plotting in two-dimensional and three-dimensional plots. C.2.1 Two-Dimensional Plotting First we start with two-dimensional plots. The plot command is used to create twodimensional plots. The simplest form of the command is plot (x,y). The arguments x and y are each a vector (one-dimensional array). The vectors x and y must have the same number of elements. When the plot command is executed a figure will be created in the Figure Window. The plot (x, y, ‘line specifiers’) command has additional optional arguments that can used to detail the color and style of the lines. Tables C.7 through C.9 show various line, point, and color types, respectively, used in MATLAB.
TABLE C.7 MATLAB Various Line Styles Line Types
MATLAB® Symbol
Solid (default) Dashed Dotted Dash-dot
-: -.
TABLE C.8 MATLAB Various Point Styles Point Type Asterisk Plus sign x-mark Circle Point Square
MATLAB Symbol * + × o . s
TABLE C.9 MATLAB Various Line Color Types Color Black Blue Green Red Yellow Magenta Cyan White
MATLAB Symbol k b g r y m c w
Appendix C: MATLAB®
479
For example, >> x=0:pi/50:2*pi;%x is a vector, 0 <= x <= 2*pi, increments of pi/50 >> y=5*sin(2*pi*x);% y is a vector >> plot(x,y,'––b')%creates the 2D plot with blue and dashed line
The command fplot(‘function’, limits, line specifiers) is used to plot a function with the form of y = f(x), where the function can be typed as a string inside the command. The limits is a vector with two elements that specify the domain x [xmin,xmax], or is a vector with four elements that specifies the domain of x and the limits of the y-axis [xmin,xmax,ymin,ymax]. The line specifiers are used the same as in the plot command. For example, >> fplot('xˆ3+2*sin(3*x)−2',[−5,5],'xr')
480
Appendix C: MATLAB®
Also, we can create a plot for a function y = f(x) using the command plot by creating a vector of values of x for the domain that the function will be plotted, then creating y with corresponding values of f(x). For example, >> x=[0:0.1:1]; >> y=cos(2*pi*x); >> plot(x,y,'ro:')
In MATLAB several graphs can be plotted at the same plot in two ways. First, using plot command with typing pairs of vectors inside the Plot command such as Plot(x, y, z, t, u, h), which will create three graphs: y vs. x, t vs. z, and h vs. u, all in the same plot. For example, the command to plot the function y = 2x 3 − 15x + 5, its first derivative y' = 6x 2 − 15, and its second derivative y" = 12x, for domain −3 ≤ x ≤ 6, all in the same plot, is as follows: >> >> >> >> >>
x=[−3:0.01:6];% vector x with the domain of the function y=2*x.ˆ3−15*x+5;% vector y with the function value at each x yd=6*x.ˆ2−15;% vector yd with the value of the first derivative ydd=12*x; %vector ydd with the value of the second t derivative plot(x,y,'−r',x,yd,':b',x,ydd,'––k')
Appendix C: MATLAB®
481
Second, using hold on, hold off commands. The hold on command will hold the first plotted graph and add to it extra figures for each time the plot command is typed. The hold off command stops the process of hold on command. For example, in the previous example, we get the same result using the following commands: >> >> >> >> >> >> >> >> >>
x=[−3:0.01:6]; y=2*x.ˆ3−15*x+5; yd=6*x.ˆ2−15; ydd=12*x; plot(x,y,'-r') hold on % the first graph is created plot(x,yd,':b') % second graph is added to the figure plot(x,ydd,'––k') % third graph is added to the figure hold off
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Appendix C: MATLAB®
Plots in MATLAB can be formatted using commands that follow the plot commands, or by using the plot editor interactively in the Figure Window. First, formatting the plot using commands can be done as follows: • Labels can be placed next to the axes with the xlabel ('text as string') for the x-axis and ylabel ('text as string') for the y-axis. • The command title ('text as string') is a title command, which can be added to the plot to place the title at the top of the figure as a text. • There are two ways to place a text label in the plot. First, using text (x,y,'text as string') command, which is used to place the text in the figure such that the first character is positioned at the point with the coordinates x, y according with the axes of the figure. Second, using gtext ('text as string') command which is used to place the text at a position specified by the user mouse in the Figure Window. • The command legend ('string1', 'string2',…,pos) is used to place a legend on the plot. The legend command shows a sample of line type of each graph that is plotted and places a label specified by the user beside the line sample. The strings in the command are the labels that are placed next to the line sample and their order corresponds to the order that the graphs were created. The pos in the command is an optional number that specifies where in the figure the legend is to be placed. Table C.10 shows the options that can be used for pos. • The command axis is used to change the range and the appearance of the axes of the plot, based on the minimum and maximum values of the elements of x and y. Table C.11 shows some common possible forms of axis command. • The command grid on is used to add grid lines to the plot and the command grid off is used to remove grid lines from the plot. TABLE C.10 Options That Can Be Used for pos pos Value −1 0 1 2 3 4
Description Place the legend outside the axes boundaries on the right side Place the legend inside the axes boundaries in a location that interferes the least with graph Place the legend at the upper-right corner of the plot (this is the default) Place the legend at the upper-left corner of the plot Place the legend at the lower-left corner of the plot Place the legend at the lower-right corner of the plot
TABLE C.11 Some Common axis Commands axis Command axis ([xmin, xmax, ymin, ymax]) axis equal axis tight axis square
Description Sets the limits of both the x and y axes (xmin, xmax, ymin, ymax are numbers) Sets the same scale for both axes Sets the axis limits to the range of the data Sets the axes region to be square
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For example, >> >> >> >> >> >> >> >>
x=0:pi/30:2*pi;y1=exp(−2*x);y2=sin(x*3); plot(x,y1,'-b',x,y2,'––r') xlabel('x') ylabel('y1, y2') title('y1=exp(−2*x), y2=sin(x*3)') axis([0,11,−1, 1]) text(6,0.6,'Comparison between y1 and y2') legend('y1','y2',0)
In MATLAB, the users can use Greek characters in the text by typing name of the letter within the string as shown in Table C.12. To get a lowercase Greek letter, the name of the letter must be typed in all lowercase. To get a capital Greek letter, the name of the letter must start with a capital letter.
TABLE C.12 Some Common Greek Characters Greek Characters in the String \alpha \beta \gamma \theta \pi \sigma
Greek Letters
Greek Characters in the String
Greek Letters
α β γ θ π σ
\Phi \Delta \Gamma \Lambda \Omega \Sigma
Φ Δ Γ Λ Ω Σ
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For example, >> >> >> >> >>
t=[0:pi/90:2*pi]; clear t=[0:pi/90:6*pi]; r=1+2*cos(t); polar(t,r,'r.')
C.2.2 Three-Dimensional Plotting MATLAB has the capability to make a graph in three-dimensional plots using line, mesh, and surface plots. The command plot3(x,y,z) is used in a three-dimensional line plot, which is a line that is obtained by connecting points in 3D space. For example, >> >> >> >>
x=linspace(0,10*pi,100); y=cos(x);z=sin(x); plot3(x,y,z,'r');grid on xlabel('x');ylabel('cos(x)');zlabel('sin(x)')
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Another example, >> >> >> >> >> >> >> >>
t = −10:0.1:10; x = (1+t.ˆ2).*sin(30*t); y = (1+t.ˆ2).*cos(30*t); z = t; plot3(x,y,z,'g') grid on xlabel('x(t)'),ylabel('y(t)'),zlabel('z(t)') title('plot3 example')
Also, the command mesh (X,Y,Z) is used in a three-dimensional plot that is applied to plotting functions z = f(x,y). This can be done by creating a grid in the x–y plane that covers the domain of the function, then calculating the value of z at each point in the grid, and then creating the plot. For example, >> >> >> >> >> >>
x=(−4:0.1:4);y=(−4:0.1:4);[X,Y]=meshgrid(x,y); Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2)); mesh(X,Y,Z) zlabel('Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2))') xlabel('X') ylabel('Y')
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Table C.15 provides other common mesh plot types. For example, >> >> >> >> >> >>
x=(−4:0.1:4);y=(−4:0.1:4);[X,Y]=meshgrid(x,y); Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2)); meshz(X,Y,Z) xlabel('X') ylabel('Y') zlabel('Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2))')
TABLE C.15 Other Common Mesh Plot Types Mesh Plot Types
Description
meshz(X,Y,Z) meshc(X,Y,Z)
Mesh curtain plot which draws a curtain around the mesh Mesh and contour plot which draws a contour plot beneath the mesh Draws a mesh in one direction only
waterfall(X,Y,Z)
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>> >> >> >> >> >>
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x=(−4:0.1:4);y=(−4:0.1:4);[X,Y]=meshgrid(x,y); Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2)); meshc(X,Y,Z) zlabel('Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2))') ylabel('Y') xlabel('X')
Another command surf(X,Y,Z) is also used in a three-dimensional plot, which is applied to plotting functions z = f(x,y) as in mesh. This can be done by the same step for the mesh.
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TABLE C.16 Provides Other Common Surface Plot Types Surface Plot Types surfl(X,Y,Z) surfc(X,Y,Z)
Description Surface plot with lighting Surface and contour plot which draws a contour plot beneath the mesh
For example, >> >> >> >> >> >>
x=(−4:0.1:4);y=(−4:0.1:4);[X,Y]=meshgrid(x,y); Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2)); surf(X,Y,Z) xlabel('X') ylabel('Y') zlabel('Z=sin(X.ˆ2+Y.ˆ2).*exp(−0.4*(X.ˆ2+Y.ˆ2))')
There are other common surface plot types as shown in Table C.16.
C.3 Programming in MATLAB So far we have used MATLAB commands executed in the Command Window. This way is fine for simple task, but for more complex ones, it becomes inconvenient and difficult as the Command Window cannot be saved and executed again. Therefore, the commands and programs can be stored into a file and the MATLAB can be instructed to get its input from the file. Such a file must be created as an M-file by clicking on File/New/scripts to open a new file in the MATLAB Editor/Debugger or simple text editor, and then typing the program and saving it by choosing save from File menu in a file with an extension “.m.”
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For example, we created a program nano1.m using M-file as follows.
Then we typed nano1 in the Command Window and then hit enter to obtain the following figure.
MATLAB uses flow control through its programs. To allow flow control in a program certain rational and logical operators are essential. These operators are shown in Tables C.17 and C.18. There are four kinds of statements used in MATLAB to control the flow through the user code. They are for loops, while loops, if, else and elseif constructions, and switch constructions.
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TABLE C.17 Rational Operators Rational Operators < > <= >= == ~=
Description Less than Greater than Less than or equal Greater than or equal Equal Not equal
TABLE C.18 Logical Operators Logical Operators ~ & |
Description NOT AND OR
C.3.1 For Loops for loops allow a group of commands to be repeated a fixed number of times. The basic form of a for loop is as follows:
for index = start: increment: stop statements end
The increment can be omitted, but MATLAB will assume the increment as 1. Also, the increment can be positive or negative. For example, >> for n=1:7 x(n)=sin(n*pi/10); end x = 0.3090 0.5878 0.8090 0.9511 1.0000 0.9511 0.8090 The general form of a for loop is:
for x = array commands... end
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For example, >> m =[1 5 8; 10 17 22] m = 1 5 8 10 17 22 >> for n = m x=n(1)−n(2) end x = −9 x = −12 x = −14
C.3.2 While Loops while loop evaluates a group of statements an indefinite number of times in conjunction with a conditional statement. The general form of while loop is as follows: while expression commands… end For example, >> n=100; >> x=[]; >> while (n>0) n=n/2−1; x=[x,n]; end >> x x = 49.0000 23.5000 10.7500 4.3750 1.1875 −0.4063
C.3.3 If, Else and Elseif The general form of an if statement is as follows: if expression statements end
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The expression can be either 1 (true) or 0 (false). The statements between the if and end statements are executed if the expression is true. If the expression is false the statements will be ignored and the execution will resume at the line after the end statement. An end keyword matches the if and terminates the last group of statements. For example, >> x= 28; >> if x>0 log(x) end ans = 3.3322
The optional elseif and else keywords provide for the execution of alternate groups of statements. The if and else can be presented as follows: if condition statements else statements end For example using just if-else statement as: >> x= −28; >> if x>0 log(x) else 'x is negative number' end ans = x is negative number
The if, else and elseif can be presented as: if condition 1 statements elseif condition 2 statements else condition 3 statements end For example using if, else and elseif statements as >> x='28'; >> if ~isnumeric(x) 'x is not a number' elseif isnumeric(x)&x<0 'x is a negative number'
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else log(x) end ans = x is not a number
C.3.4 Switch The switch statement executes groups of statements based on the value of a variable or expression. The basic form of a switch statement is as follows: switch expression case results 1 statements case results 2 statements . . . otherwise statements end The keywords case and otherwise delineate the groups of statements. These respective statements are executed if the value of expression is equal to the respective results. If none of the cases are true, the otherwise statements are done. Only the first matching case is executed. The same statements can be done for different cases by enclosing the several results in braces. For example, >> x= 9; >> switch x case 1 disp('x is 1') case {7, 8, 9} disp('x is 7, 8, and 9') case 12 disp('x is 12') otherwise disp('x is not, 1, 7, 8, 9 and 12') end x is 7, 8, and 9
Considering the following tips can be helpful in working with MATLAB: 1. The names of variables and functions are case sensitive. 2. Make comments in M-file by adding lines beginning with a % character.
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3. Use a semicolon (;) at end of each command to suppress output and the semicolon can be removed when debugging the file. 4. Retrieve previously executed commands by pressing the up (↑) and down (↓) arrow keys. 5. Use an ellipse (…) at the end of the line and continue on the next line, when an expression does not fit on one line.
C.4 Symbolic Computation In previous sections, you learned that MATLAB can be a powerful programmable and calculator. However, basic MATLAB uses numbers as in a calculator. Most calculators and basic MATLAB lack the ability to manipulate math expressions without using numbers. In this section, you now see that MATLAB can manipulate and solve symbolic expressions that make you compute with math symbols rather than numbers. This process is called symbolic math. Table C.19 shows some common symbolic commands. You can practice some of the symbolic expressions as follows: C.4.1 Simplifying Symbolic Expressions Symbolic simplification is not always straightforward; there is no universal simplification function, because the meaning of a simplest representation of a symbolic expression cannot be defined clearly. MATLAB uses the sym or syms command to declare variables as symbolic variables. Then, the symbolic can be used in expressions and as arguments to many functions. For example, to rewrite a polynomial in a standard form, use the expand function as follows: >> syms x y; % creating a symbolic variables x and >> x = sym('x'); y = sym('y'); % or equivalently >> expand (sin(x+y) ) ans = sin(x) *cos(y) + cos(x)* sin(y)
You can use subs command to substitute a numeric value for a symbolic variable or replace one symbolic variable with another. For example, >> syms x; >> f=3*xˆ2−7*x+5; >> subs(f,2) ans = 3 >> simplify (sin(x)ˆ2 + cos(x)ˆ2) % Symbolic simplification ans = 1
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TABLE C.19 Common Symbolic Commands Command diff int jacobian limit symsum taylor colspace det diag eig expm inv jordan null poly rank rref svd tril triu coeffs collect expand factor horner numden simple simplify subexpr subs compose dsolve finverse solve cosint sinint zeta ceil conj eq fix floor frac imag
Description Differentiate symbolic expression Integrate symbolic expression Compute Jacobian matrix Compute limit of symbolic expression Evaluate symbolic sum of series Taylor series expansion Return basis for column space of matrix Compute determinant of symbolic matrix Create or extract diagonals of symbolic matrices Compute symbolic eigenvalues and eigenvectors Compute symbolic matrix exponential Compute symbolic matrix inverse Compute Jordan canonical form of matrix Form basis for null space of matrix Compute characteristic polynomial of matrix Compute rank of symbolic matrix Compute reduced row echelon form of matrix Compute singular value decomposition of symbolic matrix Return lower triangular part of symbolic matrix Return upper triangular part of symbolic matrix List coefficients of multivariate polynomial Collect coefficients Symbolic expansion of polynomials and elementary functions Factorization Horner nested polynomial representation Numerator and denominator Search for simplest form of symbolic expression Symbolic simplification Rewrite symbolic expression in terms of common subexpressions Symbolic substitution in symbolic expression or matrix Functional composition Symbolic solution of ordinary differential equations Functional inverse Symbolic solution of algebraic equations Cosine integral Sine integral Compute Riemann zeta function Round symbolic matrix toward positive infinity Symbolic complex conjugate Perform symbolic equality test Round toward zero Round symbolic matrix toward negative infinity Symbolic matrix element-wise fractional parts Imaginary part of complex number
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TABLE C.19 (continued) Common Symbolic Commands Command
Description
log10 log2 mod pretty quorem real round size sort sym syms symvar fourier ifourier ilaplace iztrans laplace ztrans
Logarithm base 10 of entries of symbolic matrix Logarithm base 2 of entries of symbolic matrix Symbolic matrix element-wise modulus Pretty-print symbolic expressions Symbolic matrix element-wise quotient and remainder Real part of complex symbolic number Symbolic matrix element-wise round Symbolic matrix dimensions Sort symbolic vectors, matrices, or polynomials Define symbolic objects Shortcut for constructing symbolic objects Find symbolic variables in symbolic expression or matrix Fourier integral transform Inverse Fourier integral transform Inverse Laplace transform Inverse z-transform Laplace transform z-transform
C.4.2 Differentiating Symbolic Expressions Use diff ( ) command for differentiation. For example, >> syms x; >> f =−cos(7*x)+3; >> diff(f) ans = 7 *sin(7 x) >> y=6*sin(x)*exp(x); >> diff(y) ans = 6 *exp(x) *sin(x) + 6* exp(x)*cos(x) >> diff(diff(y))% second derivative of y ans = 12 *exp(x) *cos(x)
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An example for partial derivative is as follows: >> syms v u; >> f = cos(v*u); ∂f ∂u
>> diff(f,u)% create partial derivative ans = −sin(u v)* v >> diff(f,v) % create partial derivative ans =
∂f ∂v
−sin(u v)* u >> diff(f,u,2) % create second partial derivative ans =
∂2f ∂u 2
−cos(u v)ˆ2 *v
C.4.3 Integrating Symbolic Expressions The int(f) function is used to integrate a symbolic expression f. For example, >> syms x; >> f=cos(x)ˆ2; >> int(f) ans = 1/2 *cos(x) *sin(x) + 1/2* x >> int(1/(1+xˆ2) ) ans = atan(x)
C.4.4 Limits Symbolic Expressions The limit(f) command is used to calculate the limits of a function f. For example, >> syms x y z; >> limit( (sin(x)/x), x, 0) % lim x →0
sin x = 1 x
ans = 1 >> limit(1/x, x, 0, 'right') % lim+ x →0
1 = ∞ x
ans = infinity >> limit(1/x, x, 0, 'left') % lim x →0
ans = -infinity
1 = x
∞
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C.4.5 Taylor Series Symbolic Expressions Use taylor( ) function to find the Taylor series of a function with respect to the variable given. For example, >> syms x; N =4; N
>> taylor(exp(−x),N+1) % f(x) ≅
1
∑ n !f (0) n
n =0
ans = 1 − x + 1/2 *xˆ2 − 1/6 *xˆ3 + 1/24 *xˆ4 >> f=exp(x); >> taylor(f,3) ans = 1 + x + ½* xˆ2
C.4.6 Sums Symbolic Expressions Use symsum( ) function to obtain the sum of a series. For example, >> syms k n; n −1
>> symsum(k,0,n−1) %
∑k
= 0 + 1 + 2 + ... + n
k =0
1 =
1 2 n 2
1 n 2
ans = 1/2 nˆ2 − 1/2 n >> syms n N; N
>> symsum(1/nˆ2,1,inf) %
∑ n =0
1 π2 = 2 6 n
ans = 1/6 *piˆ2
C.4.7 Solving Equations as Symbolic Expressions Many of the MATLAB commands and functions are used to manipulate the vectors or matrices consisting of symbolic expressions. For example, >> syms a b c d; >> M=[a b;c d]; >> det(M) ans = a *d − b *c >> syms x y; >> f=solve('5*x+4*y=3','x−6*y=2'); % solve the system 5x + 4y = 3, x >> x=f.x
6y = 2
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x = 13/17 >> y=f.y y = −7/34 >> syms x; >> solve (xˆ3−6*xˆ2+11*x−6) ans = 1 2 3
Use dsolve( ) function to solve symbolic differential equations. For example, >> syms x y t; >> dsolve('Dy+3*y=8') % solve y/ + 3y = 8 ans = 8/3 -C1 +exp(−3 t) % C1 is undetermined constant >> dsolve('Dy=1+yˆ2','y(0)=1') % solve y/ = 1 + y2 with initial condition % y(0) = 1 ans = tan(t + 1/4 pi) >> dsolve('D2y+9*y=0','y(0)=1','Dy(pi)=2') % solve y // + 9y =0 with initial % conditions y(0) = 1, y(π) = 2 ans = −2/3 *sin(3 t) + cos(3 t)
