Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and Applications [1 ed.] 9781613246986, 9781608764747

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and Applications,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and Applications,

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

MICRO ELECTRO MECHANICAL SYSTEMS (MEMS): TECHNOLOGY, FABRICATION PROCESSES AND APPLICATIONS

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Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

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Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

NANOTECHNOLOGY SCIENCE AND TECHNOLOGY

MICRO ELECTRO MECHANICAL SYSTEMS (MEMS): TECHNOLOGY, FABRICATION PROCESSES AND APPLICATIONS

BRITT EKWALL Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

AND

MIKKEL CRONQUIST EDITORS

Nova Science Publishers, Inc. New York

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Micro electro mechanical systems (MEMS): technology, fabrication processes, and applications / editors, Britt Ekwall and Mikkel Cronquist. p. cm. Includes bibliographical references and index. ISBN 978-1-61324-698-6 (e-book) 1. Microelectromechanical systems. I. Ekwall, Britt. II. Cronquist, Mikkel. TK7875.M5284 2009 621--dc22 2009044341

Published by Nova Science Publishers, Inc.

New York

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

CONTENTS

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Preface

vii

Chapter 1

A Systematic Approach for Analyzing Electronically Monitored Adherence Data G.J. Knafl, K.L. Delucchi, C.A. Bova, K.P. Fennie, K. Ding and A.B. Williams

Chapter 2

Design for Reliability of Micromechatronic Structural Systems Eugenio G.M. Brusa

Chapter 3

Power MEMS: An Important Category of MEMS Siaw Kiang Chou, Kaili Zhang, Wenming Yang, Jun Li and ZhiwangLi

127

Chapter 4

Structure and Stability of Silicon Clusters Stabilized by Hydrogen at High Temperatures Alexander Y. Galashev

165

Chapter 5

Design of Optical MEMS for Transparent Biological Cell Characterization Xiaodong Zhou, Kai Yu Liu and Nan Zhang

195

Chapter 6

Nanomotors Actuated by Phonon Current Bing-Yang Cao, Quan-Wen Hou and Zeng-Yuan Guo

225

Chapter 7

Tangential Nanofretting and Radial Nanofretting Linmao Qian, Jiaxin Yu, Bingjun Yu and Zhongrong Zhou

241

Chapter 8

Adaptive Poisson Modeling of Medication Adherence among HIV-Positive Methadone Patients Provided Greater Understanding of Behavior Kevin Delucchi, George Knafl, Nancy Haug and James Sorensen

259

Chapter 9

Robust Adaptive Control for MEMS Vibratory Gyroscope J. Fei

275

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1

67

vi

Contents

Chapter 10

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor Yumin Xiang

293

Chapter 11

Portable Diagnostic Technologies for Resource Poor Environments Hideyuki F. Arata

311

Chapter 12

Ballistic Transport through Quantum Wires and Rings Vassilios Vargiamidis and Vassilios Fessatidis

317

Chapter 13

Lattice Boltzmann Model as An Innovative Method for Microfluidics Lajos Szalmás

375

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Index

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383

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PREFACE So-called top-down technologies have enabled us to manufacture and fabricate structures even smaller than the micrometer scale. MEMS (Micro Electro Mechanical Systems) technologies were developed by applying semiconductor microfabrication technologies to make three-dimensional microstructures and mechanical systems in the late 1980s. MEMS technologies offer the advantages of batch fabrication of numbers of devices as well as an ability to integrate multiple functional units in a small area, which is important for developing smart and sophisticated devices. By using top-down technology such as MEMS, material costs and the amount of waste can be reduced, thus having a potential to meet the requirements to improve global health. This book also examines a 4-step process for analyzing medication adherence data generated by MEMS and similar electronic monitoring devises. Example analyses are presented to demonstrate these methods using MEMS data HIV-positive subjects' adherence to antiretroviral medications. Other chapters in this book examine power MEMS, defined as microsystems for power generation and energy conversion, including propulsion and cooling, novel optical MEMS device that can fully characterize the transparent living cells or microparticles in real time, an adaptive sliding mode controller for a MEMS vibratory z-axis gyroscope, and the use of nanofretting in nano/microelectromechanical systems (NEMS/MEMS). In Chapter 1 the authors propose a 4-step process for analyzing medication adherence data generated by MEMS and similar electronic monitoring devices. SAS macros developed to support this analysis process are available on the Internet. An overview of these methods and macros is provided. Example analyses are presented to demonstrate these methods using MEMS data on HIV-positive subjects' adherence to antiretroviral medications. The four analysis steps are formulated including new extensions for adaptive modeling of the dispersion as well as of the expected value, i.e., variability in adherence as well as its mean. How to use the macros to conduct the example analyses is also described. The steps of the analysis process are:

1. Group MEMS opening events for each subject into opening counts and rates over disjoint intervals within that subject’s MEMS usage period. 2. Model grouped counts/rates for each subject using adaptive Poisson regression methods, fitting non-linear curves in time to the expected value and dispersion. 3. Cluster estimates of the expected value and dispersion at proportional times (e.g., every 5%) within subjects’ MEMS usage periods into adherence pattern types (e.g., high, moderate, low, improving, deteriorating).

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Britt Ekwall and Mikkel Cronquist

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4. Model membership in adherence pattern types in terms of available predictors. In Step 1, MEMS opening events are grouped using the grpevnts macro into opening counts and rates. These counts/rates are naturally analyzed using Poisson regression, but can change over time in a wide variety of complex patterns, and so non-linear models are required. In Step 2, count/rate data for each subject are adaptively modeled using the genreg macro. A heuristic search process is used identifying a non-linear model based on fractional polynomials (i.e., powers can be fractions) in time. Alternate models are compared using extended quasi-likelihood cross-validation (Q+LCV) scores, with larger Q+LCV scores indicating models more compatible with the data. Model selection is iterated over all subjects using the multsubj macro, which also generates estimates of the expected value and dispersion at proportional times, representing subjects' adherence patterns. In Step 3, these adherence patterns are clustered into adherence pattern types consisting of subjects with similar patterns using the LCVcluster macro. A wide variety of clustering alternatives corresponding to different clustering procedures under varying numbers of clusters are compared on the basis of likelihood cross-validation (LCV) scores. In Step 4, the properties of the resulting adherence pattern types are assessed by modeling associated membership variables using adaptive logistic regression methods with the genreg macro. Reliability is a key issue of Micro-Electro-Mechanical Systems (MEMS) design. Several phenomena affect microsystem life and the strength of material. Basically these effects concern electrical and mechanical behaviours, respectively. Electric reliability is experimentally investigated by a deep characterization of failure modes. Procedures are available and described in the specialised literature concerning the electronic microdevices. Structural reliability only recently began to be fully characterized by experiments at microscale. Experimental techniques and numerical methods required a long development to be suitable for an accurate analysis of the electromechanical microsystem behaviour. In Chapter 2 a preliminary overview of the main structural failure phenomena occurring in some typical MEMS is provided. Three main aspects are considered. A first analysis of the architecture and role of MEMS is proposed to identify the typical structural elements used and some critical issues of their design. Loading conditions are then investigated, together with some remarks about the available numerical tools currently used for their computation. Failure modes and reliability prediction are finally discussed with particular care for the fatigue phenomenon. Some outlines about the relevant requirements for the experimental characterization of the MEMS material and their structural behaviour are proposed at the end of this chapter. As an important category of MEMS, power MEMS are defined as microsystems for power generation and energy conversion including propulsion and cooling. Power MEMS can be classified into micro thermodynamic machines including microturbines, miniature internal combustion engines and micro-coolers; solid-state direct energy conversion including thermoelectric and photovoltaic microstructures; micro electrochemical devices including micro fuel cells and nanostructure batteries; vibration energy harvesting devices including piezoelectric, magnetic or electrostatic micro generators, and micro thrusters for propulsion. Chapter 3 will introduce extensively the micro thermophotovoltaic power generators, micro direct methanol fuel cells, solid propellant micro propulsion systems, and micro scale combustion. Then a brief introduction of other power MEMS systems will be given. Finally, some guidelines for future investigations are provided.

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Preface

ix

In Chapter 4 the authors consider how a fullerene-like cage composed of Si atoms can be obtained. The fundamental difficulty to maintain a smooth Si fullerene cage arises from the fact that not all orbitals that appear in C60 fullerene exist in hollow Si cluster. The authors also investigate stability of other silicon clusters of intermediate size at high temperature. The temperature dependence of the physicochemical properties of 60- and 73-atom silicon nanoparticles are investigated using the molecular dynamics method. The 73-atom particles have a crystal structure, a random atomic packing, and a packing formed by inserting a 13atom icosahedron into a 60-atom fullerene. They could be in vacuum and also surrounded by a "coat" from 60 atoms of hydrogen. The results indicate that crystalline nanoparticle undergoes melting at a temperature of 710 K in vacuum and the nanoassembled particle at the presence of a hydrogen "coat" has the most stable number (close to four) of Si-Si bonds per atom. The structure and kinetic properties of a hollow single-layer fullerene-structured Si60 cluster are considered in the temperature range 10 K ≤ T ≤ 1710 K. Five series of calculations are conducted, with a simulation of several media inside and outside the Si60 cluster, specifically, the vacuum and interior spaces filled with 30 and 60 hydrogen atoms with and without the exterior hydrogen environment of 60 atoms. Fullerene surrounded by a hydrogen "coat" and containing 60 hydrogen atoms in the interior space has a higher stability. Such clusters have smaller self-diffusion coefficients at high temperatures. The fullerene stabilized with hydrogen is stable to the formation of linear atomic chains up to the temperatures 270-280 K. Consecutive living cell characterization has aroused the interest of biological researchers, as the cellular investigation is turning from population to singles. Optical MEMS plays an important role in this revolutionary trend. Chapter 5 describes a novel optical MEMS device that can fully characterize the transparent living cells or microparticles in real time, with an optical aperture or an aperture array. The authors’ MEMS structure only contains an optical aperture and a microfluidic channel. When a transparent biological cell is pumped through the channel, under which an optical aperture is fabricated and a laser beam is shed on, the far-field diffraction pattern of the aperture and the cell is recorded by a camera for cell characterization. This is a threedimensional system that the cell, the aperture and the microfluidic channel are not within one plane, but with a gap between the cells and the aperture. By careful design of the gap, the image of the cells and their inclusions, such as nuclei, will appear on the diffraction pattern, and the magnification of the image can be adjusted. Because the intensity of light at each pixel of the diffraction pattern is related with the refractive indices of the cells and their inclusions, the refractive indices can be extrapolated from the image of the cells by diffraction simulation. This is the first simple optical MEMS structure to characterize the shape, size, and the refractive indices of the cells and their inclusions simultaneously, and the calculated limit of detections (LOD) of this device reach minima at the wavelength of 0.266 µm, with 9e-5 for the refractive index and 0.182 µm for the diameter of a cell, respectively. The diameter detection limit of this device is close to the resolution of conventional optical systems that suffer from diffraction caused optical limit. By recording the diffraction patterns of 5 µm-diameter polystyrene microspheres with various optical apertures in micro sizes, the authors found that their simulated diffraction patterns coincided well with the experiments, which proved the correctness of the authors’ theory and concept. Although the design of the MEMS structure is given, the whole image and characterization system with this MEMS chip has not been realized. Since the principle

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Britt Ekwall and Mikkel Cronquist

of this device has been verified and its MEMS fabrication is not complicated, the authors regard it as a feasible device for individual living cell or microparticle characterizations. As discussed in Chapter 6, design and construction of nanomotors are one of the most attractive fields in nanotechnology. Following the introduction of a novel concept of the thermomass, the relative mass of a phonon gas based on the Einstein’s energy-mass relation, the thermomass and its momentum conservation equations for the phonon gas motion are established to characterize the hydrodynamics of the phonon current in a solid. Like the gas flows in the porous mediums, the phonon current in a dielectric solid imposes a driving force on the solid framework atoms, which can be calculated quantitatively and can be applied to actuate nanomotors. The authors predict the dynamic behavior of a nanomotor made up of multi-walled carbon nanotubes in terms of molecular dynamics simulations. A shorter singlewalled carbon nanotube with a larger diameter, as a mobile part, surrounds a longer singlewalled carbon nanotube with a smaller diameter working as a shaft. When a phonon current passes through the inner shaft, the outer nanotube will translate along and/or rotate around the shaft depending on the chiralities of the carbon nanotubes. The motion traces are found to depend on the chirality pair and the system temperature regularly. This type of nanomotor may be promising because they are directly driven by thermal energy transport. Chapter 7 studies nanofretting, which refers to cyclic movements of contact interfaces with the relative displacement amplitude in nanometer scale, where the contact area and normal load are usually much smaller than those in fretting. Nanofretting widely exists in nano/microelectromechanical systems (NEMS/MEMS) and may become a key tribological concern besides microwear and adhesion. With an atomic force microscopy, the tangential nanofretting of monocrystalline Si(100) surface against spherical SiO2 and diamond tips was carried out at various displacement amplitudes (0.5~250 nm) under atmosphere and vacuum condition. It was found that: (1) the tangential nanofretting could be divided into stick regime and slip regime upon the transition criterion; (2) the adhesion force may induce the increase in the maximum static friction force and prevent the contact pair from slipping; (3) the nanofretting damage on silicon may experience two processes, namely as the generation of hillocks at low load and the formation of grooves at high load; (4) compared to those in atmosphere, the nanofretting scars in vacuum exhibited higher hillock at low load but shallower groove at high load. The radial nanofretting behaviors of several typical structural materials in NEMS/MEMS (polycrystalline copper, monocrystalline silicon and CNx films) were investigated by a nanoindenter. The results indicated that: (1) the radial nanofretting damage in copper was mainly identified as the pileup of the wrinkles around indents, and in silicon was characterized as the initiation and propagation of the cracks on the edges of plastic zone; (2) due to its higher hardness and elastic modulus, the CrNx film can effectively improve the anti-pressure ability of 40Cr substrate in radial nanofretting; (3) compared to the curvature radius, the equivalent radius of indenter played a much more important role on the nanofretting damage of material. The results in the research may be helpful to understand the nanofretting failures of components in NEMS/MEMS. The goal of Chapter 8 was to conduct an extensive analysis of medication adherence data from a 20-week randomized clinical trial designed to test the effects of using vouchers on improving medication adherence among HIV-positive methadone maintenance patients extending a method proposed by Knafl et al. [1]. Study Design and Setting: The methodology was applied and extended not only to data for the randomized subjects but also for subjects who were initially enrolled but demonstrated

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Preface

xi

sufficient adherence during the baseline period. Predicted mean adherence patterns over time from Poisson models were clustered via an extensive search among clustering methods and those results were compared to a solution based on growth mixture modeling. Results: Clustering generated high and moderately high adherence groups as well as improving and deteriorating adherence groups while growth mixture modeling clustered subjects into uniformly high, moderate, and low adherence groups. Conclusions: The results for both approaches reflected uniform levels of adherence over time with clustering also identifying changing levels of adherence over time. Results replicated the primary finding that the use of vouchers improved adherence rates. Chapter 9 presents an adaptive sliding mode controller for a MEMS vibratory z-axis gyroscope. The proposed adaptive sliding mode controller can real-time estimate the angular velocity and the damping and stiffness coefficients. The stability of the closed-loop system can be guaranteed with the proposed adaptive sliding mode control strategy. The numerical simulation for MEMS gyroscope is investigated to show the effectiveness of the proposed control scheme. It is shown that the proposed adaptive sliding mode control scheme offers several advantages such as real-time estimation of gyroscope parameters and large robustness to parameter variations and external disturbance. Based on electric energy calculation through the conformal transformations, the principle of virtual work is employed to determine the electric force exerted on the charged electrode of an inclined plate capacitor. For application in Micro Electromechanical System, the computation is aimed at the general case. The result is achieved in a board manner with aid of elliptic function. For the electrode plates, there is no restriction to dimension. Beside the parallel plate capacitor is treated as a special case, the electric force on the moving electrode is researched. Further, the deformation on the electrode plate is analyzed. The corresponding numerical simulation curves are presented in Chapter 10. Diagnostic technologies currently used in developing countries are cumbersome and unsuitable for the uses in low-resource environments ; this has been one of the major problems in improving the global health. These technologies must be operated by nonresearchers and work with poor resources. Although, MEMS-based on-chip technologies have a potential to meet such requirements, yet few literatures report technical progresses aiming those applications so far. Worse still, use of current “Lab-on-a-chip” devices is limited to the environments, such as in research laboratories, where they have huge equipments; these devices are, so to speak, “Chip-in-a-lab” devices; they are not yet appropriate for uses in the extreme resource-poor settings in developing countries. In Chapter 11, trends in this research field are explained. Instead of reviewing comprehensive literatures in this field, general ideas of portable diagnostic microchip, required technologies and possible directions for further progress are discussed. Phase-coherent electron transport through quasi-one-dimensional systems has developed into a very active and fascinating subfield of mesoscopic physics. Chapter 12 presents a review of this development focusing on ballistic conduction through quantum wires (or constrictions) and one-dimensional open rings. In quantum wires the electron conductance versus Fermi energy is quantized as a consequence of the reduced dimensionality and the subsequent quantization of transverse momentum. The presence of scatterers in otherwise “clean” wires can strongly suppress the quantum conductance, and can generate sharp resonances (which are due to quasibound states) if the scattering potential is attractive. These resonances can be of the Fano or Breit-Wigner type, depending on the size or/and strength of

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the scattering potential. Thermal effects are also considered. The scattering approach is briefly discussed in order to derive the Landauer formula, which is the basic tool for calculating the conductance of a mesoscopic sample. Scattering theory in ballistic quantum wires is formulated in terms of the Lippmann-Schwinger equation while the Feshbach coupled-channel theory is employed in order to treat Fano resonances. The occurrence of Fano resonances in strictly one-dimensional mesoscopic open rings is discussed in the last part of this review. The lattice Boltzmann model is an innovative method to simulate gaseous or liquid flows. It successfully fills the gap between the macroscopic description, the Navier- Stokes equation, and particle based methods, such as the direct simulation of Monte- Carlo. In the last years, the lattice Boltzmann model has attracted increasing attention in modeling flows in microfluidic devices. This area has developed rapidly. Microelectromechanical systems (MEMS), gaseous sensors, lab-on-chips produce significant grow in various areas of engineering and technology. In Chapter 13, the authors give an overview of the lattice Boltzmann method and its key concepts. They review recent developments related to the field of microflows and microfluidics. Particular attention is paid for non-continuum behaviours observed at the micro-scale, such as the slip and jump of macroscopic variables in a micro-device. Finally, the authors provide further ideas and directions which can serve as targets of future developments. Beyond the lattice Boltzmann model, these ideas can stimulate progress and investment in applications and technology.

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Chapter 1

A SYSTEMATIC APPROACH FOR ANALYZING ELECTRONICALLY MONITORED ADHERENCE DATA G. J. Knafl1, K. L. Delucchi2, C. A. Bova3, K. P. Fennie4, K. Ding1 and A.B. Williams4 1 2

University of North Carolina at Chapel Hill, Chapel Hill, NC, USA University of California at San Francisco, San Francisco, CA, USA 3 University of Massachusetts Worcester, Worcester, MA, USA 4 Yale University, New Haven, CT, USA

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Abstract We propose a 4-step process for analyzing medication adherence data generated by MEMS and similar electronic monitoring devices. SAS macros developed to support this analysis process are available on the Internet. An overview of these methods and macros is provided. Example analyses are presented to demonstrate these methods using MEMS data on HIVpositive subjects' adherence to antiretroviral medications. The four analysis steps are formulated including new extensions for adaptive modeling of the dispersion as well as of the expected value, i.e., variability in adherence as well as its mean. How to use the macros to conduct the example analyses is also described. The steps of the analysis process are: 1. 2. 3. 4.

Group MEMS opening events for each subject into opening counts and rates over disjoint intervals within that subject’s MEMS usage period. Model grouped counts/rates for each subject using adaptive Poisson regression methods, fitting non-linear curves in time to the expected value and dispersion. Cluster estimates of the expected value and dispersion at proportional times (e.g., every 5%) within subjects’ MEMS usage periods into adherence pattern types (e.g., high, moderate, low, improving, deteriorating). Model membership in adherence pattern types in terms of available predictors.

In Step 1, MEMS opening events are grouped using the grpevnts macro into opening counts and rates. These counts/rates are naturally analyzed using Poisson regression, but can change over time in a wide variety of complex patterns, and so non-linear models are required. In Step 2, count/rate data for each subject are adaptively modeled using the genreg macro. A heuristic search process is used identifying a non-linear model based on fractional

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

2

G. J. Knafl, K. L. Delucchi, C. A. Bova et al. polynomials (i.e., powers can be fractions) in time. Alternate models are compared using extended quasi-likelihood cross-validation (Q+LCV) scores, with larger Q+LCV scores indicating models more compatible with the data. Model selection is iterated over all subjects using the multsubj macro, which also generates estimates of the expected value and dispersion at proportional times, representing subjects' adherence patterns. In Step 3, these adherence patterns are clustered into adherence pattern types consisting of subjects with similar patterns using the LCVcluster macro. A wide variety of clustering alternatives corresponding to different clustering procedures under varying numbers of clusters are compared on the basis of likelihood cross-validation (LCV) scores. In Step 4, the properties of the resulting adherence pattern types are assessed by modeling associated membership variables using adaptive logistic regression methods with the genreg macro.

Introduction We propose a 4-step process for analyzing medication adherence data as generated by MEMS and other similar electronic monitoring devices. SAS macros developed to support this analysis process are available on the Internet. An overview of these methods and macros is provided in Section I. Example analyses are presented in Section II to demonstrate these methods using MEMS data on HIV-positive subjects' adherence to antiretroviral medications from an adherence intervention study. Adaptive regression modeling of the expected value and the dispersion (i.e., the generalized variance) is formulated in Section III and adaptive cluster analysis in Section IV. How to use the macros to conduct the example analyses is described in Section V.

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I. Analysis of MEMS Adherence Data The MEMS device is used to monitor a subject's medication adherence by recording dates and times of device openings. Each opening is assumed to coincide with the subject's taking of a dose of the medication controlled by the MEMS device. These data are downloaded from the MEMS device using manufacture-supplied software. That software supports selected analyses of MEMS data and can also generate files containing those data for use in conducting more thorough analyses. However, the analysis of MEMS data is not straightforward. We propose the following 4-step process for analyzing those data. 1. Group MEMS opening events for each subject into opening counts and rates over disjoint intervals within that subject’s MEMS usage period. 2. Model grouped counts/rates for each subject using adaptive Poisson regression methods, fitting non-linear curves in time to the expected value and dispersion. 3. Cluster estimates of the expected value and dispersion at proportional times (e.g., every 5%) within subjects’ MEMS usage periods into adherence pattern types (e.g., high, moderate, low, improving, deteriorating). 4. Model membership in adherence pattern types in terms of available predictors. Electronic devices like MEMS are increasingly being used to collect extensive data for individual subjects over time. When these data consist of event occurrences, as for MEMS data, their analysis can also be conducted using the methods proposed here.

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A Systematic Approach for Analyzing Electronically Monitored Adherence Data

3

An individual subject's MEMS data consist of dates and times for a series of discrete events corresponding to openings of that subject's MEMS device(s). MEMS data are typically aggregated into summary measures (e.g., the percent of prescribed doses taken) [13]. However, this ignores the variability in adherence over time that often exists in these data. One natural way to account for such variability over time is to group MEMS data into opening counts and rates over consecutive time intervals within that subject's MEMS usage period (Step 1). The analysis of such counts and rates is typically conducted using Poisson regression methods. However, MEMS opening counts/rates for individual subjects can vary over time in a wide variety of complex patterns, and so general non-linear models are required to analyze them. Step 2 involves non-linear modeling of the count and rate data for each subject in terms of time within that subject's MEMS usage period. Knafl et al. [5] proposed a heuristic search process for generating adaptive (i.e., adapted to the data under analysis) Poisson regression models for these purposes. These non-linear models are based on fractional polynomials [11] in time; in other words, they are based on one or more power transforms of time with the powers allowed to be fractions rather than restricted to be integers as in standard polynomials. These models are estimated using maximum likelihood. Likelihoods are based on the Poisson distribution for the opening counts together with an offset variable adjusting the model to account for the opening rates per unit time. Alternate models are compared using likelihood cross-validation (LCV) scores; models with larger LCV scores are more compatible with the data. The percent consistency between observed adherence and presumed adherence at the prescribed rate can be computed from LCV scores for these two cases. Adaptive Poisson regression models generate estimates of the expected value (i.e., mean) for opening rates over time, which represent subjects' mean adherence patterns during their MEMS usage periods. However, these models do not address variability in the opening rates. For that reason, we formulate in Section III an extended quasi-likelihood [9] approach for modeling the expected value and the dispersion in combination. Parameters are estimated using maximum extended quasi-likelihood estimation and compared using extended quasi-likelihood cross-validation (Q+LCV) scores generalizing LCV scores. Analysis of a subject's MEMS data provides useful information on that subject's adherence pattern over time, but it is also of interest to compare adherence patterns for different subjects. Step 3 addresses this issue by clustering the set of adherence patterns for all available subjects into types consisting of subjects with similar adherence patterns. Since subjects can have MEMS usage periods of varying lengths, we match adherence patterns for subjects at proportional times (e.g., 5%, 10%, ···, 100%) within their usage periods. We adaptively generate adherence pattern types on the basis of LCV scores for a wide variety of clustering alternatives corresponding to different clustering procedures under varying numbers of clusters. These LCV scores are based on mixtures of multivariate normal distributions (one for each cluster) with unstructured covariance matrices. This adaptive clustering procedure was first formulated by Delucchi et al. [1] for clustering of mean adherence patterns. An extension for clustering both the mean adherence patterns and the adherence variability patterns is provided in Section IV. Once adherence pattern types have been established, it is of interest to identify characteristics that distinguish subjects having these adherence pattern types. Step 4 addresses this issue through analyses relating membership variables for the adherence pattern types to available predictors. The separate impact on adherence pattern type membership of each

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G. J. Knafl, K. L. Delucchi, C. A. Bova et al.

available predictor can be assessed using standard χ2 and logistic regression methods. The assessment of the joint impact of multiple predictors is more complex. We use adaptive logistic regression methods to conduct these latter assessments. The same heuristic search process is used as for adaptive Poisson regression modeling, but with extended quasilikelihoods and Q+LCV scores based on the logistic (or Bernouilli) distribution rather than on the Poisson distribution. Adaptive dichotomous logistic regression can be used to model membership in one adherence pattern type versus any other type. Adaptive polytomous logistic regression can be used to model membership in several alternative adherence types. As for Poisson regression, the dispersion as well as the expected value can be modeled in terms of available predictors. Section III provides a formulation of an adaptive approach to generalized linear modeling [10] of the expected value and the dispersion, including as special cases linear, logistic, and Poisson regression assuming normal, logistic, and Poisson distributions, respectively. Estimates are computed using maximum extended quasi-likelihood estimation. Models are compared using Q+LCV scores with larger scores indicating better models. Fractional polynomial models are adaptively selected for the expected value and the dispersion using the heuristic search process of Knafl et al. [5] adapted in Section III to handle both the expected value and the dispersion in combination. Q+LCV ratio tests, generalizing LCV ratio tests and analogous to likelihood ratio tests, are used to compare models in terms of the cutoff for a substantial percent decrease (PD) in the Q+LCV score. If model 1 generates a PD compared to model 2 larger than the cutoff for a substantial PD for the data, then model 2 distinctly outperforms model 1. On the other hand, if model 1 generates a PD compared to model 2 less than that cutoff, then model 1 is a competitive alternative to model 2. SAS macros were developed to support the MEMS data analysis process and are available at http://www.unc.edu/~gknafl/EMDanalysis.html. The grpevnts macro is used to convert raw MEMS data into grouped count/rate data. The genreg macro is used to analyze the grouped count/rate data for one subject at a time using adaptive Poisson regression. The multsubj macro is used to iteratively invoke the genreg macro to analyze grouped count/rate data for all subjects. It also generates estimates of the expected value and the dispersion at equally-spaced proportional times during subjects' MEMS usage periods. The LCVcluster macro is used to cluster these estimates into adherence pattern types. The genreg macro is used to model membership variables, either dichotomous or polytomous, for adherence pattern types in terms of available predictors using adaptive logistic regression. The genreg macro also supports adaptive linear regression [3].

II. Example Analyses Knafl et al. [4] presented analyses of the MEMS data for 161 HIV-positive subjects from the Adherence through Home Education and Nursing Assessment (ATHENA) Project that tested a home-based nursing intervention for improving adherence to antiretroviral medications [16]. Subjects' usage periods were as large as 729 days or about 2 years long. In this section, we report analyses of data for a smaller period, the first 360 days or about the first year, of MEMS usage for these 161 ATHENA subjects. For these analyses, grouped opening counts/rates are based on openings within the first 360 days of MEMS usage (Step 1 generated with the grpevnts macro). For subjects with usage periods longer than 360 days,

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this generates more grouped counts/rates than restricting grouped data for all available openings to the first 360 days. The prescribed rate PR for all subjects is 2 per day. Analyses reported in this section are exploratory. Their purpose is to demonstrate the MEMS data analysis process and not to provide a formal assessment of the results of the ATHENA Project. However, they also demonstrate that this process can produce novel insights into MEMS data not otherwise identifiable.

II.1. Unit Dispersion Analyses II.1.1. Individual-Subject Analyses We fit an adaptive Poisson regression model with unit dispersion to each subject's grouped data (Step 2 using the multsubj macro, which iteratively calls the genreg macro to model each subject's data). Figure 1 contains the plot for Subject A of observed MEMS opening rates along with the fitted mean adherence curve (see equation (7)). This subject had very high mean adherence at a constant rate close to PR and with limited variability about PR. The percent consistency score (see equation (3) of [5]) for this subject was, not surprisingly, 100%. Subject B (Figure 2) had moderate levels of mean adherence varying around one half PR with percent consistency 15.5%. Subject C (Figure 3) had low mean adherence with percent consistency 1.8%. The patterns for Subjects B-C decrease somewhat with time but are relatively consistent over time. Subject D (Figure 4), on the other hand, had distinctly deteriorating mean adherence with percent consistency 4.5% while Subject E (Figure 5) had distinctly improving mean adherence with percent consistency 6.4%. The distribution for percent consistency scores (Figure 6) was bimodal with peaks at low mean adherence (percent consistency at most 10%) and high mean adherence (percent consistency over 90%), as was also the case for the full set of ATHENA MEMS data [4].

3

openings per day

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4

2

1

0 0

60

120

180

240

300

360

cum ula tive da ys w ithin study pa rticipa tion

Figure 1. Mean Adherence Pattern Based on Unit Dispersion Model for Subject A with Very High Mean Adherence.

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openings per day

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0 0

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cumulative days w ithin study participation Figure 2. Mean Adherence Pattern Based on Unit Dispersion Model for Subject B with Moderate Mean Adherence.

openings per day

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4

3

2

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0 0

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120

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240

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360

cumulative days w ithin study participation Figure 3. Mean Adherence Pattern Based on Unit Dispersion Model for Subject C with Low Mean Adherence.

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A Systematic Approach for Analyzing Electronically Monitored Adherence Data

openings per day

4

3

2

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0 0

60

120

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360

cumulative days w ithin study participation Figure 4. Mean Adherence Pattern Based on Unit Dispersion Model for Subject D with Deteriorating Mean Adherence.

openings per day

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4

3

2

1

0 0

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120

180

240

300

360

cumulative days w ithin study participation Figure 5. Mean Adherence Pattern Based on Unit Dispersion Model for Subject E with Improving Mean Adherence.

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35%

31%

30% 22%

25% 20% 15%

11%

9%

8%

10% 5%

2%

2%

4%

5%

5%

0% ≤10% 10-20%20-30%30-40%40-50%50-60%60-70%70-80%80-90% > 90%

Percent Consistency of Prescribed with MEMS Adherence Figure 6. Distribution for Percent Consistency Scores Based on Unit Dispersion Models for the 161 ATHENA Subjects.

Openings per Day

2.5 very high

2

high moderately high

1.5

moderate

1

moderately low low

0.5

very low Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Proportional Time within Study Participation

1

Figure 7. Relatively Consistent Mean Adherence Clusters Based on Unit Dispersion Models.

II.1.2. Cluster Analyses We adaptively clustered the mean adherence patterns at 5%, 10%, ···, 100% of the usage periods for all 161 subjects, considering a wide variety of clustering procedures (Section IV.3) with from 1-20 clusters. We used 10-fold LCV scores to compare these clustering alternatives. The best LCV score was generated by Ward's method using unsquared Euclidean distance with 10 clusters (Step 3 using the LCVcluster macro). There were 7 relatively consistent clusters (Figure 7) including very high mean adherence for 38 subjects (23.6%) including Subject A, high mean adherence for 18 subjects (11.2%), moderately high mean adherence for 15 subjects (9.3%), moderate mean adherence for 17 subjects (10.6%)

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including Subject B, moderately low mean adherence for 15 subjects (9.3%), low mean adherence for 20 subjects (12.4%), and very low mean adherence for 9 subjects (5.6%) including Subject C. There was 1 distinctly deteriorating mean adherence cluster (Figure 8) with mean adherence deteriorating from high to low for 9 subjects (5.6%) including Subject D. There were also 2 distinctly improving mean adherence clusters (Figure 9), including mean adherence improving from moderate to very high for 10 subjects (6.2%) and mean adherence improving from low to high for 10 subjects (6.2%) including Subject E. In contrast, Knafl et al. [4] applied the same clustering procedure, but with 7 clusters, to the full set of MEMS data and identified relatively consistent clusters as well as deteriorating clusters, but no improving clusters. These results demonstrate that a wider variety of mean adherence patterns can exist in MEMS data.

Openings per Day

2.5 2 1.5 from high to low

1 0.5 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Proportional Time within Study Participation

1

2.5 2 Openings per Day

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Figure 8. Distinctly Deteriorating Mean Adherence Cluster Based on Unit Dispersion Models.

from moderate to very high

1.5 1

from low to high

0.5 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Proportional Time within Study Participation

Figure 9. Distinctly Improving Mean Adherence Clusters Based on Unit Dispersion Models.

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II.1.3. Characterization of At Least Moderately High Mean Adherence Since high levels of adherence to antiretroviral medications are of clinical importance, we investigated (as part of Step 4) characteristics of subjects with at least moderately high mean adherence (i.e., with either moderately high, high, or very high mean adherence). We first used logistic regression to model membership in this composite set of mean adherence clusters compared to membership in any of the other less than moderately high mean adherence clusters. We tested for differences in membership for selected subject characteristics using Wald tests. Male subjects (52.2% or 84 of the 161 subjects) were significantly (p=0.012) more likely than female subjects to have had at least moderately high mean adherence with estimated odds ratio 2.26 and 95% confidence interval 1.20-4.28. White subjects (42.2% or 68 of the 161 subjects) were significantly (p

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789 K in series 1 and 4 (Figure 11a). In Figure 9a, the chains of 2-5 Si atoms can be seen, and along with the chains there are three-dimensional clouds of atoms. Thus, when the fullerene is broken down at high temperatures, linear chains of atoms are formed. Similar chains are involved in the silicyne structure.

Figure 11. Temperature dependences of (a) the average number of bonds per atom average Si-Si bond length

Lb

for the

Si 60

nb

and (b) the

fullerene. The numbers of curves 1-5 correspond to the

same conditions as in Figure 10. Symbol 6 refers to the result of molecular dynamic simulation for the

Si 200

nanoparticle [27].

12. Coefficients of Diffusion and Linear Expansion The mobility of atoms in the hollow clusters studied here is illustrated in Figure 12. At low temperatures (up to 550 K), hydrogen does not produce any noticeable effect on the selfdiffusion coefficient D of Si atoms. All five series of calculations give virtually the same dependence D(t ) . The noticeable influence of hydrogen on the kinetics of Si atoms is

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observed at T > 770 K. Particularly profound changes in D occur for the Si cluster containing 60 H atoms in the interior space. At first, the pressure created by hydrogen reduces the mobility of atoms in silicon, and then the quantity D sharply increases due to evaporation of Si atoms. For this cluster at T ~ 1500 K, the quantity D follows again the dependence D (t ) for the clusters treated in series 1 ,4, and 5. When the number of H atoms inside the Si cluster is half of that in series 3, i.e., equal to 30 (dependence 2 in Figure 12), a sharp increase in the dependence D (t ) can be seen at higher temperatures, starting from ~1370 K; this increase is also due to evaporation of Si atoms. If there is hydrogen inside and outside the cluster (series 4, 5), we can see oscillations of the function D(t ) at T > 1180 K. In this case, equalization of the numbers of H atoms inside and outside the Si cluster suppresses the oscillations of D (t ) at high temperatures. For the Si cluster in vacuum (dependence 1) at T > 1180 K, the function D(t ) oscillates with an amplitude even larger

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than that for dependences 4 and 5 in Figure 12.

Figure 12. The temperature dependence of the self-diffusion coefficient in the

Si 60

fullerene. The

numbers of curves 1-5 correspond to the same conditions as in Figure 10. Curve 6 refers to the result of molecular dynamic simulation for the

Si 200

nanoparticle [27].

Curve 6 in Figure 12 is the result of MD calculations [27] of the self-diffusion coefficient of the Si nanoparticle consisting of 6400 atoms. At the surface of that cluster, there was a monolayer of 785 H atoms. In contrast to the Si 60 cluster studied here, the Si 6400 cluster was free of voids. It can be seen that, in the temperature range 10 K ≤ T ≤ 1350 K, the selfdiffusion coefficient of the Si 6400 nanoparticle, as a rule, is noticeably smaller than the corresponding coefficient D of the clusters considered in this study. However, at T > 1500 K, the increase in the coefficient D of the Si 6400 nanoparticle becomes steeper than the increase in D of the clusters considered here. This is most likely due to the Stillinger-Weber potential used to describe the Si-Si interactions in the nanoparticles. This potential provides a

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coefficient D larger than those obtained with the potential used here because of the substantially higher mobility of surface atoms at a high temperature. The estimation of the linear expansion coefficient β for the Si 60 cluster in the temperature range 511-611 K shows that the Si fullerene filled with 60 H atoms exhibits the largest value of

β (6.09 × ⋅10 −5 K −1 ), while the Si 60 cluster covered with hydrogen shells

of 60 atoms features the smallest value (-1.51 × 10

−5

K

−1

). The negative value of

β shows

that, in the latter case, we have interplay of various factors associated with the pressure produced by hydrogen both inside and outside the cluster and with the surface tension of the fullerene in vacuum. In vacuum, the Si fullerene that has kinetic energy corresponding to the temperature 561 K is characterized by a coefficient β at least an order of magnitude larger than the average value of

β (3.5 × 10 −6 K −1 ) for crystalline silicon [56] at the same

temperature.

13. Conclusion

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Over a few last decades, the molecular dynamics method has been widely used for investigating the properties of quite different materials. These are molecular crystals, amorphous polymers, liquid crystals, zeolites, nucleic acids, proteins, superconductors, and semiconductors. In this series of studies it was demonstrated that a 73-atom crystalline nanoparticle undergoes melting at temperatures close to 710 K. The melting of a 480-atom silicon nanoparticle is observed at a temperature of 1560 K [47], which is close to the melting temperature of real silicon crystals (1665 K). It was revealed that, as the temperature increases, structural transformations in a nanoassembled silicon particle occur in a smoother manner as compared to those in a silicon nanoparticle with a random atomic packing. The results of the computer experiment demonstrated that the Si 73 clusters well adsorb hydrogen at low temperatures. The hydrogen “coat” is retained around these particles up to high temperatures (~1500–1600 K). The hydrogen atoms are repelled from each other and, thus, have a compressive effect on the cluster. The nanoparticle composed of the icosahedron and the fullerene retains the highest density up to a temperature of 1560 K. The nanoassembled particle has the most stable number (close to four) of Si-Si bonds per atom in the temperature range 35 ≤ T ≤ 1560 K. In each particle under investigation, the mean bond length decreases and the particle size Rcl increases with an increase in the temperature. Moreover, the particle shape does not deviate significantly from a spherical shape. As a result, holes should be formed inside the particles, and these holes can be occupied by hydrogen. The practical importance of our investigation is associated with the use of hydrogen as a fuel. The calculations show that the Si 60 fullerene is not a stable cluster, as can happen with its C 60 analogue. At low temperatures, the Si 60 fullerene placed in vacuum collapses, which is promoted by the formation of short chains and small bulk fragments and by the decrease in the average bond length. At high temperatures, the broken Si 60 fullerene increases its initial

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volume, and a fraction of the constituent atoms evaporate, whereas the remaining atoms are grouped into chains and ultrasmall clusters consisting of several (2-8) atoms. At lower temperatures (100 K ≤ T ≤ 800 K), the complete compensation of unpaired Si-Si bonds with hydrogen inside the fullerene produces the effect of diffusion "resistance"; in this case, the self-diffusion coefficient is nearly halved. In addition to the filling of the fullerene interior space with hydrogen, the creation of a hydrogen "coat" outside the fullerene tends to decrease the probability of evaporation of Si atoms, but does not protect the system from a sharp decrease in the number of bonds per atom at high temperatures. At a temperature close to 270-280 K, the Si 60 fullerene filled with 60 hydrogen atoms and covered with a hydrogen "coat" is close to the boundary of thermal stability. As before there is an open question as to how one can obtain a smooth fullerene-like cage 3

composed of Si atoms. Due to their sp nature, the Si atoms do not tend to bind themselves to general fullerene cages. Si N clusters ( N up to ~ 50) usually favor compact forms which are completely different from fullerene cages [57, 58]. We have shown that the structure of a Si 60 cluster in a fullerene cage is highly distorted even at not so high temperature and at the presence of hydrogen. There is a suggestion to construct Si clusters in fullerene-like cages with transition-metal atom doping [59]. First-principles calculation shows that, tungsten (W) may be a good candidate for doping of fullerene-like Si cages and WSi 12 and WSi 14 clusters are energetically most favorable. Geometric and electronic structures of metal (M) atom doped silicon (Si) clusters, MSi N (M = Ti, Hf, Mo and W), using mass spectrometry, a chemical-probe method and photoelectron spectroscopy have been studied in the work [60]. -

-

-

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In the mass spectra for all of the mixed cluster anions MSi N , both MSi15 and MSi16 were abundantly produced compared to their neighbors. Together with the result of the adsorption reactivity and photoelectron spectroscopy, it was revealed that one metal atom can be encapsulated inside a Si N cage at N ≥ 15. The molecular dynamics calculations based on the density functional theory and the ab initio theory of electronic structure revealed that liquid silicon nanoparticles containing from 274 to 323 atoms can form distorted icosahedral structures upon freezing [60]. A geometrical prerequisite for the explanation of this fact is the possibility of constructing an icosahedron from 20 slightly distorted tetrahedra shared by faces. Thus, the conclusion can be drawn that the computer simulation makes it possible to predict the structure and thermodynamic properties of new non-crystalline materials and to improve the technology used for their production.

Acknowledgments This study was supported by the Presidium of the Ural Division of the Russian Academy of Sciences within the framework of the Integration Project of the Ural Division-Far East Division of the Russian Academy of Sciences.

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[57] Honea, E. C.; Ogura, A.; Peale, D. R.; Felix, C.; Murray, C. A.; Raghavachari, K.; & Sprenger, W. O. J. Chem. Phys. 1999, 110, 12161–12171. [58] Ho, K. M.; Shvartsburg, A. A.; Pan, B.; Lu, Z.- Y.; Wang, C. Z.; Wacher, J. G.; Fye, J. L.; & Jarrold, M. F. Nature (London), 1998, 392, 582–583. [59] Hiura, H.; Miyazaki, T.; & Kanayama, T. Phys. Rev. Lett. 2001, 86, 1733 –1736. [60] Ohara, M.; Koyasu, K.; Nakajima, A.; & Kaya, K. Chem. Phys. Lett. 2003, 371, 490– 497. [61] Nishio, K.; Morishita, T.; Shinoda, W.; & Mikami, M. Phys.Rev. B: Condens. Matter. 2005, 72, 245321 (1–4).

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In: Micro Electro Mechanical Systems Editors: B. Ekwall and M. Cronquist, pp. 195-223

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Chapter 5

DESIGN OF OPTICAL MEMS FOR TRANSPARENT BIOLOGICAL CELL CHARACTERIZATION Xiaodong Zhou1, Kai Yu Liu2 and Nan Zhang1 1

Institute of Materials Research and Engineering, Agency for Science, Technology and Research, Singapore 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore

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Abstract Consecutive living cell characterization has aroused the interest of biological researchers, as the cellular investigation is turning from population to singles. Optical MEMS plays an important role in this revolutionary trend. This chapter describes a novel optical MEMS device that can fully characterize the transparent living cells or microparticles in real time, with an optical aperture or an aperture array. Our MEMS structure only contains an optical aperture and a microfluidic channel. When a transparent biological cell is pumped through the channel, under which an optical aperture is fabricated and a laser beam is shed on, the far-field diffraction pattern of the aperture and the cell is recorded by a camera for cell characterization. This is a three-dimensional system that the cell, the aperture and the microfluidic channel are not within one plane, but with a gap between the cells and the aperture. By careful design of the gap, the image of the cells and their inclusions, such as nuclei, will appear on the diffraction pattern, and the magnification of the image can be adjusted. Because the intensity of light at each pixel of the diffraction pattern is related with the refractive indices of the cells and their inclusions, the refractive indices can be extrapolated from the image of the cells by diffraction simulation. This is the first simple optical MEMS structure to characterize the shape, size, and the refractive indices of the cells and their inclusions simultaneously, and the calculated limit of detections (LOD) of this device reach minima at the wavelength of 0.266 µm, with 9e-5 for the refractive index and 0.182 µm for the diameter of a cell, respectively. The diameter detection limit of this device is close to the resolution of conventional optical systems that suffer from diffraction caused optical limit. By recording the diffraction patterns of 5 µm-diameter polystyrene microspheres with various optical apertures in micro sizes, we found that our simulated diffraction patterns coincided well with the experiments, which proved the correctness of our theory and concept. Although the design of the MEMS structure is given, the whole image and characterization

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Xiaodong Zhou, Kai Yu Liu and Nan Zhang system with this MEMS chip has not been realized. Since the principle of this device has been verified and its MEMS fabrication is not complicated, we regard it as a feasible device for individual living cell or microparticle characterizations.

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1. Introduction With gradual recognition and emphasization, cellular heterogeneity has attracted the development of cell characterization methods from population-level to cellular bases, and aroused special interest to monitor discrete and dynamic processes occurring within living cells [1]. At this stage, the tools and technologies of characterizing single cells still rely on traditional methods such as fluorescence, cytometry, scanning probe microscopy (SPM), micro-spectroscopy, with additional mechanical or optical micromanipulators or microelectromechanical systems (Bio-MEMS) to manipulate each biological sample. Such a system will inevitably be complicated and expensive. There is high demand on the development of portable lap-on-a-chip devices that can detect cells one by one with high throughput and low cost, and it is desired that the principle being used for detection is reagentless, noninvasive, and being able to detect a variety of parameters simultaneously. This chapter introduces the design of such a MEMS device, which combines the optical aperture diffraction and microfluidics to realize full characterization of transparent living cells in high throughput. Optical diffraction refers to the detour or expansion of optical waves when they encounter a grating, obstacle or aperture at a similar size to the optical wavelength. Since its discovery in 1665, it has long been regarded as a fundamental resolution limit in image formation that people strive to diminish, rather than a means to view objects. Optical diffraction has deepened people’s optical knowledge and widened optical applications, for example, grating diffraction is vastly used for dispersing wavelength and detecting microbial growth [2-4], obstacle diffraction is frequently used in X-ray diffraction for characterizing crystalline structures and scattering of microparticles for identifying their sizes [5, 6], however, it is obvious that diffraction is not the mainstream for images, and there are only a few examples utilizing aperture diffraction for image formations. In 2005, King et al. patented a technology of enlarging an image by inserting an aperture, but the image was formed by an optical lens combination, the enlarged image was deformed by the aperture and had to be corrected by software [7]. Cui et al. utilized a one-dimensional (1D) aperture array above a CCD or CMOS camera to increase the image resolution, and it was fabricated into an on-chip optofluidic microscopy (OFM) [8]. This design is more of a near-field scanning optical microscope, with the aperture array just 2.5 µm away from the camera. The disadvantage of this system is that the aperture array was arranged in a line of more than 1 mm long and a cell had to travel through in a constant speed to ascertain an authentic image, this causes the problems such as extra MEMS functions to drag the cell along the channel, killing of the living cells to avoid their tumbling and rotation, and resampling of the image when being deformed. This chapter reports that an optical aperture alone can be used to characterize the transparent cells or microparticles. For aperture diffraction, the diffracted light spot is small, thus it is difficult for the diffracted light to illuminate onto the cells; if the aperture is reduced to increase the light spot the light intensity shed on the cells will be weak. MEMS technology

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is the key for us to realize this system, as it helps to achieve the accurate positioning of the cells in front the aperture. In our method, a laser beam sheds on the optical aperture and then passes through the living cell, whose position can be confined by microfluidics, and the diffraction pattern of the aperture and cell is recorded with a CCD camera for analyses. The diffraction pattern is related with the positions, shapes and refractive indices of the cell and its inclusions. It has been found that when the cells are close to the aperture, the diffraction pattern looks chaotic that no clear image of the cells could be discerned. With the help of analysis software, only one of the parameters of the cell can be extrapolated under the condition that all other parameters are recorded correctly [9, 10]. However, by controlling the aperture size and the gap between the cell and the aperture, clear image of transparent living cell as well as its inclusion can be viewed directly in the far-field diffraction pattern, and the refractive indices of the cell and its inclusion can be further analyzed by software, thus can realize the full characterization of the cell.

Possible apertures:

Possible chambers:

X

X’

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Laser

Z

P

Y’ P’

Y

P”

Aperture Plane

Detection Plane

Figure 1. The system to detect the image and refractive indices of the cells and their inclusions by aperture diffraction, the cells are conducted and confined in the chamber by microfluidics. The aperture and the cells illuminated by laser generate a diffraction pattern, which includes the image of the cells and is recorded by a camera. Buffer in the chamber deflects the light, causing the light spot on P’ to actually appear on P”. Possible aperture and chamber shapes are drawn in the figure.

This optical diffraction system possesses several significances. Firstly, single biological cell detection can be achieved. By making the cross section of the microfluidic channel at a similar size of the cell, the cell can pass the aperture one by one and be characterized consecutively. Living cells can be detected because the light intensity after aperture

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diffraction is low. Secondly, even the cell is totally transparent, no special techniques, such as fluorescence, polarized light, phase contrast, differential interference contrast, or confocal microscopy, is required for detection, because the diffraction image is resulted from the effective optical path difference besides the optical absorption of the cell, it can capture the image of the transparent objects by the variation of refractive index. Thirdly, the diffraction aperture acts as a simplified microscope to view the image of the cells and their inclusions, with its size resolution approximate to the limit of a conventional optical microscope. The refractive indices of the cells and their inclusions can be characterized simultaneously by software, which is an advantage over a microscope. Because the hardware of this system only includes a laser and a camera, an inexpensive miniaturized and portable cytometry can be fabricated to characterize living cells, as well as micro-sized transparent polymer spheres and insoluble droplets in aqueous solution. This chapter comprises ten sections: introduction; design and fabrication process of the MEMS device; formulas for calculating the diffraction patterns; influence of the gap between the cells and the apertures; the effect of different shapes of the aperture; the effect of different kinds of chamber; method of extrapolating the refractive index of the cells; calculation of the limits of detection (LOD) of the device; preliminary experiments conducted to prove the correctness of our simulations; and conclusion for further research. Aperture Central Line

Aperture Central Line

Aperture Central Line

Aperture Central Line

γ

Side view

γ

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Top view

Circular chamber (up to 3 layers)

Pyramidal chamber (up to 3 layers)

Rectangular chamber (up to 3 layers)

Conical chamber

Figure 2. Fabrication errors cause misalignments between the aperture and different kinds of chambers. Several layers stacked in a chamber help the positioning of the cells and result in more misalignments, they have to be simulated to predict and reduce their influences to the diffraction image of the cells.

2. Device Design The diffraction system consists of a laser to shed the light onto the aperture, transparent cells in microfluidic chambers and above the aperture, and a camera to record the far-field diffraction pattern, as shown in Figure 1. The MEMS device in Figure 1 only comprises an aperture layer, a microfluidic chamber at the center of the aperture, and microfluidic channels to conduct the cells into and out off the device. A microfluidic pump might be built in the device to help the constant flow of the cells, but it is not essential.

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Figure 3. Fabrication process of the MEMS device with an optical aperture and a two-layer stacked pyramidal chamber. (a) wet etch a pyramidal silicon tip; (b) wet etch a wider pyramidal chamber and a channel on silicon; (c) fabricate the aperture by metal lift-off; (d) spincoat a layer of PDMS on the glass and stamp the PDMS with silicon mold to form the microfluidics; (e) bond PDMS with an inlet/outlet predrilled top glass cover.

The aperture can be circular or rectangular. A rectangular aperture functions similarly to a circular one except its diffraction pattern turns ellipsoidal instead of circular. The circular aperture can also be central blocked to form a ring; a ring shape enhances the contrast and profile of the cells. An aperture array can also be used, it increases the light intensity as well as the illumination area, thus eases the capture of the cells. Four kinds of microfluidic chambers are designed and included in our simulation software, namely cylindrical, cuboidal, pyramidal, or conical chamber. Pyramidal and conical chambers can trap the cells in front of the aperture better by slow injection. Similar functions can be implemented by stacking cylindrical or cuboidal chambers in several layers with size steps. However, more layers may introduce more misalignments and process complexity in fabrication. As illustrated in Figure 2, the device fabrication introduces misalignments between the aperture and each layer of the chamber. Since each combination of the aperture and chamber produces a different diffraction pattern which forms the background of the cells’ image, the shapes of the aperture and chamber are analyzed in following sections.

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In MEMS fabrication, the optical aperture is fabricated on a glass substrate by metal liftoff. Microfluidics is fabricated on the aperture by spincoating a layer of transparent polydimethylsiloxane (PDMS) polymer and stamping a microfluidic mold fabricated by silicon substrate on the PDMS. After microfluidics being stamped, the top PDMS layer is activated by O2 plasma treatment to bond to a glass cover with inlet/outlet openings predrilled. After fabrication, the misalignment of each layer can be checked under a microscope. Figure 3 exemplifies the fabrication process of a two-layer pyramidal chamber. The spincoat speed of PDMS adjusts the gap between the cells and the aperture. For cylindrical or cuboidal chambers, the silicon mold can be fabricated by silicon deep ion reactive etching (DIRE), and the molds for pyramidal and conical chambers are achieved by silicon KOH wet etching or silicon tip fabrication methods.

3. Theory Diffraction analysis is among the difficult problems encountered in optics, and rigorous and accurate solutions are rare. Since the aperture, the chamber, the buffer and the cells are not in one plane, they must be regarded as a whole three-dimensional system, and we use the Fresnel-Kirchhoff diffraction integral to calculate the diffraction pattern [11]. The benefit of this integral is that because the phase of the light ray is accurately considered in calculations, the simulated diffraction pattern applies not only to far-field, but also to some diffractions that

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do not exactly fulfill the far-field condition r

2

Lλ i )

⎣

⎦

(19)

where VR(r) and VA(r) denote the pair-additive repulsive and attractive interactions, respectively, and bij represents a many-body coupling between the bond from atom i to atom j and the local environment. The interwall interactions between the carbon atoms are via the Lenard-Jones potential in the form of

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⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ [ V ( r ) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥ , ⎝ r ⎠ ⎥⎦ ⎣⎢⎝ r ⎠

(20)

in which ε = 2.968 meV, σ = 0.3407 nm. [49]

Figure 1. Schematic of the nanomotor constructed by a DWNT.

Table 1. Structural parameters of three DWNTs based nanomotors

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Case 1 2 3

Chirality pair (5, 5) / (10, 10) (13, 0) / (22, 0) (8, 2) / (17, 2)

Length (nm) 24.9 / 7.5 25.9 / 5.2 26.4 / 5.2

Radius (nm) 0.344 / 0.688 0.516 / 0.873 0.364 / 0.718

Interwall distance (nm) 0.344 0.357 0.354

3.2. Operation Behaviors Three DWNTs based nanomotors with different chirality pairs are simulated as tabulated in table 1. The interwall distance in all the systems is about 0.34 nm which ensures the stability of the structure and movement. The potential patterns of the three DWNT systems are plotted in figure 2. The two degrees of freedom of motion are taken to be sliding along the nanotube axis and rotation around the nanotube axis. A darker color denotes a lower potential energy. Allowing for the difference in DWNTs’ lengths, the potential patterns calculated in this paper are quantitatively in agreement with the results in Ref. [49] and [50]. If the outer tube is completely released, it will fall into the potential valleys with the lowest energy. Driven by a small force, it will choose an easy way, which is just the minimum energy track, to move. The minimum energy tracks of the (5, 5) / (10, 10), (13, 0) / (22, 0) and (8, 2) / (17, 2) nanomotors are along the tube axis, circumference and helix, respectively, as illustrated in figure 2. Between two neighbor minimum energy tracks is a potential barriers U, which is the restriction ability from changing tracks for the outer tube. The barriers of case 1, 2 and 3 in our simulations are 0.024 eV, 0.15 eV and 0.12 eV, respectively.

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Figure 2. Interwall potential patterns for DWNTs with different chirality pairs: (a) (5, 5) / (10, 10), (b) (13, 0) / (22, 0), (c) (8, 2) / (17, 2).

With the two ends of the inner tube being at 600 K and 400 K, the outer tube will speed up, namely the thermally driven nanomotor will run. The center of the mass (CoM) and the rotation angle along the circumference of the outer tube varying with time are shown in figure 3. Driven by the temperature difference, the outer tubes of all the three nanomotors move towards the specific directions. This phenomenon is similar to that observed in the experiments and simulations by Barreiro et al [16]. Its actuation mechanism can be understood by considering the drag on the outer tube induced by the phonon current transporting in the nanomotor as stated above. For the (5, 5) / (10, 10) nanomotor as shown in figure 3(a), the outer tube translates along the axis and oscillates in the circumferential direction. The maximum oscillation amplitude is about 15 degree, less than the separation angle between two neighbor minimum energy tracks in the (5, 5) / (10, 10) DWNTs. It indicates that the movement of the outer tube is confined in the minimum energy tracks as shown in figure 2(a). The same conclusion can be drawn for the (13, 0) / (22, 0) DWNT (figure 3(b)) in which the outer tube rotates around the axis with very small oscillations in the axis direction (less than 0.06 nm). In this case, however, we should point out that because of the perpendicularity between the minimum energy track and the thermal driving force, the rotation direction of the outer tube, that is clockwise or anticlockwise, depends on the initial actuation. The minimum energy track in the (8, 2) / (17, 2) DWNT has been demonstrated to be a helix above. For this nanomotor, figure 3(c) shows the trace of the outer tube, and the

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inset gives the rotation angle as a function of the translational distance in a polar coordinate system. The rotational angle is proportional to the translational distance, indicative of a helix trace. The helix angle of the trace is 810±9.8 degree/nm, which agrees with the helix angle of the minimum energy tracks, 829 degree/nm obtained by figure 2(c). The small deviation may arise from the deformation of the tubes at different temperatures. Therefore, the motion in this system is also consistent with the minimum energy tracks. Under the current conditions, the thermally driven nanomotors run along the minimum energy tracks with specific directions depending on the DWNT chirality pairs. From this point of view, the directional control of DWNTs based nanodevices can be realized. Can the outer tubes escape from the minimum energy tracks? Imagining that the outer tube is running in the potential patterns, the potential barriers play a dominant role in restricting its traces. If the outer tube goes to escape, it must have enough energy to overcome the potential barriers. According to the energy equipartition theorem, each degree of freedom of a particle has a kinetic energy kBT/2 (kB is the Boltzmann constant). If this energy is less than the potential barrier, the particle will be confined in a potential valley. However, if the kinetic energy is larger than the barrier, the particle is able to escape from the restriction of the minimum energy tracks. According to the potential barrier U, we can define a critical temperature

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[ Tc =

2ΔU / kB ,

(21)

which determines whether the motion is along the minimum energy track or not. For the three chirality pairs simulated above, the critical temperatures are 557 K, 3478 K and 2783 K, respectively. Since the average system temperature, 500 K, is less than the critical temperatures, the motion of the outer tubes can be confined in the minimum energy tracks. For the (5, 5) / (10, 10) nanomotor, since the system temperature is very close to the critical temperature, obvious oscillations in the circumferential direction can be observed as shown in figure 3(a). To verify the statements above, additional MD simulations, setting the system average temperatures of the (5, 5) / (10, 10) nanomotor at 800 K and 1000 K with a temperature difference of 200 K, are performed. We run twice for each average temperature. For all the calculation cases the outer tubes are found to move towards the cold ends, as observed above. However, the motions in the circumferential direction are quite different. Figure 4 shows the rotational angle as a function of time. At 800 K and 1000 K, the outer tube can rotate clockwise or anti-clockwise, even change rotational directions during the motion. The random behaviors imply that the outer tubes are not confined in the minimum energy tracks. The critical temperature serves as a crucial criterion for the directional control of nanomotors based on DWNTs or MWNTs. To confine the relative motion in the minimum energy tracks, the temperature of the system should be lower than the critical temperature defined by the potential barrier between two neighbor potential valleys. The higher the critical temperature is, the better direction controllability the system has. In order to realize the precise control of the motion, high potential barrier is required. For commensurate DWNTs, the potential barrier is proportional to the length of the contact surface, so we can increase the potential barrier by increasing the length of the tubes. For incommensurate DWNTs, the potential barrier only fluctuates near an average value when the tube length increases.

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Figure 3. The center of mass and the rotation angle of the outer tube varying with time for: (a) (5, 5) / (10, 10), (b) (13, 0) / (22, 0), (c) (8, 2) / (17, 2). The inset in figure 3(c) plots the trace of the motion in a polar coordinate system whose polar axis and angle units are nanometer and degree, respectively.

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Figure 4. The rotation angle of the outer tube in the (5, 5) / (10, 10) DWNT varying with time. We run twice for each average temperature, marked by R1 and R2.

In this case, chirality pairs with large potential barriers and high critical temperatures are preferred in practice. Therefore, knowing the chirality pair and the critical temperature can enable the realization of the direction controllability of the thermally driven nanomotors based on DWNTs.

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4. Conclusion Based on introducing a novel concept called thermomass, the relativistic mass of a phonon gas in light of the Einstein’s energy-mass relation, the mass and momentum conservation equations for the phonon gas motion are established to characterize the hydrodynamics of the phonon current in a solid. There are three kinds of force acting on the phonon gas simultaneously: driving force, inertial force and resistance. The counteracting force of the resistance, like the gas flows in the porous mediums, gives rise to a driving force on the solid framework atoms, which can be calculated quantitatively and can be applied to actuate nanomotors. We predict the dynamic behavior of a nanomotor made up of multi-walled carbon nanotubes in terms of molecular dynamics simulations. A shorter single-walled carbon nanotube with a larger diameter, as a mobile part, surrounds a longer single-walled carbon nanotube with a smaller diameter working as a shaft. When a phonon current passes through the inner shaft, the outer nanotube will translate along or rotate around the shaft depending on the chiralities of the carbon nanotubes. The motion traces are found to depend on the chirality pair and the system temperature regularly, which implies that the motion direction of the CNTs based nanomotors can be controlled. This makes such a type of nanomotor more promising.

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Acknowledgment Financial supports from National Natural Science Foundation of China (No. 50606018) and Tsinghua National Laboratory for Information Science and Technology (TNList) Crossdiscipline Foundation are gratefully acknowledged.

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[27] Guo Z. Y.; Cao B. Y.; Zhu H. Y.; Zhang Q. G. Acta Phys. Sin. 2007, vol. 56, 33063312. [28] Guyer R. A.; Krumhansi J. A. Phys. Rev. 1966, vol. 148, 778-788. [29] Cimmelli V. A.; Frischmuth K. Physica B 2007, vol. 400, 257-265. [30] Alvarez F. X.; Jou D.; Sellitto A. J. Appl. Phys. 2009, vol. 105, 014317. [31] Cattaneo C. Atti. Sem. Mat. Fis. Univ. Modena 1948, vol. 3, 83-101. [32] Vernotte P. C. R. Acad. Sci. 1958, vol. 246, 3154-3155. [33] Lloyd J. R. Semicond. Sci. Technol. 1997, vol. 12, 1177-1185. [34] Tu K. N. J. Appl. Phys. 2003, vol. 94, 5451-5473. [35] Denbigh K. G. The Thermodynamics of the Steady State; Methuen: London, UK, 1951. [36] Soret C. H. Arch. De Geneve 1879, vol. 3, 48-61. [37] Goldhirsch I.; Ronis D. Phys. Rev. A 1983, vol. 27, 1616-1634. [38] Stokes H. T. Solid State Physics; Allyn and Bacon: Boston, US, 1987. [39] Huntington H. B. J. Phys. Chem. Solids 1968, vol. 29, 1641-1651. [40] Berber S.; Kwon Y. K.; Tomanek D. Phys. Rev. Lett. 2000, vol. 84, 4613-4616. [41] Hou Q. W.; Cao B. Y.; Guo Z. Y. Acta Phys. Sin. 2009, vol. 58 (In press). [42] Pop E.; Mann D.; Wang Q.; Goodson K.; Dai H. Nano Lett. 2006, vol. 6, 96-100. [43] Fujii M.; Zhang X.; Xie H. Q.; Ago H.; Takahashi K.; Ikuta T.; Abe H.; Shimizu T. Phys. Rev. Lett. 2005, vol. 95, 065502. [44] Cumings J.; Zettl A. Science 2000, vol. 289, 602-604. [45] Yu M. F.; Yakobson B. I.; Ruoff R. S. J. Phys. Chem. B 2000, vol. 104, 8764-8767. [46] Servantie J.; Gaspard P. Phys. Rev. Lett. 2003, vol. 91, 185503. [47] Allen M. P.; Tildesley D. J. Computer simulation of liquids; Clarendon: Oxford, UK, 1987. [48] Brenner D. W. Phys. Rev. B 1990, vol. 42, 9458-9471. [49] Saito R.; Matsuo R.; Kimura T.; Dresselhaus G.; Dresselhaus M. S. Chem. Phys. Lett. 2001, vol. 348, 187-193. [50] Lozovik Y. E.; Minogin A.; Popov A. M. Phys. Lett. A 2003, vol. 313, 112-121.

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ISBN: 978-1-60876-474-7 © 2010 Nova Science Publishers, Inc.

Chapter 7

TANGENTIAL NANOFRETTING AND RADIAL NANOFRETTING Linmao Qian*, Jiaxin Yu, Bingjun Yu and Zhongrong Zhou Tribology Research Institute, National Traction Power Laboratory, Southwest Jiaotong University, Chengdu, China

Abstract

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Nanofretting refers to cyclic movements of contact interfaces with the relative displacement amplitude in nanometer scale, where the contact area and normal load are usually much smaller than those in fretting. Nanofretting widely exists in nano/microelectromechanical systems (NEMS/MEMS) and may become a key tribological concern besides microwear and adhesion. With an atomic force microscopy, the tangential nanofretting of monocrystalline Si(100) surface against spherical SiO2 and diamond tips was carried out at various displacement amplitudes (0.5~250 nm) under atmosphere and vacuum condition. It was found that: (1) the tangential nanofretting could be divided into stick regime and slip regime upon the transition criterion; (2) the adhesion force may induce the increase in the maximum static friction force and prevent the contact pair from slipping; (3) the nanofretting damage on silicon may experience two processes, namely as the generation of hillocks at low load and the formation of grooves at high load; (4) compared to those in atmosphere, the nanofretting scars in vacuum exhibited higher hillock at low load but shallower groove at high load. The radial nanofretting behaviors of several typical structural materials in NEMS/MEMS (polycrystalline copper, monocrystalline silicon and CNx films) were investigated by a nanoindenter. The results indicated that: (1) the radial nanofretting damage in copper was mainly identified as the pileup of the wrinkles around indents, and in silicon was characterized as the initiation and propagation of the cracks on the edges of plastic zone; (2) due to its higher hardness and elastic modulus, the CrNx film can effectively improve the anti-pressure ability of 40Cr substrate in radial nanofretting; (3) compared to the curvature radius, the equivalent radius of indenter played a much more important role on the nanofretting damage of material. The results in the research may be helpful to understand the nanofretting failures of components in NEMS/MEMS.

*

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1. Introduction Nanofretting refers to cyclic movements of contact interfaces with the relative displacement amplitude in nanometer scale, where the contact area and normal load are usually much smaller than those in fretting [1]. Due to the excellent mechanical and physical behaviors, monocrystalline silicon has been widely used as structural material in nano/microelectromechanical systems (NEMS/MEMS) [2-3]. Because of the temperature variation and mechanical vibration, nanofretting of monocrystalline silicon may exist in the contact interfaces of these microdevices. Therefore, with the development in NEMS/MEMS, the understanding and control of the nanofretting behavior of monocrystalline silicon has become an important issue of concern [4-6]. After the concept of nanofretting was proposed by Zhou and Qian [7] in 2003, people have performed tangential and radial nanofretting tests on kinds of materials. In 2005, Varenberg et al [8] reported their studies on nanoscale fretting wear behavior of monocrystalline silicon (100). With a scanning probe microscopy, the grooves were observed on Si(100) surface even though the displacement amplitude of tangential nanofretting was as small as 29 nm. In 2007, Qian et al [1] presented their research results on the tangential nanofretting behaviors of NiTi shape memory alloy. They found that nanofretting was different from fretting in aspects of the variation of tangential force versus number of nanofretting cycles, the value of friction coefficient, and the wear mechanism. These differences were further attributed to the single-asperity contact in nanofretting and multiasperity contact in fretting. More recently, Yu et al [9, 10] reported that the adhesion force revealed a strong effect on the regimes of tangential nanofretting. Different from fretting, the tangential nanofretting damage on silicon may experience two processes, namely as the generation of hillocks at low load and the formation of grooves at high load. In 2006, Zhang et al [11] indicated that the radial nanofretting damage in copper was mainly identified as the pileup of the wrinkles around indents, and in silicon was characterized as the initiation and propagation of the cracks on the edges of plastic zone. In 2008, Zhang et al [12] reported that due to its higher hardness and elastic modulus, the CrNx film could effectively improve the anti-pressure ability of 40Cr substrate in radial nanofretting. More recently, the effect of curvature radii on radial nanofretting was discussed by Zhang et al [13]. It was found that compared to the curvature radius, the equivalent radius of indenter played a much more important role on the nanofretting damage of material.

2. Tangential Nanofretting All the tangential nanofretting tests and in situ topography scanning were performed by an atomic force microscopy (AFM) equipped with a vacuum chamber. Figure 1 schematically shows the nanofretting of spherical tips against Si(100) surface. The spherical tips included SiO2 (Novascan Technologies, USA) and diamond microsphere (Microstar Technologies, USA), respectively. Their radii R ranged from 0.15 μm to 1.0 μm. During the nanofretting, the spherical tips moved horizontally on silicon surface with a displacement amplitude D under a normal load Fn. The applied normal loads Fn were varied between 0.5 μN and 70 μN. The displacement amplitudes D were ranged from 0.5 nm to 250 nm. The frequency was 2 Hz and the number of nanofretting cycles N was varied between 1 and 500. After nanofretting,

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the topography of scars was scanned by a sensitive silicon nitride tip, which has a curvature radius of 20 nm and a nominal normal spring constant of 0.1 N/m. All the nanofretting tests were performed in atmosphere with a relative humidity of 50%-60%, or in vacuum with a pressure below 5.0×10-6 torr (6.7×10-4 Pa).

Figure 1. The schematic illustration showing the tangential nanofretting test.

2.1. The Effect of Adhesion Force on the Regimes of Tangential Nanofretting [9]

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Figure 2 shows the frictional logs (variation of the tangential force Ft and displacement d with number of cycles N, or Ft–d–N curves) of Si(100) against SiO2 tip in vacuum. Similar to fretting in vacuum and in atmosphere [14, 15], the shape of Ft–d curves in nanofretting was found to vary from line shape to parallelogram with the increase in displacement amplitude D, which corresponds to the transition from the stick regime to slip regime.

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Figure 3. The transition displacement Dr between stick regime and slip regime versus (a) adhesion force Fa and (b) Fa+Fn curves.

While fretting operates in partial slip regime, the maximum tangential force Ft-max will increase with the increase in displacement amplitude D. However, when fretting operates in gross slip regime, the Ft-max will reach a constant value [16]. It suggested that the turning point Dr in the Ft-max -D curves is the transition condition of two regimes. In order to understand the role of adhesion force in the nanofretting behavior of material, the transition displacement Dr was obtained through three SiO2 tips (R=0.43 μm) with the adhesion forces of 0.4 μN, 2.2 μN and 3.3 μN on Si(100) surface, respectively. Figure 3 shows the variation of transition displacement Dr between stick regime and slip regime versus adhesion force Fa and Fa+Fn. It clearly reveals that with the increase in adhesion forces, the transition displacement amplitude Dr would move towards high values of displacement amplitude in Figure 3a. During the nanofretting process of Si(100)/SiO2 pair, the adhesion force may induce the increase in the maximum static friction force and prevent the contact pair from slipping. For instance, when the adhesion force varied from 0.4 μN to 3.3 μN, the maximum static friction force increased from 0.46 μN to 1.33 μN for an applied normal load of 2 μN. As a result, with the increase in adhesion force Fa, the tip was more difficult to slip.

2.2. The Damage Mode of Tangential Nanofretting [10] As shown in Figure 4a, for the tip with R =0.15 μm and under Fn =5 μN, no obvious damage was observed in stick regime (D≤ 5 nm). With the increase in D, the hillocks were generated in the contact area of Si(100) surface. However, when the normal load Fn attained 30 μN, the nanofretting scars was not identified as the hillock. Instead, the grooves were observed in the wear area of Si(100) surface both in atmosphere and in vacuum, as shown in Figure 4b and Figure 4d. Finally, as shown in Figure 4c, when the normal load Fn was increased to 70 μN, wider and deeper grooves were observed on Si(100) surface. The results indicated that the nanofretting damage of Si(100) surface was strongly depended on the displacement amplitude and normal load. With the increase in normal load, the nanofretting damage would undergo an evolvement from the generation of hillock to groove.

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(a) 5 μN

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Figure 4. AFM images of the scars on Si(100) surface after nanofretting at various displacement amplitudes D and normal loads Fn in atmosphere (a-c) and in vacuum (d). R= 0.15 μm and N= 100.

While the blunt tip with R=0.9 μm was used, no grooves were observed in the damage area of silicon surface under various normal loads. As shown in Figure 5, even the normal load attained 70 μN and the number of nanofretting cycle was 500, only hillocks were formed during the nanofretting process both in atmosphere and in vacuum.

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Vacuum

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Figure 5. AFM images of the scars on Si(100) surface by blunt tip with R=0.9 μm. Fn= 70 μN and N = 500.

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Figure 6. AFM images of the scars on Si(100) surface after nanofretting under various normal load Fn in atmosphere. R=0.15 μm, D = 100 nm and N = 100.

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2.3. The Transition between Two Damage Modes During the fretting wear either in atmosphere or in vacuum, the generation of groove was usually observed, but the formation of hillock has never been reported [17, 18]. Different from fretting wear, both the hillock and groove were observed in nanofretting under various experimental conditions. To understand the nanowear process of Si(100), the nanofretting of Si(100)/diamond pair was conducted at various normal loads in atmosphere. As shown in Figure 6, the transition of surface damage of silicon from hillock to groove was observed with the increase in normal load Fn. The hillock was generated on Si(100) surface for Fn 15 μN, the surface height was below the original surface and the depth of groove attained 9.4 nm for Fn =70 μN. As shown in Figure 7, the nanofretting of Si(100)/diamond pair was also conducted after various number of nanofretting cycles N in atmosphere. Even all the nanofretting scars reveal hillock, the shape of the hillocks may exhibit the evidence for the formation of groove when N > 200. With the increase in N, the height of hillock increased and attained 1.6 nm after 200 nanofretting cycles. After that, the height of hillock revealed a little decrease.

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Figure 7. AFM images of the scars on Si(100) surface after various number of nanofretting cycles N in atmosphere. R = 0.15 μm, Fn =5 μN and D =100 nm.

Table. 1 Calculation of the Hertzian contact pressure of Si(100)/diamond pair in nanofretting. The damage mode was identified based on Figure 6 R (μm) Fn (μN) P (GPa) Damage mode

0.9 30 4.83

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Fn (μN) (b) R=0.9 μm Figure 8. The tangential force Ft versus the normal load Fn curves for two diamond tips against Si(100) in atmosphere. N=100 and D=100 nm.

Clearly, the nanofretting damage of Si(100) surface may successively experience two progresses: hillocks and grooves. The hillock was usually generated under low normal load or in the initial nanofretting cycles. However, the groove was always formed under high normal load or after a large number of nanofretting cycles. Table. 1 shows the Hertzian contact pressures P and damage modes corresponding to various experimental conditions [19]. It indicates that the damage mode may transform from hillock to groove with the increase in contact pressure. The critical contact pressure for the transition of damage mode should be between 12.67 GPa and 15.96 GPa under the given condition.

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In order to ascertain the critical condition between the two damage modes more accurately, the variation of tangential force Ft versus the normal load Fn curves was plotted in Figure 8. For the tip with R=0.15 μm, an inflexion was observed in the Ft−Fn curve from Figure 8a. At the same time, the shape of Ft–d curves also revealed a transition from an inerratic parallelogram to a warped one. It suggests that the inflexion in the Ft−Fn curve may be associated to the transition of damage mode. Since while the tip slipped into groove, the tip would touch the groove wall at the end of slip, which in turn induced an increase in the tangential force. For the tip with R=0.9 μm, since the transition of damage mode did not occur under the given normal loads (2 μN-70 μN), no inflexion was observed in the Ft−Fn curve as shown in Figure 8b. Therefore, we may estimate the transition contact pressure according to the inflexion in Figure 8a. Since the normal load corresponding to the inflexion was between 22 μN and 26 μN, the critical Hertzian contact pressure P could then be estimated as 14.3~15.2 GPa, which was very close to the hardness of Si(100) 13~14 GPa [20-22].

2.4. Comparison of Tangential Nanofretting and Fretting [1] Figure 9 shows the comparison of Ftmax–N curves of NiTi in tangential nanofretting and fretting. As shown in Figure 9a, during nanofretting process, the Ftmax almost kept unchanged with the increase in the number of nanofretting cycles. However, in traditional fretting, the Ftmax usually increased sharply in the initial number of cycles and then kept stable or revealed a decrease (Figure 9b). This difference can be attributed to the different contact modes between nanofretting and fretting. Since the contact radius in fretting is usually in the micrometer scale, the contact should be the multi-asperity contact mode. However, because the contact radius in nanofretting is usually in the nanometer scale, the contact can then be considered as the single-asperity contact.

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Figure 9. The maximum tangential force Ftmax in a cycle vs. the number of cycle N (Ftmax–N) curves in (a) nanofretting of NiTi/diamond pair corresponding to various values of D, Fn=10 mN, R=50 μm, and (b) traditional fretting of NiTi/GCr15 steel ball pair corresponding to various values of D, Fn=100 N, R=20 mm.

Therefore, due to the multi-asperity contact in fretting, the local contact pressure in contact area may be much higher than the nominal value and the surface is easy to be damaged. As a result, the real contact area and the maximum tangential force Ftmax will increase with the increase in N during the initial fretting cycles. However, due to the single-asperity contact in nanofretting, the surface is difficult to be broken and Ftmax may keep constant with the increase in N while the maximum contact pressure is below the yielding stress of material.

2.5. Comparison of Nanofretting in Atmosphere and in Vacuum Some differences were found when nanofretting operated in atmosphere and in vacuum. Firstly, compared to in vacuum, the tangential force is higher and the tips are more difficult to slip in atmosphere. The reason may be explained as the formation of water layer on silicon surface in atmosphere. Qian et al [23] reported that the friction force on silicon would increase by a factor of 2 as the relative humidity increased from 5% to 55% in atmosphere. Since the contact angle of silicon was measured as 39º and the relative humidity in atmosphere was ranged from 50% to 60%, the water layer may form on the silicon surface and induce relatively higher tangential force in atmosphere than in vacuum [24]. Moreover, due to the higher friction in atmosphere, the tip is more difficult to slip than in vacuum. Secondly, the nanofretting damage in vacuum is also different from that in atmosphere. At low load, the height of hillocks in vacuum is a little higher than that in atmosphere as shown in Figure 5. One possible reason may be explained as the “soft coating” effect of oxide layer. Since the elastic modulus of silicon oxide is 72 GPa [25], which is much smaller than

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that of Si(100) (130 GPa) [4]. When the load was too low to destroy the oxidation layer on silicon, the soft oxidation layer can then reduce the contact pressure and prevent the formation of hillock to some extent in the contact area. Due to the absence of the oxygen in vacuum, the oxidation layer formed in nanofretting in vacuum is thinner than that in atmosphere. As a result, the “soft coating” effect of the oxide layer will be stronger in atmosphere and the formed hillocks would be lower than that in vacuum. At high load, the depth of groove generated during the nanofretting in vacuum is smaller than that in atmosphere as shown in Figure 4. This may also be attributed to the “soft coating” effect of oxide layer. Under high load, while the contact pressure was higher than the yielding stress of silicon, the plough would dominate the wear process of nanofretting on Si(100) surface. Compare to the silicon substrate, the soft oxide layer is easier to be ploughed with the increase in nanofretting cycles. Since the oxidation is limited in vacuum due to the absence of oxygen, the grooves generated in atmosphere would be deeper than that in vacuum.

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3. Radial Nanofretting The radial nanofretting tests were made by a nanoindenter at room temperature. As shown in Figure 10, the indenters were diamond tips with varies radii. During a test, the diamond tip was pressed into specimen till a given depth or load and then removed, synchronously the load on, as well as the displacement of, the indenter were recorded. The resolution of the normal load is 1 μN and of the displacement is 0.3 nm. Four peak forces (Fmax), 1 mN, 5 mN, 10 mN and 100 mN, were used in the radial nanofretting tests. At each test, the radial nanofretting was run from 0.1Fmax (for Fmax=1~10 mN) or 1mN (for Fmax=100 mN) to Fmax for different number of cycles. In each nanofretting cycle, the loading rate was adjusted to keep 30 seconds loading period, 2 seconds delay at peak load and 30 seconds unloading period. The topography of the radial nanofretting area was characterized by a scanning electron microscope (SEM) and laser confocal scanning microscopy. All nanofretting experiments were performed under unlubricated condition and a relative humidity of 50%~60%. At each condition, the nanofretting tests were repeated at least two times to ensure the repetition of data.

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Figure 10. Continued on next page.

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50 μm (c) Spherical indenter (R=20 μm) Figure 10. SEM images of the diamond indenters.

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3.1. Radial Nanofretting on Silicon and Copper [11] Figure 11 shows the normal force F vs. displacement d (or F-d) curves in radial nanofretting of a indenter against copper and Si(110), respectively. Here, the indenter is a Berkovich diamond tip, whose radius at the top was estimated by a SEM as about 150 nm. To show the F-d curve clearly, the start point of the curve has been shifted to different values of displacement. Since the displacement amplitudes of the F-d curves were in nano scale (10~200 nm) from the second cycle, the cyclic movements between diamond tip and samples were typical radial nanofretting. Their displacement amplitudes were much smaller than those obtained in the radial fretting of a 52100 steel ball against 1045 steel plate (2~7 µm) [26]. From Figure 11, it was observed that: (i) the residual deformation depth hr in a nanofretting cycle quickly decreased to zero with the increase in the number of nanofretting cycles N; (ii) as hr becomes zero, the F-d curves show hysteresis loops; (iii) the energy dissipation is the highest in the first cycle and sharply decreases to a constant after 20 cycles. To characterize the damage process of copper and silicon in radial nanofretting, Figure 12 shows the SEM images of radial nanofretting damage on two materials after various numbers of nanofretting cycles and under a Fmax of 100 mN. The results indicated that the projected area of indents on two samples attained to constant after an increase in the first several cycles. This was consistent with the results of the F-d curves shown in Figure 11, where the residual depth in a nanofretting cycle quickly decreased to zero after the first several cycles. Under high loads, the main nanofretting damage of two materials was plastic deformation. However, with the increase in the number of nanofretting cycles, the typical nanofretting damage in polycrystalline copper was identified as the pileup of the wrinkles on the edge of indents. The reason may be attributed to the low yield stress of copper. In radial nanofretting, plenty of dislocations initiated in the subsurface of the area around the indent of copper. Some of these dislocations could not propagate on the surface and form the wrinkles around the indents [27]. In addition, the typical nanofretting damage in silicon was characterized as the initiation and propagation of the cracks on the edges of plastic zone of indents. During the radial nanofretting cycle, the silicon may suffer the stress-induced phase transition deformation and elastic recovery. As a result, the cracks may be initiated by the shear stress on the edges of indents [28].

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Figure 12. SEM images of the indents in copper and Si(110) after various numbers of nanofretting cycles.

3.2. Radial Nanofretting on 40Cr Steel and its CrNx Coating [12] Figure 13 shows F-d curves in radial nanofretting of the diamond tip against 40Cr steel and its CrNx coating, respectively. The curvature radius of the diamond indenter is 20 μm. It was found that the F-d curves of two materials are open in the first nanofretting cycle and become closed later on. Under the same experimental condition, the residue indentation depth on CrNx coating (130nm) was far below than on 40Cr steel (160nm). This suggested that due to its higher hardness and elastic modulus, the CrNx film can effectively improve the antipressure ability of 40Cr substrate in radial nanofretting.

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Figure 14 shows the SEM images of the indents on 40Cr and CrNx film after various numbers of nanofretting cycles and under various peak indentation forces. As Fmax=10 mN, no clear damage was found on CrNx film (Figure 14a). As Fmax attained 50 mN, the radial nanofretting damage of CrNx film was characterized as the initiation and propagation of the ring cracks, which might be raised from the radial and hoop tensile stress on the interface of CrNx film and substrate (Figure 14c). However, the radial nanofretting damage of 40Cr was identified as the pileup of the material around indents (Figure 14d). Under the same conditions, the indents on CrNx film were much smaller than those on 40Cr substrate, which further confirmed that the CrNx film revealed better anti-pressure ability than 40Cr substrate in radial nanofretting.

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Figure 14. SEM images of the indents on 40Cr and CrNx film after various N and under various Fmax.

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3.3. Effect of Equivalent Radius of Indenter on Radial Nanofretting [13] Figure 15 revealed the F-d curves in the first nanofretting cycle by three diamond indenters with various radii. Clearly, the loading and unloading curves by 20 μm spherical indenter were repeatable and the deformation of silicon was elastic. Even the curvature radius of Berkovich indenter was much smaller than that of 2 μm spherical indenter, both the maximum indentation depth and the residue depth of the F-d curve obtained by Berkovich indenter were smaller than those obtained by 2 μm spherical indenter. The nanofretting damage of Si(100) surface by different indenters were shown in Figure 16. Similarly, even the Berkovich indenter was sharper than 2 μm spherical indenter, the nanofretting damage of Si(100) surface induced by Berkovich indenter was weaker than that induced by 2 μm spherical indenter. Figure 17 shows the contact modes of the indenter on samples. While the contact depth hc is smaller than the critical depth h0, the contact can be considered as sphere-plane contact. However, while hc ≥ h0, the contact should be translated to cone-plane contact. As shown in Figure 17b, we may define an equivalent radius of indenter Re, which is the radius of the maximum inscribed sphere of the cone. Since the equivalent radius Re of Berkovich indenter is larger than that of 2 μm spherical indenter under the load of 50 mN, the nanofretting damage induced by Berkovich indenter is weaker than that induced by 2 μm spherical indenter.

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4. Conclusions The tangential nanofretting of monocrystalline Si(100) surface against spherical SiO2 and diamond tips was carried out by an atomic force microscopy. The radial nanofretting behaviors of several typical structural materials in MEMS (polycrystalline copper, monocrystalline silicon and CNx films) were investigated by a nanoindenter. The main conclusions can be summarized as following: (1) The tangential nanofretting could be divided into stick regime and slip regime upon the transition criterion. The adhesion force may induce the increase in the maximum static friction force and prevent the contact pair from slipping. (2) The nanofretting damage on silicon may experience two processes, namely as the generation of hillocks at low load and the formation of grooves at high load. Compared to those in atmosphere, the nanofretting scars in vacuum exhibited higher hillock at low load but shallower groove at high load, which could be explained as the “soft coating” effect of oxide layer on Si(100) surface. (3) The radial nanofretting damage in copper was mainly identified as the pileup of the wrinkles around indents, and in silicon was characterized as the initiation and propagation of the cracks on the edges of plastic zone. Due to its higher hardness and elastic modulus, the CrNx film can effectively improve the anti-pressure ability of 40Cr substrate in radial nanofretting. (4) Compared to the curvature radius, the equivalent radius of indenter played a much more important role on the nanofretting damage of material.

Acknowledgments

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The authors are grateful for the financial support from the Natural Science Foundation of China (50625515, 50821063).

References [1] Qian, L. M., Zhou, Z. R., Sun, Q. P., Yan, W. Y.: Nanofretting behaviors of NiTi shape memory alloy. Wear 263, 501-507 (2007). [2] Tanaka, M.: An industrial and applied review of new MEMS devices features. Microelectron. Eng. 84, 1341-1344 (2007). [3] Ko, W. H.: Trends and frontiers of MEMS. Sensor. Actuat. A 136, 62-67 (2007) . [4] Bhushan, B.: Modern Tribology Handbook, Volume One (CRC Press LLC, Florida, USA) (2001). [5] Williams, J. A., Le, H. R.: Tribology and MEMS. J. Phys. D 39, R201-R214 (2006). [6] Kaneko, R., Umemura, S., Hirana, M., Andoh, Y., Miyamoto, T., Fukui, S.: Recent progress in microtribology. Wear 200, 296-304 (1996). [7] Zhou, Z. R., Qian, L. M.: Tribological size effect and related problems. Chinese Journal of Mechanical Engineering 39(8), 22-26 (2003). [8] Varenberg, M., Etsion, I., Halperin, G.: Nanoscale fretting wear study by scanning probe microscopy. Tribol. Lett. 18, 493-498 (2005).

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[9] Yu, J. X., Qian, L. M., Yu, B. J., Zhou, Z. R.: Nanofretting behavior of monocrystalline silicon (100) against SiO2 microsphere in vacuum. Tribol. Lett. 34, 31-40 (2009). [10] Yu, J. X., Qian, L. M., Yu, B. J., Zhou, Z. R.: Nanofretting behaviors of monocrystalline silicon (100) against spherical diamond tips in atmosphere and vacuum. Wear, in press (2009). [11] Zhang, J. Y., Qian, L. M., Zhou, Z. R.: Radial nanofretting of single crystal copper and silicon under high load. Tribology 26(1), 1-6 (2006) . [12] Zhang, S., Qian, L. M., Mo, J. L.: Radial nanofretting behavior of chromium nitride (CrNx) Film. Tribology 28(4), 316-321 (2008). [13] Zhang, Z. W., Qian, L. M.: Effect of equivalent radius of indenter on the radial nanofretting damage of monocrystal Silicon. Chinese Journal of Mechanical Engineering, in pressing (2009). [14] Ramalho, A., Merstallinger, A., Cavaleiro, A.: Fretting behaviour of W–Si coated steels in vacuum environments. Wear 261, 79-85 (2006) . [15] Qian, L. M., Sun, Q. P., Zhou, Z. R.: Fretting wear behavior of superelastic nickel titanium shape memory alloy. Tribol. Lett. 18 (4), 463-475 (2005). [16] Fouvry, S., Kapsa, Ph., Vincent, L.: Analysis of sliding behavior for fretting loadings: determination of transition criteria. Wear 185, 35-46 (1995). [17] Bhushan, B.: Nano- to microscale wear and mechanical characterization using scanning probe microscopy. Wear 251, 1105–1123 (2001). [18] Bhushan, B., Kulkarni, A.V., Bonin, W., Wyrobek, J.T.: Nanoindentation and picoindentation measurements using a capacitive transducer system in atomic force microscopy. Philos. Mag. 74, 1117-1128 (1996). [19] Johnson, K. L.: Contact Mechanics (Cambridge University Press, Cambridge UK) (1985). [20] Qian, L. M., Li, M., Zhou, Z. R., Yang, H., Shi, X. Y.: Comparison of nano-indentation hardness and micro hardness. Surf. & Coat. Tech. 195(2-3), 264-271 (2005). [21] Jang, J., Lance, M. J., Wen, S. Q., Tsui, T. Y., Pharr, G. M.: Indentation-induced phase transformations in silicon: influences of load, rate and indenter angle on the transformation behavior. Acta. Mater. 53, 1759-1770 (2005). [22] Weppelmann, E. R., Field, J. S., Swain, M. V.: Observation, analysis, and simulation of the hysteresis of silicon using ultra-micro-indentation with spherical indenters. J. Mater. Res. 8, 830-840 (1993). [23] Qian, L. M., Tang, F., Xiao, X. D.: Tribological properties of self-assembled monolayers and their substrates under various humid environments. Tribol. Lett. 15, 169-176 (2003). [24] Yu, B. J., Qian, L. M., Yu, J. X., Zhou, Z. R.: Effects of tail group and chain length on the tribological behaviors of self-assembled dual-layer films in atmosphere and in vacuum. Tribol. Lett. 34, 1-10 (2009). [25] International Standard. Instrumented indentation test for hardness and materials parameters. ISO 14577-1: 2002. [26] Zhu. M. H., Zhou. Z. R.: An experimental study on radial fretting behavior. Tribol. Int 34, 321-326 (2001). [27] Knap J., Ortiz M.: Effect of indenter 2radius on Au (001) nanoindentation. Phys. Rev. Lett. 90(22), 226102 – 226106 (2003). [28] Hill, M. J., Rowcliffe, D. J.: Deformation of silicon at low temperatures. J. Mater. Sci. 9, 1569–1576 (1974).

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In: Micro Electro Mechanical Systems Editors: B. Ekwall and M. Cronquist, pp. 259-273

ISBN: 978-1-60876-474-7 © 2010 Nova Science Publishers, Inc.

Chapter 8

ADAPTIVE POISSON MODELING OF MEDICATION ADHERENCE AMONG HIV-POSITIVE METHADONE PATIENTS PROVIDED GREATER UNDERSTANDING OF BEHAVIOR Kevin Delucchi1, George Knafl2, Nancy Haug1 and James Sorensen1 1

2

University of California, San Francisco, CA, USA University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

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Abstract Objective The goal of this study was to conduct an extensive analysis of medication adherence data from a 20-week randomized clinical trial designed to test the effects of using vouchers on improving medication adherence among HIV-positive methadone maintenance patients extending a method proposed by Knafl et al. [1]. Study Design and Setting The methodology was applied and extended not only to data for the randomized subjects but also for subjects who were initially enrolled but demonstrated sufficient adherence during the baseline period. Predicted mean adherence patterns over time from Poisson models were clustered via an extensive search among clustering methods and those results were compared to a solution based on growth mixture modeling. Results Clustering generated high and moderately high adherence groups as well as improving and deteriorating adherence groups while growth mixture modeling clustered subjects into uniformly high, moderate, and low adherence groups. Conclusions The results for both approaches reflected uniform levels of adherence over time with clustering also identifying changing levels of adherence over time. Results replicated the primary finding that the use of vouchers improved adherence rates.

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Introduction Over the last several years a variety of effective methods have been developed to improve medication adherence, each with advantages and disadvantages [2]. Such methods include contingency management such as the use of vouchers [3,4], adherence counsellors [5,6], and reminder packaging [7]. But one of the main challenges in studies using these methods is the measurement of the level of adherence—mainly because the behavior of interest, self-administration of the medication, is not directly observed (except, of course, in directly-observed therapy studies). Equally challenging is making sense of measured adherence data. Good measurement and analysis is especially important in the treatment for HIV as research has demonstrated that antiretroviral therapy is effective if medication is taken as prescribed [8,9]. Of special concern are HIV-positive drug users who have difficulty adhering to the complex highly active antiretroviral therapy regimens now used [8-12]. One method of measurement, using an electronic monitoring device (EMD) like the Medication Event Monitoring System (MEMS), has been used in this population of drug users. EMDs produce accurate rates of medication container opening (presumed to correlate with medication taking) but also generate a wealth of complex data. For example, the data used in this study contained over 15,000 device openings. But it is not obvious how best to analyze such a large set of adherence data, especially when collected on a limited number of patients, a common feature of studies of HIV-positive drug abusers, who may be divided into groups such as experimental and control conditions in a study. More data does not necessarily mean more useful information and pulling the signal out of the noise becomes more difficult. Knafl et al. [1] have proposed methods for analyzing adherence data from EMDs based on adaptive Poisson regression modeling. The objective of these methods is to capitalize on the large amount of data provided by EMDs while avoiding the problem of over-fitting the model to the data. The aim is to be able to accurately describe adherence levels, summarize those levels in a meaningful way, and to allow one to test for differences in adherence among groups. What is not known is the extent to which this computer-intensive methodology can be applied to data from an intervention trial among HIV-positive drug abusers. So in this study, we examined the use of this new methodology in an extensive analysis of data from a 20-week randomized clinical trial designed to test the effects of vouchers on improving medication adherence among HIV-positive methadone maintenance patients. A unique feature of the original study allowed us to include data from an adherent group not randomized to treatment condition. The goals of this work were to expand the method of analysis, compare the results from two methods for clustering generated mean adherence patterns, and contrast findings to prior results based on summary measures.

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Methods

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Health Incentives Project The Health Incentive Project was a randomized clinical trial testing the effect of incentives in the form of vouchers on improving antiretroviral medication adherence in HIVpositive drug users on methadone maintenance. Full details on the study can by found in Sorensen et al. [13]. Briefly, the study was a two-arm randomized clinical trial comparing the use of a contingency management intervention designed to improve medication adherence among HIV-positive methadone maintenance patients. The design encompassed three phases with three study groups in the first phase and two groups in subsequent phases. The first phase consisted of a 4-week baseline period. Subjects who demonstrated sufficient levels of medication adherence at the end of the four weeks were not randomized and not measured further. Eligible subjects (n=66), i.e., those with low levels of baseline adherence, were randomly assigned to: (a) medication coaching sessions every other week to assist with adherence strategies (control group) or (b) medication coaching plus voucher reinforcement for opening electronic medication devices on time (intervention group). To measure medication adherence, subjects were given pill bottles equipped with EMDs, which recorded dates and times for openings. Each opening is presumed to correspond to medication taking. The pill bottle contained the subject’s primary HIV medication. Data were collected over a 4-week baseline phase, as noted, plus a 12-week intervention phase and a 4week follow-up phase. All study subjects should have opened the bottle twice a day and done so within a 2-hour window. Ideally, a total of 84 days with 168 openings should have been observed for each person during the intervention phase. For the published analysis [13], medication adherence was measured as the percentage of scheduled opening times when the device was opened on time. The mean percentage was then compared between randomized groups within each of the three study phases. That analysis indicated that the use of vouchers resulted in greater levels of medication adherence. The concern which led to this work was that such an approach does not evaluate consistency in adherence over time and may discard useful information on changes in adherence over time. So, for example, while two patients may both take 60% of their medications on time, it would be informative to know that the one in the intervention group did so on an increasingly more consistent basis improving with the length of time exposed to the intervention, as opposed, say, to the one in the control group with adherence deteriorating over time.

Overview of the Modeling Process The goal of the analysis was to characterize mean adherence over time using the analysis process of Knafl et al. [14]. This was achieved by first estimating each subject's mean adherence pattern as represented by estimated mean EMD opening rates over time. Each of these patterns was then assessed for how consistent it was with adherence at the prescribed rate. Next, subjects were clustered into groups with similar mean adherence patterns to characterize the complete set of such patterns. Finally, membership in identified clusters was

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related to study group membership. These methods can be used to characterize mean adherence for the complete study participation period or within phases of study participation. In this study, individual-subject mean adherence patterns (i.e., estimated mean adherence curves) were identified by first reducing the raw EMD data to counts/rates for opening events. Then, an adaptive modeling process was applied to the reduced data to identify Poisson regression models based on fractional polynomials in time within study participation (i.e., the powers for time can be fractions). Overall adherence was assessed using percent consistency of observed adherence with presumed adherence at the prescribed rate. Next, individualsubject mean adherence patterns were clustered into mean adherence pattern types consisting of subjects with similar patterns. Finally, membership in the mean adherence types was related to study group membership to assess the impact of the intervention on adherence. Computations were conducted in SAS, using specialized macros written primarily in the matrix language PROC IML [14], or in M-Plus.

Data Reduction

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The raw EMD data consisted of over 15,000 openings. These were grouped over each subject's EMD usage period into more than 3,300 counts/rates using the following data reduction process: 1) partition time in the study for a subject into equal-size (Δt) intervals, 2) count events (ΔN) in these intervals, 3) compute event rates (r=ΔN/Δt) for these intervals. A total of 100 intervals was used for a subject if the associated length Δt was at least 3 days long, so that each interval would equal 1% of the usage period for a subject. Otherwise, intervals of just over 3 days each were used to avoid small-sized intervals. Interval sizes were also adjusted when subjects used multiple devices. The resulting counts/rates can be analyzed for one subject at a time or combined together for any group of subjects, such as those in the intervention group, for the full length of the study or during different study phases.

Data Modeling Poisson regression was used since the data consisted of counts/rates. Models of event counts ΔN for time intervals of length Δt in terms of cumulative times t by the end of those intervals were estimated using the natural log link function with offset variable log(Δt) so that they are equivalent to modeling event rates r. ΔN|t was treated as Poisson distributed with log expected value a function of a vector x(t) of multiple transforms of time t with associated parameter vector β, i.e., log(E(ΔN|t))=βT⋅x(t)+log(Δt) so that log(E(r|t))=βT⋅x(t). Covariates can also be included in the predictor vector x when modeling data for multiple subjects.

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Model Evaluation Models were evaluated and compared using k-fold likelihood cross-validation (LCV). Observations (indexed by i) were assigned randomly to folds F with equal assignment probabilities 1/k. Likelihoods L were computed using deleted parameter estimates β(Fc), i.e., parameters used in likelihood terms for data in fold F were estimated using data in the complement Fc of the fold. Parameters were estimated using maximum likelihood. Formally, LCV scores for individual-subject data at observed times ti, 1≤i≤m, satisfy LCV=ΠFΠi∈FL(ΔNi,xi,Δti,ti;β(Fc))1/m. Deleted likelihood terms are transformed by taking mth roots so that LCV scores are geometric average deleted likelihoods. Larger LCV scores indicate better models, more compatible with the data. Scores for multiple-subject data are defined similarly. The number k of folds was set to 10, which works well for individual-subject data with m≤100, but k may need to be adjusted for larger data sets.

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Model Selection The model with the best overall LCV score need not be the best choice; it might include extraneous terms providing negligible improvements. A possible alternative approach is to allow a limited reduction in the LCV score to obtain a more parsimonious model. Considerations like this form the basis for the rules controlling the heuristics for searching through alternative models described in Knafl et al. [1]. The search proceeds through two phases, an expansion starting from a base model (often the constant model with only an intercept) followed by a contraction. The model produced by the expansion is the base model of the contraction. All terms of the model including the intercept are subject to removal in the contraction. The selected model is the one generated by the contraction. This process is similar to standard variable selection procedures applied to a fixed set of transforms. The expansion is analogous to forward selection while the contraction resembles backward elimination. However, each expansion step considers any possible power transform to add to the model, rather than just those in a fixed set. Moreover, remaining transforms have their powers adjusted with each removal during contraction. Rules based on percent changes in LCV scores are used to guide the model selection process, since exhaustive search is impractical. Tolerance parameters control the rules, e.g., the expansion and contraction phases stop when percent changes in LCV scores first exceed specific tolerances. Recommended settings for these parameters are provided in Knafl et al. [1] for analyzing individual-subject adherence data that typically produce models with highly significant coefficients, usually selecting parsimonious models, i.e., with all selected predictor variables providing substantive predictive benefits (as measured by substantive decreases in LCV scores with those variables removed from the model).

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Overall Adherence Assessment The percent of prescribed doses taken (PDT) is commonly used as a summary measure for electronically monitored adherence [15], assuming openings correspond to dose taking, and was used in the published analysis of these data. Since percent PDT is computed from the average number of openings per unit time, it is based on the implicit assumption that the adherence pattern is constant over time, an assumption which often does not hold. To account for the more general situation, consistency between prescribed and observed adherence was measured by taking the ratio of the LCV score for the model corresponding to prescribed adherence (in this case, 2 medications per day for all subjects) divided by the LCV score for the model selected using a subject's count/rate data, converted to a percentage. For small sample sizes, the LCV score is based on even smaller subsets of the data and so can be unreliable. Consequently, for sample sizes less than 5 we compute percent consistency scores using likelihoods rather than with LCV scores.

Intervention Phase Mean Adherence Clusters

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Adaptively selected Poisson regression models were used to generate estimates of individual-subject mean adherence at 10 equally spaced proportions (10%, 20%, ···, 100%) of time during the intervention phase. The analysis was restricted to the intervention phase since that was judged to be of primary importance for assessing subject adherence. Ten estimates were used rather than a larger number since usage periods were relatively small (at most 12 weeks). These estimates were clustered into mean adherence pattern types in two ways. First, an exhaustive search was conducted over a wide variety of clustering procedures, each with from 1 to 10 clusters, to identify the choice with the best LCV score under a multivariate normal mixture model. Covariance matrices Σc for clusters indexed by c were allowed to have different variances but common correlations. In other words,

Σc=Vc½ ⋅R⋅Vc½ where Vc is the diagonal matrix of variances for cluster c and R is the unstructured correlation matrix common to all clusters. Computations were conducted in SAS. Second, growth mixture modeling [16] was used to detect latent adherence patterns, computed in M-Plus. Both of these approaches can be applied to cluster mean adherence patterns over any phase of the study or over all phases combined together.

Results Individual-Subject Overall Mean Adherence Patterns A variety of individual-subject overall mean adherence patterns were identified. Figures 1-4 provide a selection of these including a high mean adherence pattern (Figure 1) very close to the prescribed rate with 99.9% consistency with the prescribed rate, an improving mean adherence pattern (Figure 2) with 87.7% consistency, a concave mean adherence pattern with

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moderate adherence (Figure 3) at 58.1% consistency, and a low mean adherence pattern (Figure 4) with 2.9% consistency. Individual-subject mean adherence patterns were also generated for each of the three phases of the study. Figure 5 provides an example for the improving overall mean adherence pattern of Figure 2 broken down into phases. This subject had moderate mean adherence at 50.6% consistency during the baseline phase which improved substantially to high mean adherence of 99.1% consistency in the intervention phase and dropped a little from that to 96.6% consistency during the follow-up phase. This example demonstrates the importance of considering adherence within phases for such a design as this rather than just globally for all phases combined. This subject's overall mean adherence had only a moderately high consistency score due to having moderate baseline mean adherence, but his/her post-baseline mean adherence was substantially higher. This subject was in the intervention group, and so the improvement in mean adherence from the baseline phase reflects the beneficial impact of the intervention for this subject.

Summary Adherence Measures: Percent Consistency versus Percent Prescribed Doses Taken (PDT)

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We found that subjects were more likely to have had a high level (>90%) of percent consistency over all phases combined than a high level (also >90%) of percent PDT. All subjects with high percent PDT also had high percent consistency. The chance of high percent consistency was 1.6 times that of high percent PDT (39% vs. 24%). Hence, percent consistency provides a more extensive identification of highly adherent subjects than percent PDT. Poorly adherent subjects also were identified more extensively, for example, 18% of the subjects had 30 percent consistency or less compared to only 5% of the subjects with 30 percent PDT or less.

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Association of Summary Adherence Measures with Study Group The chance of high percent consistency over all phases combined was greater for the intervention group than for the control group and closer to that of the non-randomized group (Figure 6). The combined intervention and non-randomized groups had a significantly (χ2 p=0.045, n=83) greater chance of high percent consistency than the control group: 47% (24/51) versus 25% (8/32). The chance of high percent PDT over all phases combined was about the same for the intervention and control groups and less than that of the nonrandomized group (Figure 7). The combined intervention and control groups had a significantly (χ2 p=0.013, n=83) smaller chance of high percent PDT than the nonrandomized group: 18% (12/66) versus 47% (8/17). Thus, percent consistency can identify intervention adherence effects not identifiable with percent PDT. non-randomized

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High Percent Consistency (>90%) by Treatment Group… Figure 6. Association of Percent Consistency with Treatment Condition. non-randomized

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Figure 7. Association of Percent Prescribed Doses Taken (PDT) with Treatment Condition.

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Intervention Phase Adherence Within the intervention phase, the chance of having high percent consistency was significantly (χ2 p=0.013, n=64) greater for the intervention group than for the control group: 67% (22/33) versus 35% (11/31). The chance of high percent PDT was also significantly (χ2 p=0.014, n=64) greater for the intervention group than for the control group: 48% (16/33) versus 19% (6/31). Hence, percent consistency and percent PDT agree for the assessment of intervention effects on adherence during the intervention phase. These intervention phase differences did not persist into the follow-up period using both summary measures (χ2 p>0.14, n=59).

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Intervention Phase Mean Adherence Pattern Types Given the identification of each subject's mean adherence pattern, we turned next to the identification of common, underlying mean adherence pattern types. We restricted this analysis to patterns within the intervention phase. Not only did we consider this phase to be of primary importance, we also wanted to avoid negative baseline adherence effects (e.g., compare Figures 2 and 5). This was addressed by modeling mean adherence patterns in two alternate ways: using clustering methods and using growth mixture modeling. Mean adherence patterns were represented by vectors of estimates of mean adherence at 10 equally spaced proportional times within the intervention phase (i.e., 10%, 20%, ···, 100%). The clustering of mean adherence patterns was addressed by fitting multiple models looking for the best, robust fit. A wide variety of procedures supported by SAS PROC CLUSTER and PROC FASTCLUS [17] were considered. First, 1- and 2-stage nearest neighbor procedures based on from 2 to 7 nearest neighbors were estimated and then the average, centroid, complete, EML, flexible, k-means, McQuitty, median, single, and Ward procedures. The average, centroid, median, and Ward procedures were considered with distance measured both by Euclidean distance and the default of squared Euclidean distance. Altogether, a total of 26 procedures were considered. The 2-stage nearest neighbor approach using 4 nearest neighbors generated the best overall LCV score (computed as described above) under the 4-cluster solution, including high mean adherence (n=20; 31.3%), moderately high mean adherence (n=13; 20.3%), improving mean adherence (n=8; 12.5%), and deteriorating mean adherence (n=23; 35.9%). These mean adherence pattern types are plotted in Figure 8. Membership in the mean adherence pattern types was significantly related to study group membership (p=0.041, using Fisher's exact test since there were cells with low expected counts). The relationship is displayed in Figure 9. Compared to the control group, the intervention group was more likely to have high adherence (42% vs. 19%) and moderately high adherence (24% vs. 13%) and less likely to have deteriorating adherence (27% vs. 45%) and improving adherence (6% vs. 23%). These relationships demonstrate the validity of the adaptive clustering procedure and also provide further support of the effectiveness of the intervention.

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2.5

Openings per Day

2

1.5

1

0.5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportional Time within Study Participation high

moderately high

improving

deteriorating

Figure 8. Mean Adherence Pattern Types.

50% 45%

45% 42%

40% 35%

27%

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30% 25% 20%

24%

23%

19%

intervention control

13%

15% 10%

6%

5% 0% high

moderately high

deteriorating

improving

mean adherence pattern type

Figure 9. Adaptive Cluster Membership versus Group Membership.

For the growth mixture approach we fit models using 2-4 latent classes. As this was not a very large sample, we did not consider larger numbers of classes but just compared growth mixture results to clustering results. Growth mixture results pointed towards a 3-class model; high adherence (n=33; 51.6%) averaging at about 1.8 meds per day, a little below the prescribed rate of 2 meds per day, intermediate adherence (n=27; 42.2%) averaging close to 1.5 meds per day and low adherence (n=4; 6.3%) averaging below 1 med per day. When we

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cross-classified cluster membership with latent class membership, we found very good agreement. The highest adhering class consisted of members from the two highest adhering clusters, the intermediate class contained members from the two lower adhering clusters and the small lowest adhering class took all four of its members from the lowest deteriorating adhering cluster. This consistency between results for the growth mixture and clustering approaches provides validation of the clustering results.

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Conclusion The flood of data from electronic sources such as EMDs is growing and is bound to accelerate as technology advances. Studies using such methods as well as other methods like electronic diaries[18], ecological momentary assessments [19], and the timeline follow back [20]—not to mention gene-expression and imaging data—are becoming more common, and so methods of analysis for such data need to match their sophistication. The concern is that simpler methods of summarizing such data, while easier to understand and use, may not fully describe study results. In this study, we have demonstrated the analysis of individual-subject EMD adherence data using adaptive Poisson regression methods and compared the summary adherence measures percent consistency and percent prescribed doses taken (PDT). Percent consistency can identify group effects not identifiable with percent PDT. Percent consistency also provided a more extensive identification of highly adherent and poorly adherent subjects than percent PDT. Furthermore, modeling adherence over individual study phases can provide insights into adherence not identifiable by modeling it globally over the full course of a study. Extending these computer-intensive methods, we also demonstrated clustering of individualsubject mean adherence patterns using standard clustering methods selected through LCV as well as with growth mixture modeling. We found that mean adherence during the intervention phase can be at uniform levels over time as well as improving or deteriorating over time. Taken together, these results both confirm the findings in the original analysis demonstrating a positive effect for vouchers as a means of improving adherence in this population as well as providing greater detail about that effect. In closing, we note that some aspects of this particular clinical trial may be unique so that the results might not generalize to other studies, although similar results would be expected in most cases. Another concern is that this methodology is not as easily explained as less complex approaches to analyzing EMD data. But the findings are understandable and good science requires that the most powerful methods be used to analyze data. Also, further work is needed to more fully characterize adherence for these subjects. Our analyses only characterized mean adherence while assuming unit dispersion as is inherent in Poisson regression modeling. Analyses of these data are needed that address variability in adherence in terms of non-unit dispersion as well as mean adherence [14]. Furthermore, adaptive Poisson regression treats opening counts/rates as independent (a property of Poisson processes), and so extensions are needed to account for possible temporal correlation in opening counts/rates. It also only addresses the frequency domain for device opening processes. Analogous methods are needed to model times between openings to characterize regularity in the timing of adherence.

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Acknowledgments This work was supported in part by Award Number P50 DA009253 from the National Institute of Drug Abuse (NIDA), Award Number R01 AI057043 from the National Institute of Allergy and Infectious Diseases (NIAID), and Award Number R03 MH086132 from the National Institute of Mental Health (NIMH). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA, the NIAID, the NIMH, or the National Institutes of Health. Drs. Delucchi and Knafl both developed and executed the analyses presented here. Dr. Sorensen was the principal investigator for the parent study. He designed that study and directed its implementation, including quality assurance and control. Dr. Haug supervised the field activities and data collection. Parts of this work were presented at the meeting of the College on Problems of Drug Dependence, Scottsdale, AZ, June, 2006 and at the Joint Statistical Meeting, Seattle, WA, August, 2006.

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References [1] Knafl, G. J., Fennie, K. P., Bova, C., Dieckhaus, K., & Williams, A. B. (2004). Electronic monitoring device event modelling on an individual-subject basis using adaptive Poisson regression. Statistics in Medicine, 23, 783-801. [2] Osterberg, L., & Blaschke, T. Drug therapy - Adherence to medication. New England Journal of Medicine, 353, 487-497. [3] Rosen, M. I., McMahon, T., Valdes, B., Petry, N. M., Cramer, J., & Rounsaville, B. (2007). Improved adherence with contingency management. AIDS Patient Care and STDs, 21, 30-40. [4] Carroll, R. B. (2007). A perfect platform: Combining contingency management with medications for drug abuse. The American Journal of Drug and Alcohol Abuse, 33, 343-365. [5] Scheid, T. L. (2007). Specialized adherence counselors can improve treatment adherence: Guidelines for specific treatment issues. Journal of HIV/AIDS & Social Services, 6, 121-138. [6] Cooperman, N. A., Chabon, B., Berg, K. M, & Arnsten, J. H. (2007). The development and feasibility of an intervention to improve HAART adherence among HIV-positive patients receiving primary care in methadone clinics. Journal of HIV/AIDS & Social Services, 6, 101-120. [7] Heneghan, C. J., Glasziou, P. P., & Perera, R. (2005). Reminder packaging for improving adherence to self-administered long-term medications. Cochrane Database of Systematic Review, 2, DOI 10.1002/14651858.CD005025.pub2. [8] Low-Beer, S., Yip, B., O'Shaughnessy, M. V., Hogg, R. S., & Montaner, J. S. G. (2000). Adherence to triple therapy and viral load response. Journal of Acquired Immune Deficiency Syndromes, 23, 360-361. [9] Paterson, D. L., Swindells, S., Mohr, J., Brester, M., Vergis, E. N., Squier, C., Wagener, M. M., & Singh, N. (2000). Adherence to protease inhibitor therapy and outcomes in patients with HIV infection. Annals of Internal Medicine, 133, 21-30.

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[10] Batki, S. L., Ferrando, S. J., Manfredi, L., London, J., Pattillo, J., & Delucchi, K. (1996). Psychiatric disorders, drug use, and medical status in injection drug users with HIV disease. American Journal on Addictions, 5, 249-258. [11] Freeman, R. C., Rodriguez, G. M., & French, J. F. (1996). Compliance with AZT treatment regimen of HIV-seropositive injection drug users: A neglected issue. Aids Education and Prevention, 8, 58-71. [12] Fogarty, L., Roter, D., Larson, S., Burke, J., Gillespie, J., & Levy R. (2002). Patient adherence to HIV medication regimens: A review of published and abstract reports. Patient Education and Counseling, 46, 93-108. [13] Sorensen, J. L., Haug, N. A., Delucchi, K. L., Gruber, V., Kletter, E., Batki, S. L., Tulsky, J. P., Barnett, P., & Hall, S. (2006). Voucher reinforcement improves medication adherence in HIV-positive methadone patients: A randomized trial. Drug and Alcohol Dependence, 88, 54-63. [14] Knafl, G. J., Delucchi, K. L., Bova, C. A., Fennie, K. P., & Williams, A. B. (2010). A systematic approach for analyzing electronically monitored adherence data. In B. Ekwall & M. Cronquist (Eds.), Micro Electro Mechanical Systems (MEMS) technology, fabrication processes and applications, Chapter 1, p. 1-66 Hauppauge, NY: Nova Science Publishers. [15] Sereika, S. M., & Dunbar-Jacobs, J. (2001). Analysis of electronic event monitored adherence. In L. E. Burke & I. S. Ockene (Eds.), Compliance in healthcare and research (pp. 139-162). Armonk, NY: Futura Publishing. [16] Muthén, B., & Shedden. K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463-469. [17] SAS Institute. (2004). SAS/STAT 9.1 user's guide. Cary, NC: SAS Institute. [18] Burton, C., Weller, D., & Sharpe M. (2007). Are electronic diaries useful for symptoms research? A systematic review. Journal Of Psychosomatic Research, 62, 553-561. [19] Moskowitz, D. S., & Young, S. N. (2006). Ecological momentary assessment: What it is and why it is a method of the future in clinical psychopharmacology. Journal of Psychiatry and Neuroscience, 31, 13-20. [20] Searles, J. S., Helzer, J. E., Rose, G. L., & Badger, G. J. (2002). Concurrent and retrospective reports of alcohol consumption across 30, 90 and 366 days: Interactive voice response compared with the timeline follow back. Journal of Studies on Alcohol, 63, 352-362.

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

In: Micro Electro Mechanical Systems Editors: B. Ekwall and M. Cronquist, pp. 275-291

ISBN: 978-1-60876-474-7 © 2010 Nova Science Publishers, Inc.

Chapter 9

ROBUST ADAPTIVE CONTROL FOR MEMS VIBRATORY GYROSCOPE J. Fei Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology, and College of Computer and Information, Hohai University, Changzhou, China

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Abstract This chapter presents an adaptive sliding mode controller for a MEMS vibratory z-axis gyroscope. The proposed adaptive sliding mode controller can real-time estimate the angular velocity and the damping and stiffness coefficients. The stability of the closed-loop system can be guaranteed with the proposed adaptive sliding mode control strategy. The numerical simulation for MEMS gyroscope is investigated to show the effectiveness of the proposed control scheme. It is shown that the proposed adaptive sliding mode control scheme offers several advantages such as real-time estimation of gyroscope parameters and large robustness to parameter variations and external disturbance.

1. Introduction Gyroscopes are commonly used sensors for measuring angular velocity in many areas of applications such as navigation, homing, and control stabilization. Vibratory gyroscopes are the devices that transfer energy from one axis to the other through Coriolis forces. Fabrication imperfections result in some cross stiffness and cross damping effects that may hinder the measurement of angular velocity of MEMS gyroscope. Other noise sources such as time varying system parameters, thermal, mechanical noise and sensing circuitry noise also affect the performance. The angular velocity measurement and minimization of the cross coupling between two axes are challenging problems in vibrating gyroscopes. Based on the sliding mode concepts, this chapter investigates a novel adaptive sliding mode controller, which can real-time estimate the angular velocity and the damping and stiffness coefficients of the gyroscope. The contribution of this chapter is that a sliding mode control algorithm is

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276

J. Fei

incorporated into the adaptive control system and the feasibility of adaptive sliding mode control with sliding mode control in the presence of the model uncertainties and external disturbance is investigated. The angular velocity and all gyroscope parameters including coupling stiffness parameter are determined. Sliding mode control is a robust control technique which has many attractive features such as robustness to parameter variations and insensitivity to disturbance. The sliding mode controller is composed of an equivalent control part that describes the behavior of the system when the trajectories stay over the sliding manifold and a variable structure control part that enforces the trajectories to reach the sliding manifold and prevent them leaving the sliding manifold. Sliding mode control is insensitive to system uncertainties and disturbances when they are within expected limits. But it also has some limitation such as chattering or high frequency oscillation in practical applications. Adaptive control is an effective approach to deal with parameter variations. Adaptive sliding mode control has the advantages of combining the robustness of variable structure methods with the tracking capability of adaptive control strategies. Utkin [1] introduced the variable structure system and sliding mode control and showed that variable structure control is insensitive to parameters perturbations and external disturbances. Narendra [2], Astrom [3] proposed the model reference adaptive control. In the last few years, many applications have been developed using sliding mode control and adaptive control. Lee [4] developed a variable structure augmented adaptive controller for a gyro platform. Wang [5] proposed an adaptive sliding mode controller for a microgravity isolation system. Song [6] developed a smooth robust compensator. Sam [7] presented a class of proportional and integral sliding mode control with application to active suspension system. Lin [8] and Chou [9] proposed an integral sliding surface and derived an adaptive law to estimate the upbound of uncertainties. Some control algorithms have been proposed to control the MEMS gyroscope. Batur [10] developed a sliding mode control for MEMS gyroscope system. Leland [11] presented an adaptive controller for tuning the natural frequency of the drive axis of a vibrational gyroscope. An adaptive controller for a MEMS gyroscope is reported in [12] [13], which drived both axes of vibration and controls the entire operation of the gyroscope. Our controller is different from [12] in that a sliding mode control algorithm is incorporated into the adaptive control system. The motivation of this chapter is to propose an adaptive sliding mode controller to estimate the angular velocity and all gyroscope parameters including cross stiffness and damping coefficients in the presence of the model uncertainties and external disturbance. A smooth sliding mode compensator is used to reduce control chattering. An adaptive law to update the parameters of adaptive sliding mode controller is derived. The chapter is organized as follows. First, dynamics of MEMS gyroscope is described. Second, an adaptive sliding mode control scheme is developed and its stability is proved. Then, the simulation results of MEMS gyroscope model are discussed.

2. Dynamics of MEMS Gyroscope A z-axis MEMS gyroscope is depicted in Figure 1. A typical MEMS vibratory gyroscope includes a proof mass suspended by springs, an electrostatic actuation and sensing

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Robust Adaptive Control for MEMS Vibratory Gyroscope

277

mechanisms for forcing an oscillatory motion and sensing the position and velocity of the proof mass. We assume that the table where the proof mass is mounted with a constant velocity; the gyroscope is rotating at a constant angular velocity Ω z over a sufficiently long time interval; 2

2

the centrifugal forces mΩ z x , mΩ z y are assumed to be negligible; gyroscope undergoes rotation about the z axis only, and thereby Coriolis force is generated in a direction perpendicular to the drive and rotational axes. Referring to [13], with these assumptions, the dynamics of gyroscope becomes

k xy

mx + d xx x + d xy y + k xx x + k xy y = u x + 2mΩ z y

(1)

my + d xy x + d yy y + k xy x + k yy y = u y − 2mΩ z x .

(2)

Fabrication imperfections contribute mainly to the asymmetric spring and damping terms, and d xy . The x and y axes spring and damping terms k xx , k yy , d xx and d yy are mostly

known, but have small unknown variations from their nominal values. The proof mass can be determined very accurately, and u x , u y are the control forces in the x and y direction.

y

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k xx

k yy

d yy

k xx

m Proof Mass

d xx

d xx k yy

x

d yy

Angular Velocity Ω z Figure 1. A simplified model of a MEMS z-axis gyroscope.

Dividing (1) and (2) by the reference mass and rewriting the gyroscope dynamics in vector forms result in

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J. Fei

K D u q + a q = − 2Ωq m m m

q +

where

⎡ 0 −Ωz ⎤ ⎡ux ⎤ ⎡d xx ⎡x⎤ q=⎢ ⎥ , u = ⎢ ⎥ , Ω= ⎢ , D=⎢ ⎥ ⎣Ωz 0 ⎦ ⎣y⎦ ⎣uy ⎦ ⎣d xy

(3)

d xy ⎤ ⎡k xx , Ka = ⎢ ⎥ d yy ⎦ ⎣k xy

k xy ⎤ . k yy ⎥⎦ 2

∗

Using non-dimensional time t = w0 t , and dividing both sides of (3) by w0 and the reference length q 0 give the final form of the non-dimensional equation of motion as

K q q D q u Ω q + + a2 = −2 . 2 q 0 mw0 q 0 mw0 q 0 mw0 q 0 w0 q 0

(4)

Defining a set of new parameters as follows:

d xy Ω q ∗ ∗ , d xy = , Ωz = z , q0 mw0 w0

q∗ =

ux

∗

ux =

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

wx =

2

mw0 q 0

k xx mw0

2

, wy =

uy

∗

,uy =

2

mw0 q 0

k yy mw0

2

(5)

, and

, w xy =

(6)

k xy mw0

2

.

(7)

Ignoring the superscript (*) for notational clarity, the nondimensional representation of (1) and (2) is

q + Dq + K b q = u − 2Ωq

⎡wx 2 where K b = ⎢ ⎣⎢ w xy

(8)

wxy ⎤ 2⎥. w y ⎦⎥

3. Adaptive Sliding Mode Controller 3.1. Adaptive Sliding Mode Controller Design and Stability Analysis This section proposes an adaptive sliding mode control strategy for MEMS gyroscopes. The control target is to achieve real-time compensation for fabrication imperfections and

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Robust Adaptive Control for MEMS Vibratory Gyroscope

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closed-loop identification of the angular velocity. The block diagram of an indirect adaptive sliding mode control for a MEMS gyroscope is shown in Figure 2, the tracking error between reference state and gyroscope state comes to the indirect adaptive sliding mode controller. The adaptive sliding mode controller is proposed to control the MEMS gyroscope. Angular velocity can be estimated by adaptive estimator. Referring to (8), we consider the dynamics with parametric uncertainties and external disturbance as

q + ( D + 2Ω + ΔD)q + ( K b + ΔK b )q = u + d

(9)

where ΔD is the unknown parameter uncertainties of the matrix D + 2Ω , ΔK b is the unknown parameter uncertainties of the matrix K b , d is an uncertain extraneous disturbance and/or unknown nonlinearity of the system . Rewriting (9) as

q + ( D + 2Ω)q + K b q = u + f

(10)

where f represents the matched lumped uncertainty and disturbance which is given by

f = d − ΔDq − ΔK b q .

(11)

Disturbance

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Input

Reference qm Model -

e

q

up

Indirect Sliding Mode Controller

MEMS Gyroscope

+

Adaptive Law

Estimation of Angular Rate

ΩZ

Figure 2. Block diagram of an indirect adaptive sliding mode control for a MEMS gyroscope.

We make the following assumption: The lumped uncertainty and

disturbance

f

is

bounded

f ≤ α 1 q + α 2 q + α 3 , where α 1 , α 2 and α 3 are known positive constants .

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

such

as

280

J. Fei

Suppose that a reference trajectory is generated by an ideal oscillator and the control objective is to make the trajectory of the gyroscopes to follow that of the reference model. The reference model is defined as

{

where K m = diag w1

2

2

qm + K m q m = 0

(12)

e = q − qm

(13)

s (t ) = e + λe

(14)

}

w2 .

The tracking error is defined as

The sliding surface is defined as

where

λ is a positive definite constant matrix to be selected, i.e. λ = diag {λ1 λ 2 }.

The derivative of the sliding surface is

s = e + λe = q − qm + λ (q − q m ) = u + f − ( D + 2Ω)q − K b q + λ (q − q m ) + K m q m .

⎡wx 2 d xy ⎤ ⎡ 0 −Ωz ⎤ , Ω= ⎢ ⎥ and K b = ⎢ d yy ⎥⎦ ⎣Ωz 0 ⎦ ⎢⎣ w xy

⎡d xx Substituting D = ⎢ ⎣d xy

(15)

wxy ⎤ 2 ⎥ into (15) w y ⎥⎦

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yields

d xy − 2Ω z ⎤ ⎡ q1 ⎤ ⎡wx 2 −⎢ d yy ⎥⎦ ⎢⎣q 2 ⎥⎦ ⎣⎢ wxy

⎡ d xx s = u + f − ⎢ ⎣d xy + 2Ω z

wxy ⎤ ⎡ q1 ⎤ 2 ⎥⎢ ⎥ + λ (q − q m ) + K m q m . (16) w y ⎦⎥ ⎣q 2 ⎦

Rewriting (16) yields

⎡q s = u + f − ⎢ 1 ⎣0

q 2 q1

0 q 2

− 2q 2 2q1

q1 0

q2 q1

⎡ d xx ⎤ ⎢d ⎥ ⎢ xy ⎥ ⎢ d yy ⎥ 0 ⎤⎢ ⎥ Ω z ⎥ + λ (q − q m ) + K m q m . ⎥ ⎢ q2 ⎦ ⎢wx 2 ⎥ ⎥ ⎢ ⎢ w xy ⎥ ⎢w 2 ⎥ ⎣ y ⎦

Defining

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(17)

Robust Adaptive Control for MEMS Vibratory Gyroscope

⎡q Y =⎢ 1 ⎣0

[

θ * = d xx

d xy

d yy

Ωz

q 2 q1 wx

2

0 q 2

− 2q 2 2q1

w xy

wy

q1

q2

0

q1

0⎤ , q 2 ⎥⎦

] , and Q = λ (q − q

281

(18)

2 T

m

) + K m qm

(19)

Then, (17) becomes

s = u + f − Yθ * + Q where Y ( q, q , q d , q d ) is an 2 × 7 matrix of known functions and

(20)

θ * contains unknown

system parameters. We assume both positions and velocities are measurable. Setting s = 0 to solve equivalent control u eq gives

u eq = Yθ * − Q − f .

(21)

The adaptive controller u is proposed as

u = Yθ − Q + us

= Yθ − Q − ρ sgn( s )

(22)

⎡ u1 ⎤ ⎡ s1 ⎤ * , s = ⎢ ⎥ , θ is the estimation of θ , ⎥ u s ⎣ 2⎦ ⎣ 2⎦

where u = ⎢

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⎡u ⎤ ⎛ρ u s = ⎢ s1 ⎥ = − ρ sgn( s ) = −⎜⎜ 1 ⎝0 ⎣u s 2 ⎦

0 ⎞⎛ sgn( s1 ) ⎞ ⎟ is the sliding mode signal. ⎟⎜ ρ 2 ⎟⎠⎜⎝ sgn( s 2 ) ⎟⎠

Substituting (22) into (20) yields

~ s = Yθ + f − ρ sgn( s ) .

~

(23)

where θ = θ − θ . Define a Lyapunov function to analyze the stability of (23) as *

V=

1 T 1~ ~ s s + θ T τ −1θ 2 2

where τ = τ are positive definite matrix. Differentiating V with respect to time yields T

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(24)

282

J. Fei

~ ~ V = s T s + θ T τ −1θ ~ ~ = s T (Yθ − Q − Yθ * + Q − ρ sgn( s ) + f ) + θ T τ −1θ ~ ~ ~ = s T Yθ − ρ ( s1 + s 2 ) + s T f + θ T τ −1θ ~ ~ ~ = − ρ ( s1 + s 2 ) + s T f + ( s T Yθ + θ T τ −1θ ) .

(25)

To make V ≤ 0 , we choose an adaptive law

~

θ (t ) = θ(t ) = −τY T s(t ) with

(26)

θ (0) being arbitrary. This choice yields V = − ρ ( s1 + s 2 ) + s T f ≤ − ρ s + s f

(27)

≤ − s ( ρ − α 1 q − α 2 q − α 3 ) ≤ 0 . With the choice of

ρ ≥ α 1 q + α 2 q + α 3 + η , where η is a positive constant, V

becomes negative semi-definite, i.e., V ≤ −η s . This implies that the trajectory reaches the sliding surface in finite time and remains on the sliding surface. V is negative definite implies

~

that s and K converge to zero. V is negative semi-definite ensures that V , s and bounded. It can be concluded from (23) that s is also bounded.

~

θ are all

Barbalat’s lemma can be used to prove that lim s (t ) = 0 .The inequality V ≤ −η s Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

t →∞

implies that s is integrable as

∫

t

0

s dt ≤

1

η

[V ( 0 ) − V (t ) ]. Since

V (0) is bounded and

t

V (t ) is nonincreasing and bounded, it can be concluded that lim ∫ s dt is bounded. Since t →∞ 0

t

lim ∫ s dt is bounded and s is also bounded, according to Barbalat’s lemma, s (t ) will t →∞ 0

asymptotically converge to zero, lim s (t ) = 0 . t →∞

Remark 1. Definition of Persistence of Excitation (PE) :A vector v ∈ R to be persistence of excitation if there exist positive constants

t > 0,

∫

t +T

t

q

q ≥ 1 is said

α and T such that for all

v(τ )v T (τ )dτ ≥ αI .

PE is a notion of a time signal that contains sufficient richness so that the v(τ )v (τ ) T

matrix is nonsingular. It requires that v(τ ) varies in such a way with time that the integral of

the matrix is positive definite over any time interval [t

t + T ].

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Robust Adaptive Control for MEMS Vibratory Gyroscope

283

~

To make conclusions about θ = 0 , other than the fact that they are bounded, we need to make the persistence of excitation argument. From the adaptive law

~

θ (t ) = θ(t ) = −τY T s(t ) , according to [14], if Y is persistently exciting signal, then ~

~

θ (t ) = −τY T s (t ) guarantees that θ → 0 , θ will converges to its true values. Because s → 0 implies e → 0 , (13) determines that q1 = A1 sin( w1t ) , q1 = w1 A1 cos(w1t ) , q2 = A2 sin(w2t ) , q2 = w2 A2 cos(w2t) . It can be shown that there exist some positive scalar constants

α and T such that for all t > 0 ,

∫

t +T

t

Y T Ydτ ≥ αI .

where

⎡ q12 − 2q1q2 q1q2 q1q1 q1q2 0 0 ⎤ ⎢ ⎥ q12 + q22 q1q2 − 2q22 + 2q12 q1q2 q2 q2 + q1q1 q1q2 ⎥ ⎢ q1q2 ⎢ 0 q2 q1 q22 q2 q1 q2q2 ⎥ 2q1q2 0 ⎢ ⎥ T 2 2 2 2 Y Y = ⎢− 2q1q2 − 2q2 + 2q1 2q1q2 4q2 + 4q1 − 2q1q2 − 2q2q2 + 2q1q2 2q1q2 ⎥ ⎢ q q − 2q1q2 q1q2 q12 q1q2 0 0 ⎥ ⎢ 1 1 ⎥ − 2q2q2 q2 q2 q1q2 q22 0 0 ⎥ ⎢ q2 q1 ⎢ 0 − 2q2 q1 q1q2 q2 q2 q1q2 q22 ⎥⎦ 0 ⎣ From (12) and (18) it can be shown that Y Y has full rank if w1 ≠ w2 , i.e. the excitation T

frequencies on x and y axes should be different. In other words, excitation of proof mass

~

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should be persistent[13]. Since θ → 0 , then the unknown angular velocity as well as all other unknown parameters can be consistently estimated. In consequence, angular velocity 2

2

Ω z and all gyroscope parameters such as d xx , d xy , d yy , wx , w xy and w y converge to their true values. In summary,

if

persistently

~

exciting

drive

signals,

x m = A1 sin( w1t )

and

ym = A2 sin( w2t ) are used, then θ (t ) , s (t ) and e(t ) all converge to zero asymptotically. ˆ (t ) = Ω . Consequently the unknown angular velocity can be determined as lim Ω t →∞

z

z

However it is difficult to establish the convergence rate. Remark 1: In the adaptive control system design, the persistent excitation condition is an important factor to estimate the angular velocity Ω z correctly. The reference trajectory that the gyroscope must follow is generated such that the resonance frequency of the x-axis is different from that of the y-axis which satisfies the persistent excitation condition.

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J. Fei

3.2. Comparison with Standard Adaptive Controller A standard adaptive controller which has an addition term K f s is proposed as

u = Yθ − Q − K f s where θ is the estimated parameter of

(28)

θ * , K f is positive definite matrix.

Then (20) becomes

s = u + f − Yθ * + Q ~ = Yθ − K f s + f .

(29)

Define a Lyapunov function

V= ~

1 T 1~ ~ s s + θ T τ −1θ 2 2

(30)

where τ = τ > 0 , θ = θ − θ . Differentiating V with respect to time yields

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T

*

~ ~ V = s T s + θ T τ −1θ ~ ~ ~ = s T (Yθ − K f s + f ) + θ T τ −1θ ~ ~ ~ = − s T K f s + s T f + ( s T Yθ + θ T τ −1θ ) .

(31)

To make V ≤ 0 , we choose an adaptive law as

~

θ (t ) = θ(t ) = −τY T s(t ) .

(32)

If f = 0 , therefore V becomes

V = − s T K f s ≤ 0

(33)

which implies that the stability of closed-loop system can be guaranteed. If f ≠ 0 , the stability of closed-loop system cannot be guaranteed. Remark 1. Such an adaptive controller would be inadequate to address the control system where there exist appreciable non-parametric uncertainties which include unmodelled dynamics, external disturbance and other imperfections in the estimates of gyroscopes

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Robust Adaptive Control for MEMS Vibratory Gyroscope

285

parameters. Therefore, the proposed adaptive sliding mode controller incorporates the capability to maintain stable performance in the presence of model uncertainties and external disturbance. Remark 2. In order to eliminate the control discontinuities, a smooth sliding mode control tanh(as ) that can reduce chattering problem is introduced. The parameter a determines the slope of tanh(as) function at s = 0 . Therefore the smooth sliding mode controller is proposed as

u = Yθ − Q − ρ tanh(as)

(34)

3.3. Adaptive Sliding Mode Design under Asymmetric Coupling Term If gyroscope system does not have same coupling damping and spring constant, the gyroscope system dynamics can be written as

q + Dq + K b q = u − 2Ωq ⎡u x ⎤ ⎡0 ⎡ x⎤ where q = ⎢ ⎥ u = ⎢ ⎥ Ω = ⎢ ⎣ y⎦ ⎣Ω z ⎣u y ⎦

(35)

⎡ wx 2 d xy ⎤ K =⎢ d yy ⎥⎦ b ⎢⎣ w yx

− Ωz ⎤ ⎡ d xx , D=⎢ ⎥ 0 ⎦ ⎣d yx

wxy ⎤ 2⎥. w y ⎦⎥

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The dynamics of sliding surface can be derived as

⎡q q 0 0 − 2q 2 q1 q2 0 s = u + f − ⎢ 1 2 0 0 q1 ⎣ 0 0 q1 q 2 2q1

⎡ d xx ⎤ ⎢d ⎥ ⎢ xy ⎥ ⎢ d yx ⎥ ⎢ ⎥ d yy 0 ⎤⎢ ⎥ ⎢ Ω z ⎥ + λ (q − q m ) + K m qm q2 ⎥⎦ ⎢ 2 ⎥ ⎢ wx ⎥ ⎢w ⎥ ⎢ xy ⎥ ⎢wyx ⎥ ⎢ w2 ⎥ ⎣ y⎦ (36)

Define

⎡q Y =⎢ 1 ⎣0

q 2 0

0 q1

0 q 2

− 2q 2 2q1

q1

q2

0

0

0

q1

0⎤ q 2 ⎥⎦

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(37)

286

J. Fei

θ * = [d xx

d xy

d yx

d yy

Ωz

wx2

wxy

w yx

w y2

]

T

Q = λ (q − q m ) + K m q m ,

(38) (39)

Similarly as Lyapunov analysis before, the adaptive law is derived as

~

θ (t ) = θ(t ) = −τY T s(t )

(40)

Therefore, all the system parameter including unsymmetrical coupling damping and spring parameters such as d xy , d yx , w xy and w yx can be consistently estimated. Remark 3. The motion of a mode-unmatched gyroscope, in which the resonance frequency of the x-axis is different from that of the y-axis, has sufficient persistence of excitation to permit the identification of all major fabrication imperfections as well as angular velocity. A MEMS gyroscope, suitable for the adaptive mode of operation requires equal movements in the x and y axes. Thus, there is no specific drive and sense axis in the sense of conventional MEMS gyroscopes. It should be noted that a conventional gyroscope structure is normally designed based on the assumption that the movement of the proof mass in the drive axis (x-axis) is relatively large, but the movement in the sense axis (y-axis) is very small. The proposed gyroscope design consists of a proof mass, four hairpin type spring suspensions and several pairs of parallel electrodes for actuation and sensing located at both x and y axes. Conclusion: With the control law (22) and the parameter adaptation law (26), if the gyroscope is controlled to follow the mode-unmatched reference model, the persistent excitation condition is satisfied , i.e. w1 ≠ w2 , and all unknown gyroscope parameters,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

including the angular velocity, are estimated correctly.

4. Simulation of MEMS Gyroscope We will evaluate the proposed adaptive sliding mode control with a sliding mode observer on the lumped MEMS gyroscope model [10] using MATLAB/SIMULINK. The control objective is to design an adaptive sliding mode controller so that the trajectory of X (t ) can track the state of reference model X m (t ) . In the simulation, we allowed ± 2% parameter variations for the spring and damping coefficients and further assumed ± 1% magnitude changes in the coupling terms i.e. d xy and ω xy The external disturbance is a random variable with zero mean and unit variance. Parameters of the MEMS gyroscope are as follows:

m = 0.57e − 8 kg, d xx = 0.429e − 6 N s/m,

d xy = 0.0429e − 6 N s/m, d yy = 0.687e − 6 N s/m k xx = 80.98 N/m, k xy = 5 N/m, k yy = 71.62 N/m

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287

w0 = 1kHz , q 0 = 10 −6 m . The initial conditions

θ (0) = 0.95θ * , angular velocity Ω = 5.0 rad/s . The desired

motion trajectories are xm = sin(w1t ) and y m = 1.2 sin( w 2 t ) , where w1 = 4.17 kHz and

w2 = 5.11kHz. The sliding gain of (25) is ρ = diag {200 200} , the adaptive gain of (28)

is chosen as τ = diag{200 200 200 200 200 200 200} . The sliding mode parameter of (22) is

λ = diag {4 4} and a = 5 in the smooth sliding mode controller tanh(as).

The tracking error and sliding surface are shown Figure 3 and Figure 4. It is shown that both tracking error and sliding surface converge to zero. Figure 5 and Figure 6 show that the estimation of spring and damping coefficients converge to their true values with persistent sinusoidal reference signal.

Tracking Error e1

0.1 0 -0.1 -0.2 -0.3

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

Tracking Error e2

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0.2 0.1 0 -0.1 -0.2

Figure 3. Convergence of the tracking error e(t).

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J. Fei

Sliding Surface s1

10 5 0 -5 -10

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

Sliding Surface s2

5 0 -5 -10

Figure 4. Convergence of the sliding surface s(t). -3

dxx

2 0 -2

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

-3

dxy

2

x 10

0 -2

0 -4

5 dyy

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x 10

x 10

0 -5

0

Figure 5. Adaptation of damping coefficients of gyroscope.

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289

Wx

81 80.5 80

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0.035

0.04

Wxy

8 6 4

Wy

72 71 70

Figure 6. Adaptation of spring constants of gyroscope. 25 20 15

Angular Rate(rad/s)

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10 5 0 -5 -10 -15 -20 -25

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

Figure 7. Convergence of the estimated angular velocity with smooth sliding mode controller.

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290

J. Fei 20

15

Angular Rate(rad/s)

10

5

0

-5

-10

-15

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

Figure 8. Convergence of the estimated angular velocity with bang-bang type sliding mode controller.

Sliding Mode Control Force us1 Sliding Mode Control Force us2

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Figure 7 and Figure 8 compare the angular velocity estimation between smooth sliding mode controller and bang-bang type sliding mode controller sgn(s) under the sinusoidal reference signal. Both figures show that estimation of angular converges to its true values. It is observed that the estimated angular velocity with smooth sliding mode controller has better convergence performance. Figure 9 depicts smooth sliding mode control force of the adaptive sliding mode controller. It is shown from Figure 9 that adaptive sliding mode system with smooth sliding mode controller can reduce chattering significantly. 100 50 0 -50 -100

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

0

0.005

0.01

0.015 0.02 0.025 Time(Second)

0.03

0.035

0.04

100 50 0 -50 -100

Figure 9. Smooth sliding mode control force of the adaptive sliding mode controller.

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Robust Adaptive Control for MEMS Vibratory Gyroscope

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5. Conclusion This chapter investigated the design of adaptive control with sliding mode controller for the gyroscope system. New adaptive sliding mode controller was formulated for MEMS gyroscopes with two unmatched oscillatory modes which have sufficient persistence of excitation to permit the identification of all gyroscope parameters including the damping and stiffness coefficients and angular velocity. The proposed adaptive sliding mode controller incorporates the capability to maintain stable performance in the presence of model uncertainties and external disturbance. Numerical simulations show that the proposed adaptive sliding mode control has satisfactory performance and robustness in the presence of model uncertainty and external disturbance.

References

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[1]

Utkin, V.I. 1977, “Variable structure systems with sliding modes,” IEEE Transaction on Automatic Control, 22, pp. 212-222. [2] Narendra, K. S., Annaswamy, A. M., 1989. Stable Adaptive Systems. Prentice-Hall, Englewood Cliffs, NJ. [3] Astrom, K. J., Wittenmark, B., 1989. Adaptive Control. Addison-Wesley Publishing . [4] Wang, Y. P. , Sinha, A., 1998, “Adaptive sliding mode control algorithm for a microgravity isolation system”, Acta Astronautica, 43(7-8), pp. 377-384. [5] Lee, T. H. Tan, K. K., and. Lee, M. W., 1998, “A variable structure-augmented adaptive controller for a gyro mirror line of light stabilization platform”, Mechatronics, 8, pp.47-64. [6] Song, G., Mukherjee, R., 1998, “ A comparative study of conventional nonsmooth time-invariant and smooth time-varying robust compensator”, IEEE Transactions on Control System technology, 6(4) , pp. 571-576. [7] Sam, Y., Osman, J.H., and Ghani, M.R., 2004, “A class of proportional-integral sliding mode control with application to active suspension system,” Systems & Control Letters 51(3/4), 217–224. [8] Lin, F., Chiu, S., and Shyu, K.,1998, “ Novel sliding mode controller for synchronous motor drive”, IEEE Transactions on Aerospace and Electronic Systems, 34(2), pp. 532541. [9] Chou, C., Cheng, C., 2003. “A decentralized model reference adaptive variable structure controller for large-scale time-varying delay systems”, IEEE Transactions on Automatic Control, 48(7), pp. 1213-1217. [10] Batur, C., Sreeramreddy , T., and Khasawneh, Q., 2005, “Sliding mode control of a simulated MEMS gyroscope”, Proceedings of the 2005 American Control Conference, pp. 4160 – 4165. [11] Leland, R. P, 2003, “ Adaptive mode tuning for vibration gyroscopes”, IEEE Transactions on Control Systems Technology, 11(2), pp.242-247 . [12] S. Park; Horowitz, R., 2004,” New adaptive mode of operation for MEMS gyroscopes, Transactions of the ASME: Journal of Dynamic Systems, Measurement and Control, 126(4), pp. 800-810.

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[13] S. Park, 2000, “Adaptive control strategies for MEMS gyroscope,” Ph.D. dissertation, University of California, Berkeley. [14] P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: PrenticeHall, 1996.

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

In: Micro Electro Mechanical Systems Editors: B. Ekwall and M. Cronquist, pp. 293-310

ISBN: 978-1-60876-474-7 © 2010 Nova Science Publishers, Inc.

Chapter 10

THE ELECTRIC FORCE ON THE MOVING ELECTRODE OF AN INCLINED PLATE CAPACITOR Yumin Xiang* Department of Physics, Sichuan University of Science and Engineering, Sichuan, China

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Abstract Based on electric energy calculation through the conformal transformations, the principle of virtual work is employed to determine the electric force exerted on the charged electrode of an inclined plate capacitor. For application in Micro Electromechanical System, the computation is aimed at the general case. The result is achieved in a board manner with aid of elliptic function. For the electrode plates, there is no restriction to dimension. Beside the parallel plate capacitor is treated as a special case, the electric force on the moving electrode is researched. Further, the deformation on the electrode plate is analyzed. The corresponding numerical simulation curves are presented.

Keywords: conformal mapping; elliptic function; principle of virtual work; electric force PACS: 41.20Cv, 62.20.Fe, 02.30.Gp

Introduction In Micro Electromechanical System [1] the electric force on the plate of charged capacitor is essential factor to be controlled precisely. The capacitor commonly used is considered having two parallel plates. Actually, the manufacturing constraints make an electrode plate not parallel to the other. This article deals with the field region mapping to determine the electric force on the inclined plate in general case without any restrictive assumption for the geometric dimension of the two plates. The case for parallel plate *

E-mail address: [email protected]. Tel: (813) 5505892; Fax: (813) 5505800; Address for Correspondence and Proofs: Yumin Xiang. Department of Physics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, P.R.China

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294

Yumin Xiang

capacitor is taken as a special example. Further, the electric force on the plate moving with translation and rotation is presented. The deformation distribution on the plate is displayed and analyzed. The numerical simulation is carried out.

zI l1 r1 0

A C

ϕ

r2

O

B

D

zR

l2

Figure 1. The cross section of inclined plate capacitor in z-plane.

2. Region Mapping and Energy Calculation A capacitor has two non-parallel conducting plates. Figure 1 shows its cross section in zplane. The angle between plates AB and CD is ϕ which vertex O is taken as the origin. Let l1 and l 2 denote the length of plate AB and CD. The distance OA and OC are r1 and

r2 , respectively. The potential difference between the two electrode plates is V.

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The longitudinal dimension of the plates is far more than l and r. Under such condition, the problem can be treated as two dimensional in the cross section plane [2]. The electric field in the z-plane is almost confined by two electrode plates to the interior of the angle ∠AOC . That region can be mapped onto the upper half of t-plane of Figure 2 by [3] π ϕ

t = Mz + M 0

(1)

Find the coordinates of B and D in the two planes. Substituting them into Eq.(1) we obtain

M =

2 π ϕ

( r1 + l1 ) + ( r2 + l2 )

π ϕ

and M 0 =

π ϕ

π ϕ

π ϕ

π ϕ

( r1 + l1 ) − ( r2 + l2 ) ( r1 + l1 ) + ( r2 + l 2 )

(2)

The coordinate of A and C in the t-plane are denoted byα andβ, respectively. Using Eqs.(1)and (2), we determine them as follows:

α=

π ϕ

π ϕ

− 2r1 + ( r1 + l1 ) − ( r2 + l 2 ) π ϕ

( r1 + l1 ) + ( r2 + l 2 )

π ϕ

π ϕ

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(3)

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

β=

π ϕ

π ϕ

2r2 + ( r1 + l1 ) − ( r2 + l 2 ) π ϕ

( r1 + l1 ) + ( r2 + l2 )

295

π ϕ

(4)

π ϕ

Successively, the field region in the t-plane is mapped into the

ζ -plane of Figure 3 by

the fractional linear transformation

ζ =

(1 − α )(1 + t ) 2( t − α )

(5)

The modulus k can be computed by substituting t = β and ζ = 1 / k 2 at point C into Eq.(5). The result is

k=

π π π ⎛ ϕπ ⎞⎛ ⎜ r1 + r2ϕ ⎟⎜ ( r1 + l1 ) ϕ + ( r2 + l2 ) ϕ ⎜ ⎟⎜ ⎝ ⎠⎝ π π π ⎛ ϕ ⎞⎛ π ⎜ r1 + ( r2 + l 2 ) ϕ ⎟⎜ r2ϕ + ( r1 + l1 ) ϕ ⎜ ⎟⎜ ⎝ ⎠⎝

2( β − α ) = (1 − α )(1 + β )

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tI

B

-1

A

α

0

C

D

β

1

tR

Figure 2. The t-plane.

ζ

I

0

A

B

1

1/ k 2

D

C

ζR

A

Figure 3. The ζ -plane.

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⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

(6)

296

Yumin Xiang

uI 1

-K+iK'

-K

iK' C

A

D

B 0

uR 1

Figure 4. The u-plane.

Referring to Fig 4, finally, the upper half of ζ -plane is mapped into the interior of the rectangle ABDC in the u-plane by Schwarz-Crystoffel transformation 1

1

1

− − du 1 − = N ζ 2 (ζ − 1) 2 (ζ − 2 ) 2 dζ k

(7)

Integration of Eq.(7) leads to

u = kN ∫

dζ

ζ

0

ζ (1 − ζ )(1 − k 2ζ )

+ N0

(8)

Based on the definition of the Jacobian elliptic functions sn( v, k ) , cn ( v, k ) and

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dn( v, k ) [4], the foregoing equation is replaced by ⎛ u − N0 ⎞ ,k⎟ ⎝ kN ⎠

ζ = sn 2 ⎜

Substituting the coordinates of B and D in u-plane and

(9)

-pane into Eq.(9), we get

N 0 = 0 and kN = 1

(10)

ζ = sn 2 (u , k )

(11)

Hence, Eq.(9) is rewritten as

Now the field confined by the rectangle ABDC is uniform. So the capacitance per unit longitudinal length of the plates is [5]

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The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

C = ε0

K ′( k ) K (k )

297 (12)

where K ( k ) is the complete elliptic integration of the first kind, and

K ′( k ) = K ( k ′)

(13)

in which k ′ is referred to as the complementary modulus of k , e.g. π π π π ⎛ ⎞⎛ ⎞ ϕ ϕ ϕ ϕ ⎜ ( r1 + l1 ) − r1 ⎟⎜ ( r2 + l 2 ) − r2 ⎟ ⎜ ⎟⎜ ⎟ (1 + α )(1 − β ) ⎝ ⎠ ⎝ ⎠ ′ = k = π π π π (1 − α )(1 + β ) ⎛ ⎞⎛ ⎞ ⎜ ( r1 + l1 ) ϕ + r2ϕ ⎟⎜ ( r2 + l2 ) ϕ + r1ϕ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

(14)

Then the electric energy per unit longitudinal length of the capacitor is represented by

1 W = CV 2 2

(15)

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3. Electric Force on the Electrode Plate As illustrated in Figure 5, the electric force F1 on the electrode plate AB has two components, F1ϕ perpendicular to the plate and F1r in the plate. Making use of the principle of the virtual work [6], F1ϕ can be calculated as

F1ϕ =

∂W ∂ h1

= V =c.

1 2 ∂C 1 ∂C ∂k V = V2 2 ∂ h1 2 ∂ k ∂ h1

(16)

where dh1 represents the infinitesimal distance of the plate AB moved in the normal direction by the electric force. Figure 5 shows

d h1 = d r1 tan ϕ = dr2 sin ϕ

(17)

Substituting Eq.(17) into Eg.(16) produces

F1ϕ =

⎞ ∂ k ∂ r2 ⎞ 1 2 ∂ C ⎛ ∂ k 1 2 ∂ C ⎛ ∂ k ∂ r1 ∂k ⎜⎜ ⎟⎟ = V ⎜⎜ + V cot ϕ + csc ϕ ⎟⎟ (18) ∂ k ⎝ ∂ r1 ∂ h1 ∂ r2 ∂ h1 ⎠ 2 ∂ k ⎝ ∂ r1 2 ∂ r2 ⎠

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Yumin Xiang

zI

zI

B

l1 r1

A A

dr1

dh1 O dr2 O ϕ

B

ϕ

F1rθ1 F1ϕ F1 F 2 F2ϕ F2rθ 2

C

l2 D

zR

Figure 5. The differential elements and forces in z-plane.

Differentiating Eq.(12) with respect to the modulus k, one arrives at

∂C ∂ ⎛ K ′( k ) ⎞ = ε0 ⎟ ⎜ ∂k ∂ k ⎜⎝ K ( k ) ⎟⎠

(19)

∂ ⎛ K ′( k ) ⎞ π ⎟⎟ = − ⎜⎜ 2 ∂ k ⎝ K (k ) ⎠ 2k k ′ K 2 ( k )

(20)

πε0 ∂C =− ∂k 2k k ′2 K 2 ( k )

(21)

Note that [7]

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Eq.(19) then becomes

Differentiation of Eq.(6) with respect to r1 and r2 gives, respectively

∂ k π kk ′ 2 = τ1 ∂ r1 2ϕ

(22)

∂ k πkk ′ 2 = τ2 ∂ r2 2ϕ

(23)

where

⎛ ⎜ π π π ϕ ϕ ϕ ⎜ r (l + r ) l1 r1 (l1 + r1 ) 1 ⎜ τ1 = + 1 π 1 1π + π π π π (l1 + r1 ) ⎜ ⎛ ϕ ⎞ ⎛ ϕ ϕ ϕ ⎟ ϕ ϕ ⎜ ⎜ l r l r ( ) ( ) + + + r r r r l r r + ( + ) − 2 2 ⎜⎜ 1 1 1 1 1 2 ⎜ ⎟ 1⎜ 1 1 ⎝ ⎠ ⎝ ⎝

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

⎞ ⎟ ⎟ − 1⎟ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠

(24)

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

299

and

⎛ ⎜ π π π ϕ ϕ ϕ ⎜ r (l + r ) l 2 r2 (l 2 + r2 ) 1 ⎜ τ2 = + 2 π2 2π + π π π π (l 2 + r2 ) ⎜ ⎛ ϕ ⎞ ⎛ ϕ ϕ ϕ ⎟ ϕ ϕ ⎜ ⎜ r2 r1 + r2 r (l + r ) − r2 ⎜⎜ (l1 + r1 ) + ( l 2 + r2 ) ⎜ ⎟ 2⎜ 2 2 ⎝ ⎠ ⎝ ⎝

⎞ ⎟ ⎟ − 1⎟ (25) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠

Substituting Eqs.(22)-(25) into Eq.(18) results in

F1ϕ = −

π 2 ε 0V 2 (τ 1 cos ϕ + τ 2 ) 8ϕ sin ϕ K 2 ( k )

(26)

where the minus sign indicates the force is attractive. Similarly, with the aid of Eqs.(21),(22) and (24), F1r is computed by

F1 r =

∂W ∂ r1

= V =c.

π 2 ε 0τ 1 1 2 ∂C 1 ∂C ∂k = V2 =− V V2 2 2 ∂ r1 2 ∂ k ∂ r1 8ϕ K ( k )

(27)

So the magnitude of the resultant force F1 is

F1 =

F12r + F1ϕ2 =

π 2ε 0 V 2 τ 12 + τ 22 + 2τ 1τ 2 cos ϕ 8ϕ sin ϕ K 2 ( k )

(28)

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The direction of F1 is determined by

tan θ 1 = F1ϕ / F1 r = where

τ 1 cos ϕ + τ 2 τ 1 sin ϕ

(29)

θ1 is the angle between F1and AB.

4. Characteristics of the Force Using symbolic symmetry, the simple and direct method to achieve the electric force on another electrode plate CD is exchanging the subscripts 1 and 2 in the results of Eqs.(26)(29). Therefore

F2 ϕ = −

π 2 ε 0V 2 (τ 2 cos ϕ + τ 1 ) 8ϕ sin ϕ K 2 ( k )

(30)

π 2 ε 0τ 2 V2 8ϕ K 2 ( k )

(31)

F2 r = −

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

300

Yumin Xiang

F2 =

π 2ε 0 V 2 τ 12 + τ 22 + 2τ 1τ 2 cos ϕ 2 8ϕ sin ϕ K ( k ) tan θ 2 =

τ 2 cos ϕ + τ 1 τ 2 sin ϕ

(32)

(33)

It has been seen that

F1 = F2 = F

(34)

tan θ 1 + tan θ 2 = − tan ϕ 1 − tan θ 1 tan θ 2

(35)

Using Eq.(29) and (33) we have

tan( θ 1 + θ 2 ) = which means

θ1 + θ 2 = π − ϕ

(36)

Eqs.(34) and (36) implies that a pair of F1 and F2 have the same magnitude and opposite direction. Namely, (37) F1 = − F2 Introduce a dimensionless force as

F

=

π 2ε 0 8 ( l1 + r1 )( l 2 + r2 )

V

2

( l1 + r1 )( l 2 + r2 ) (τ 12 + τ 22 + 2τ 1τ 2 cos ϕ ) ϕ sin ϕ K 2 ( k )

10 8

~ F

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~ F =

6 4 2 0.2

0.4

0.6

0.8

1

ϕ ~

Figure 6. The curve of F versus ϕ. ( r1 : l1 : r2 : l 2 = 2.0 : 1.0 : 2.5 : 1.5 ).

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(38)

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

301

Under the condition of r1 : l 1: r2 : l 2 = 2.0 : 1.0 : 2.5 : 1.5 , Figure 6 shows the function

~ of F versus ϕ .

5. Further Discussion δ 1 are introduced to replace r1 and r2 . As depicted in Figure 7, h1 is the distances from A to CD and δ 1 = ∠ACZ R . The following For convenience, the parameters h1 and

geometric relations are applied:

h 2 = h1 + l1 sin ϕ ; r1 =

h1 ; sin ϕ

r1 + l1 =

h2 ; sin ϕ

cot δ 2 = r2 =

h1 cot δ 1 + l1 cos ϕ − l 2 ; h1 + l1 sin ϕ

h1 sin( δ 1 − ϕ ) ; sin ϕ sin δ 1

r2 + l 2 =

(39)

h 2 sin( δ 2 − ϕ ) ; sin ϕ sin δ 2

Substituting Eqs.(39) into Eqs.(6),(24) and (25) leads to

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k=

π ⎛ ⎜ ⎛ sin(δ 1 − ϕ ) ⎞ ϕ ⎟⎟ ⎜1 + ⎜⎜ sin δ 1 ⎠ ⎜ ⎝ ⎝

π ⎞⎛ ⎞ ⎟⎜ ⎛ sin(δ 2 − ϕ ) ⎞ ϕ ⎟ ⎜ ⎟ + 1 ⎟⎜ ⎜ sin δ ⎟ ⎟ 2 ⎠ ⎟ ⎟⎜ ⎝ ⎠⎝ ⎠ π π ⎛ ⎞⎛ ⎜ ⎛ ( h1 + l1 sin ϕ ) sin(δ 2 − ϕ ) ⎞ ϕ ⎟⎜ ⎛ h1 sin(δ 1 − ϕ ) ⎞ ϕ ⎟⎟ ⎟⎜1 + ⎜⎜ ⎟⎟ ⎜1 + ⎜⎜ h1 sin δ 2 ⎠ ⎟⎜ ⎝ ( h1 + l1 sin ϕ ) sin δ 1 ⎠ ⎜ ⎝ ⎝ ⎠⎝

⎞ ⎟ ⎟ ⎟ ⎠

⎛ ⎞ ⎜ ⎟ l1 l1 1 + sin ϕ sin ϕ ⎜ ⎟ h1 h1 sin ϕ ⎜ 1 ⎟ τ1 = − − + 1 π π π ⎟ h2 ⎜ ϕ ⎞ϕ ⎛ ⎛ sin(δ 1 − ϕ ) ⎞ ϕ l1 ⎜ 1 + ⎛⎜ sin(δ 2 − ϕ ) ⎞⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ϕ − + 1 + 1 1 sin ⎟ ⎟ ⎜ h ⎟ ⎜ sin δ ⎜ ⎜ sin δ ⎟ 1 1 2 ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ ⎝ ⎠

(40)

(41)

⎛ ⎞ ⎜ ⎟ ⎛ l1 ⎞ sinδ 1 sin(δ 2 − ϕ ) ⎛ l1 ⎞ sinδ 1 sin(δ 2 − ϕ ) ⎜⎜1 + sinϕ ⎟⎟ ⎜⎜1 + sinϕ ⎟⎟ −1 ⎜ ⎟ sinϕ sinδ 2 ⎜ 1 ⎠ sinδ 2 sin(δ 1 − ϕ ) ⎝ h1 ⎠ sinδ 2 sin(δ 1 − ϕ ) − ⎝ h1 ⎟ − τ2 = + 1 π π π ⎟ h2 sin(δ 2 − ϕ ) ⎜ ⎛ ⎛ l1 ⎛ sinδ 1 ⎞ ϕ ⎜ ⎛ sinδ 2 ⎞ ϕ ⎞ sinδ 1 sin(δ 2 − ϕ ) ⎞ ϕ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ + 1 + 1 ⎟ ⎜ − + ϕ 1 1 sin ⎜ ⎜ sin(δ − ϕ ) ⎟ ⎟ ⎜ sin(δ − ϕ ) ⎟ ⎟ sinδ sin(δ − ϕ ) ⎟ ⎜⎜ h 1 2 ⎠ ⎝ ⎠ ⎝ 1 2 1 ⎠ ⎝⎝ ⎠ ⎝ ⎠

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(42)

302

Yumin Xiang The parallel plate capacitor is a special case as

ϕ approaches zero. Keep in mind that

π

π ⎛ sin( δ − ϕ ) ⎞ ϕ lim ⎜ = exp( − π cot δ ) and lim (1 + m sin ϕ )ϕ = exp( m π ) (43) ⎟ ϕ →0 ϕ →0⎝ sin δ ⎠

As

ϕ → 0 , from Eqs.(3 9) − (42) we have cot δ 2 = cot δ 1 +

h 2 = h1 = h ;

k=

l1 − l 2 h

(44)

(1 + exp( −π cot δ 1 ) )(1 + exp( −π cot δ 2 ) )

(45)

⎛ l ⎛ l ⎞⎞ ⎛ ⎞ ⎞⎛ ⎜⎜1 + exp⎜ − π ⎛⎜ 1 + cot δ 1 ⎞⎟ ⎟ ⎟⎟⎜⎜1 + exp⎜ π ⎛⎜ 2 − cot δ 1 ⎞⎟ ⎟ ⎟⎟ ⎠⎠⎠ ⎝h ⎠ ⎠ ⎠⎝ ⎝ ⎝h ⎝ ⎝ τ1 +τ1 =

l1 sin 2 ϕ ⎛ l l ⎞ ⎜ coth π 1 + coth π 2 ⎟ 2 2h ⎝ 2h 2h ⎠

(46)

Applying the three above equations to Eqs.(34) and (28) then taking the limit as

ϕ → 0,

we derive

F =

π 2 ε 0 l1

l l ⎞ ⎛ V 2 ⎜ coth π 1 + coth π 2 ⎟ 16 h K ( k ) 2h 2h ⎠ ⎝ 2

A typical case for the parallel plate capacitor is that l1 = l 2 = l and Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(47)

2

δ1 =

π 2

. Hence,

from Eq.(44) and (45) we obtain

δ2 =

π 2

and k = sech π

l 2h

(48)

zI l1

B

A

r1

ϕ

O

δ1

r2 C

h1

l2

h2 δ2

zR

D

Figure 7. Geometric relation of the lines and angles in the z-plane.

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

303

Eq.(47) then reduces to

l 2h V 2 F = l ⎞ ⎛ 8 h 2 K 2 ⎜ sech π ⎟ 2h ⎠ ⎝

π 2 ε 0 l coth π

l >> h . Consequently coth π l → 1 and

In commonly used case we have

sech π

(49)

2h

l → 0 . Using K ( 0 ) = π / 2 , Eq.(49) becomes 2h

F =

ε 0l 2h 2

V2

(50)

which has been well known by us.

zI

zI

l1

B

A v

r1

ξ

r1o

A

ϕo

ϕo Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

B

r2 O r2o C

O

l2 D

zR

Figure 8. Moving plate in translation.

6. Moving Plate In micro electromechanical system capacitor with moving plate is applied. Say the plate CD is fixed while AB movable in the z-plane[8]. Firstly, suppose AB only moves with translation velocity v in time interval Δ t as shown in Figure 8. The angle between v and AB is and

ϕo

ξ

. Labeling the initial value of

ri and ϕ as rio

respectively, we have

r1 = r1o +

v Δ t sin( ξ − ϕ o ) v Δ t sin ξ and r2 = r2 o + sin ϕ o sin ϕ o

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(51)

304

Yumin Xiang Then assume A with angular velocity

r1 = ( l1 + r1o )

ω

around B in rotation only. Figure 9 illustrates

sin ϕ o ( l + r ) sin ω Δ t − l1 and r2 = r2 o − 1 1o sin( ϕ o + ω Δ t ) sin( ϕ o + ω Δ t )

zI

zI

B A ω

r1o ϕo

O

l1 A

r1 ϕ

r2o

(52)

O

r2 C

l2 D

zR

Figure 9. Moving plate in rotation.

Since the plate motion is combination of translation and rotation around B, from Eqs.(51) and (52) we obtain the general expression

r1 =

1 ( r1o sin ϕ o − l1 (sin( ϕ o + ω Δ t ) − v Δ t sin ξ ) + v Δ t sin ξ ) sin( ϕ o + ω Δ t )

r2 = r2o −

(53)

⎞ ⎛ vΔ t 1 ⎜ (l1 + r1o ) sin ωΔt − (cos(ωΔt − ξ + 2ϕ o ) − cos(ωΔt − ξ ) ⎟⎟ sin(ϕ o + ωΔt ) ⎜⎝ sin ϕ o ⎠

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

(54) Substituting Eqs.(53) and (54) into (6),(24),(25),(28) and (38) yields the result for the capacitor with moving plate.

7. Deformation of the Plate The deformation of the plate AB is a 2-dimendional problem in the z-plane. Since the electric field E is perpendicular to the metallic electrode, no shear stress presents and the normal stress has only the component perpendicular to AB in the z-plane. That indicates [9]

1 2

2

σ ϕ = ε 0 E and σ r = 0 Then the tensile strain expressions for the plate are [10]

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

(55)

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

305

eϕ =

2 1 +ν ((1 − ν )σ ϕ − νσ r ) = (1 − ν ) ε 0 E 2 Y 2Y

(56)

er =

1 +ν ((1 − ν )σ r − νσ ϕ ) = − ν (1 + ν ) ε 0 E 2 Y 2Y

(57)

and

where Y is Yang’s modulus and ν is Poisson’s ratio of the metallic electrode plate. Figure 4 indicates that the complex potential in the u-plane is

U=

K ′( k ) uV K (k )

(58)

Thus, the electric field in the z-plane is [11]

E=−

dU dU du dζ dt K ′( k ) du dζ dt =− =− V dz du dζ dt dz K ( k ) dζ dt dz

(59)

Taking differentiation of Eq.(11) with respect to ζ yields [12]

du 1 1 = = dζ 2 sn (u , k )cn (u , k )dn (u , k ) 2 ζ (1 − ζ )(1 − k 2ζ )

(60)

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In much the same way, from Eqs. (5) and (1) we have

dt dζ (1 − α ) (1 + α ) = and =− 2 dz dt 2 (t − α )

2π z

π ϕ

π π ⎛ ϕ ϕ ⎜ ϕ z ( r1 + l1 ) + ( r2 + l2 ) ⎜ ⎝

⎞ ⎟ ⎟ ⎠

(61)

Substitution of Eqs. (61),(60) and (1)-(6) into Eq.(59) we obtain, after simplifying

E=−

where the dimensionless factor

Λ=

z

π ϕ

ϕ

Λ

2π VK ′(k ) Λ K (k ) z

(62)

is

π π ⎛ ϕπ ⎞⎛ π ⎞ ⎜ r + (r + l )ϕ ⎟⎜ rϕ + (r + l )ϕ ⎟ 1 2 2 2 1 1 ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ Fi π π π π π π π π π π π π π ⎛ ϕ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎜ 2r1 − (r1 + l1)ϕ + (r2 + l2 )ϕ + 2zϕ ⎟⎜ 2r2ϕ − (r2 + l2 )ϕ + (r1 + l1)ϕ − 2zϕ ⎟⎜(r1 + l1)ϕ + (r2 + l2 )ϕ − 2zϕ ⎟⎜(r2 + l2 )ϕ + (r1 + l1)ϕ + 2zϕ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠

(63)

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

306

Yumin Xiang Notice on the plate AB

z = r exp(iϕ )

( r1 < r < r1 + l1 )

(64)

then π π ⎛ ϕπ ⎞⎛ π ⎞ ⎜r1 + (r2 + l2 )ϕ ⎟⎜r2ϕ + (r1 + l1)ϕ ⎟ ⎜ ⎟⎜ ⎟ r ⎝ ⎠⎝ ⎠ ΛAB = − π Fi π π π π π π π π π π π π π ϕ ⎛ ϕ ⎞ ⎞⎛ ⎛ ⎞ ⎛ ⎞ ⎜2r1 − (r1 + l1)ϕ + (r2 + l2 )ϕ − 2rϕ ⎟⎜2r2ϕ − (r2 + l2 )ϕ + (r1 + l1)ϕ + 2rϕ ⎟⎜(r1 + l1)ϕ + (r2 + l2 )ϕ + 2rϕ ⎟⎜(r2 + l2 )ϕ + (r1 + l1)ϕ − 2rϕ ⎟ ⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎜ ⎠ ⎠⎝ ⎠⎝ ⎠⎝ ⎝ π ϕ

(65) Substituting Eqs. (63), (64) and (65) into Eqs.(56) and (57), we achieve the deformation distribution on the metallic plate AB 2 2π 2ε 0 (1 − ν 2 )V 2 K ′2 ( k ) Λ AB ( r1 < r < r1 + l1 ) eϕ = Y K 2 (k ) r2

(66)

2 2π 2ε 0ν (1 + ν )V 2 K ′2 ( k ) Λ AB ( r1 < r < r1 + l1 ) er = − Y K 2 (k ) r2

(67)

Normalizing

and

(67)

with

2π 2ε 0 (1 − ν 2 )V 2 K ′2 ( k ) Y K 2 ( k ) l12

and

2π 2ε 0 ν (1 + ν )V 2 K ′ 2 ( k ) respectively, the same result is defined as dimensionless strain Y K 2 ( k ) l12

0.14 0.12 0.1

∼ σ

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−

Eqs.(66)

0.08 0.06 0.04 0.02 0.2

0.4

0.6

0.8

1

( r-r1 ) / l (a)

r1 : l1 : r2 : l 2 = 2.2 : 1.0 : 2.0 : 0.95 Figure 10. Continued on next page.

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The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

0.3

∼ σ

0.25 0.2 0.15 0.1 0.05 0.2

0.4

0.6

0.8

1

( r-r1 ) / l (b)

r1 : l1 : r2 : l2 = 1.95 : 1.0 : 2.0 : 0.95

2.5

∼ σ

2 1.5 1 0.5

0.2

0.6

0.8

1

( r-r1 ) / l r1 : l1 : r2 : l2 = 1.70 : 1.0 : 2.0 : 0.95

3 2.5

∼ σ

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(c)

0.4

2 1.5 1 0.5 0.2

0.4

0.6

0.8

1

( r-r1 ) / l (d)

r1 : l1 : r2 : l2 = 1.45 : 1.0 : 2.0 : 0.95 Figure 10. Continued on next page.

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

307

308

Yumin Xiang

0.012 0.01

∼ σ

0.008 0.006 0.004 0.002 0.2

0.4

0.6

0.8

1

( r-r1 ) / l (e)

r1 : l1 : r2 : l2 = 1.2 : 1.0 : 2.0 : 0.95

Figure 10. The curves of

~ e

versus

( r − r1 ) / l1

with

ϕ = 0.5 .

l2 e~ = 12 Λ2AB ( r1 < r < r1 + l1 ) r

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

e versus ( r − r1 ) / l1 are plotted in Figure 10 (a)- (e) with The five curves of ~

(68)

ϕ = 0.5 .

The conditions of r1 : l1 : r2 : l 2 are 2.2 : 1.0 : 2.0 : 0.95 for (a), 1.95 : 1.0 : 2.0 : 0.95 for (b), 1.70 : 1.0 : 2.0 : 0.95 for (c), 1.45 : 1.0 : 2.0 : 0.95 for (d), and 1.2 : 1.0 : 2.0 : 0.95 for (e), respectively. The maximum strain area on the electrode plate could be obtained by letting the denominator of the square root in Λ AB be zero. From Eq. (65) the possible solutions satisfy π π π π π π π π π ⎛ Fiϕ ⎞ ⎛ Fi ⎞ ⎛ ⎞ ⎜2r1 − (r1 + l1 ) ϕ + (r2 + l 2 ) ϕ − 2r ϕ ⎟ = ⎜ 2r2ϕ − (r2 + l2 ) ϕ + (r1 + l1 ) ϕ + 2r ϕ ⎟ = ⎜(r2 + l2 ) ϕ + (r1 + l1 ) ϕ − 2r ϕ ⎟ = 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(69) Scrutinizing Eq.(69) it is found the variation of the maximum strain area with the plate motion. When r1 + l1 > r2 + l 2 , only the maximum strain area (MSA) No.1 appears on AB. Supposing the plate moves along BO to O, MSA No.1 moves towards B. When r2 + l2 = r1 + l1 , MSA No.1 and MSA No.2 locate at the two terminals B and A respectively. MSA No.2 moves on AB and MSA No.1 moves out AB as r1 + l1 < r2 + l 2 . Gradually decreasing r1 may cause MSA No.3 and MSA No.2 on AB simultaneously. Further reduction of r1 produces that MSA No.2 leaves from AB and MSA No.2 drifts to B. Finally, MSA No.3 moves out AB. Figure 10 (a)-(e) do reflect the above analyses visually.

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

The Electric Force on the Moving Electrode of an Inclined Plate Capacitor

309

8. Conclusion The electric force exerts on the moving electrode of a charged inclined plate capacitor and causes deformation of the plate. Computation of the force is complicated due to the diverse dimension, position and motion of the electrode plates. However, the conformal mapping and the virtual work principle provide a concise way to calculate. The exact result is expressed with the elliptic functions. The parallel plate capacitor only is the special case as the inclined angle approaches zero. Based on the general result the electric force on the moving plate and the strain distribution on the electrode are analyzed. The function curves of numerical simulation are plotted..

Acknowledgments This research is supported by the Education Commission of Sichuan Province of China.

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References [1] Luo A J C, Wang F Y. Nonlinear Dynamics of a Micro-Electro-Mechanical System with Time-Varying Capacitors. J. of Vibration and Acoustics 2004; 1: 77-83. [2] Xiang Y. The electrostatic capacitance of an inclined plate capacitor. J. of Electrostatics, in press. [3] Lang S. Complex analysis , 2ed ed. New York: Springer-Verlag; 1985. [4] Patric D V. Elliptic functions and elliptic curves. London: Cambridge; 1973. [5] Xiang Y, Lin W. A study of electrostatic force on the walls of N-regular polygonmultifin line. J. of Electrostatics 2001; 50: 119-28. [6] Lin W, Xiang Y. Electrostatic force on the walls of a rectangular coaxial line. J. of Electrostatics 1998; 43: 275-83. [7] Bowman F. Introduction to elliptic functions. New York: Dover; 1961. [8] Luo A J C, Wang F Y. Chaotic motion in a micro-electro-mechanical system with nonlinearity from capacitors. CNSNS 2002; 7: 31-49. [9] Xiang Y, Lin W. The electrically induced stress in the dielectric of a transmission coaxial line. CNSNS 2005; 10: 157-168. [10] Timoshenko S P, Goodie J N. Theory of elasticity, 3rd ed. New York: McGraw-Hill; 1970. [11] Shen C, Kong J K. Applied electromagnetism, New York: PWS Engineering; 1987. [12] Greenhill A G. The application of elliptic functions. New York: Dover; 1959.

Reviewed by Albert C. Luo. He has been an Editor for the journal Communications in Nonlinear Science and Numerical Simulation since 2002, an Editor for two book series on Nonlinear Science and Complexity (Elsevier) since 2004 and one book series on Complexity, Nonlinearity and Chaos (World Scientific) since 2007. He has been an Associate Editor for ASME Journal of Computational and Nonlinear Dynamics since 2006. He has been a member of editorial board for IMeCh E Part K Journal of Multi-body Dynamics since 2002 and

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

310

Yumin Xiang

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Journal of Vibration and Control since 2005. Dr. Luo also organized over 18 international symposiums and conferences on Dynamics and Control, including 14 ASME Symposiums and conferences. He served as the Technical Program Chair for ASME 2005 IDETC. He joined the faculty of the Department of Mechanical and Industrial Engineering at Southern Illinois University Edwardsville in 1998. (http://www.asme.org/Governance/Honors/ Fellows/ Fellows_Listing.cfm)

Micro Electro Mechanical Systems (MEMS): Technology, Fabrication Processes and Applications : Technology, Fabrication Processes and

In: Micro Electro Mechanical Systems Editors: B. Ekwall and M. Cronquist, pp. 311-315

ISBN: 978-1-60876-474-7 © 2010 Nova Science Publishers, Inc.

Chapter 11

PORTABLE DIAGNOSTIC TECHNOLOGIES FOR RESOURCE POOR ENVIRONMENTS Hideyuki F. Arata Curie Institute, Paris, France and Japan Society for the Promotion of Science, Japan

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Abstract Diagnostic technologies currently used in developing countries are cumbersome and unsuitable for the uses in low-resource environments ; this has been one of the major problems in improving the global health. These technologies must be operated by nonresearchers and work with poor resources. Although, MEMS-based on-chip technologies have a potential to meet such requirements, yet few literatures report technical progresses aiming those applications so far. Worse still, use of current “Lab-on-a-chip” devices is limited to the environments, such as in research laboratories, where they have huge equipments; these devices are, so to speak, “Chip-in-a-lab” devices; they are not yet appropriate for uses in the extreme resource-poor settings in developing countries. In this commentary, trends in this research field are explained. Instead of reviewing comprehensive literatures in this field, general ideas of portable diagnostic microchip, required technologies and possible directions for further progress are discussed.

Introduction In developing countries, only the diagnostic technologies that are cumbersome and unsuitable for uses in low-resource settings are currently available; this has been one of the major issues in improving global health. Most of the developing countries now urgently need to incorporate genetic approaches, including DNA diagnosis, into their health services [1]. Simple, accurate and stable diagnostic tests are essential to improve global health, but they are usually unavailable or inaccessible to those who need them in developing countries. However, many of the research studies in genomics and other related biotechnologies, such as modified molecular technologies for diagnosis of infectious diseases, recombinant technologies to develop vaccines against infectious diseases, and technologies for drug and

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vaccine delivery systems, have been concerned with resource-rich environments in industrialized nations, and yet a limited number of studies have shown that these technologies could help improving health in developing countries [2]. Although affordable and simple diagnosis of infectious diseases is one of the most promising molecular biotechnologies, only PCR, monoclonal antibodies, and recombinant antigens are already in use or being tested [2]. This technological challenge for global health has attracted intensive social interests [3] and fundings [4,5] only at the beginning of this century. These technologies must be portable, easy to be operated by non-experts and work with poor resources rather than in research laboratories. On the other hand, so-called top-down technologies have enabled us to manufacture and fabricate structures even smaller than micrometer scale [6]. The MEMS (Micro-ElectroMechanical-Systems) technology has been developed by applying semiconductor microfabrication technologies to make three-dimensional microstructures and mechanical systems in the late 1980s [7]. The MEMS technology offers the advantages in batch fabrication of numbers of devices as well as an ability to integrate multiple functional units in a small area, which is important for developing smart and sophisticated devices. Multiple functional units such as microfluidics, electrical circuits, and optical systems can be integrated on a microchip to realize µTAS (Micro-Total-Analysis-Systems) [8-11]. It allows parallel processing of rapid and highly sensitive biological or chemical analyses with a small amount of samples thanks to the merits of miniaturization. Handling a small amount of samples in the microscale realizes short diffusion distances, high surface-to-volume ratio, and small heat capacity [12]. Small volumes also reduce the time taken to synthesize and analyze products, and the unique behaviors of liquids at the microscale allow better control of handling molecular concentrations and interactions. Material costs and the amount of waste can also be reduced. Therefore, top-down technologies such as MEMS or µTAS have a potential to meet the requirements for improving the global health discussed above. Importance and impacts of the miniaturization of analytical procedures for point-of-care (POC) immunoassays [13] or diagnostics [14], applying top-down technology, such as microchip, microarrays and biochips, has been intensively discussed from the beginning of this century [15,16]. Hence, several literatures have reported technological progress aiming at this application. Integration of various analytical functions was applied to cell analyses and clinical diagnoses utilizing human serum samples [17]. Some examples of complete microfluidic devices which can function entirely without any moving parts and external power sources have been reported [18]. Microfluidic diagnostic systems which demonstrate practical integration of sample preparation, analyte enrichment, and optical detection have now become possible [19]. However, in spite of the technological progress in current “Labon-a-chip” devices, their usage is still limited to the environments where they have enough equipment such as in research laboratories; these devices are, so to speak, “Chip-in-a-lab” devices. They are not yet appropriate for the use in extreme resource-poor settings of developing countries [20]. Possible solutions and directions to overcome these limitations must be considered.

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Possible Future Trends Techniques using microfluidics and micro/nano particles might be two major applicable techniques to overcome the limitations discussed above. Because they can be operated without cumbersome equipments such as electricity, optical resources, and computer control, they have a higher potential for realizing portable diagnostic chip utilizing advantages of simple operation and self assembly. This chip must analyze biomaterial, such as a patient’s blood, preferably without any treatment. Output must be simple such as Yes/No signal (Fig. 1). Magnetic particles in the fluidic channel can be manipulated from outside of the chip even by classical permanent magnet. Using pillars biofunctionalized magnetic beads [21] to immobilize biomaterials on a specific area in the channel can be one of the applicable solutions. This also allows us to perform the following chemical processes in series. The pillars of magnetic beads can be built and disassembled electrically using an electric magnet constructed outside of the channel. In addition, pillars of magnetic beads can be disassembled easily to collect materials for following processes. Separation between magnetic beads and samples will be a subsequent issue. Designing an appropriate geometry of micro fluidics devices enables self sorting or self aligning of micro- nano- particles [22,23]. In a narrow channel, a sample in the water solution can automatically flow into the hydrophilic channel by its surface tension. These can be controlled by geometry of the channel and surface treatments, thus enabling us to program automatic flow and initiation of reaction. To design the geometries and the surface conditions of fluidics, a numerical simulation might play an important role because in such a small scale, not all the effects and phenomena are directly measurable. The assays using microparticles and magnetic particles are also appropriate techniques which can be incorporated into the micro-fluidic-systems. Colloidosomes [24] and encapsulation technique by droplets [25] can be suitable techniques for on-chip applications in pharmaceutics and diagnosis because their size and characteristics are somewhat designable by currently developed techniques. Using magnetic microparticles enables additional external control without changing the geometry of the fluidic system and it is also compatible with most of the materials generally used to manufacture micro-fluidic-devices. Even a typical permanent magnet, such as a neodymium magnet, of around one centimeter in size is appropriate to attract and manipulate magnetic beads in a channel.

Figure 1. General idea of a portable diagnostic microchip.

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Microdevices by those techniques can be operated with minimum equipments available in developing countries. Therefore, they have a higher potential for portable diagnostic chip utilizing advantages of simple operation and self-assembly. Those techniques may lead us to overcome the limitations discussed in the first paragraph.

Outlook The fundamental features of microchip technology are well fitted for POC diagnostic devices. Spontaneously, they have been applied to develop portable chips aiming for POC diagnostics. However, most of the devices developed so far seem to be focusing on miniaturization and on-chip integration of currently existed techniques in the macroscale. A device requires more cumbersome facilities as integration advances. This is contradictory to the requirements of POC even if the analytic accuracy advances. It seems that researches in microfluidic POC devices are becoming too application oriented, though, basic science still remains to be studied. One of the breakthroughs for POC microchip may come from a simple device of self operation such as microfluidics, which enables programmable sample handlings [22]. Because the POC market is estimated to be over 10 billion U.S. dollars, with doubledigit growth in some areas such as cardiac and infectious diseases [26], this field seems to continue growing and attracting social and industrial interests.

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References [1] [2] [3] [4] [5] [6] [7]

A. Alwan and B. Modell, Nat. Rev. Gen., 2003, 4, 61. A. S. Daar, et al., Nat. Gen., 2002, 32, 229. T. Acharya, A. S. Daar and P. Singer, Nat. bioth., 2003, 21, 1434. H. Varmus, et al., Science, 2003, 302, 398. D. Mabey, et al., Nat. Rev. MicroBiol., 2004, 2, 231. M. T. Bohr, IEEE Trans. Nanotech., 2002, 1, 56. For an overview, see W. S. Trimmer, Micromechanics and MEMS—Classic and Seminal Papers to 1990. Piscataway, NJ: IEEE Press, 1997. [8] D. R. Reyes, D. Iossifidis, P.-A. Auroux, and A. Manz, Anal. Chem, 2002, 74, 2623. [9] P.-A. Auroux, D. Iossifidis, D. R. Reyes, and A. Manz, Anal. Chem., 2002, 74, 2637. [10] G.M. Whitesides, Nature, 2006, 442, 368. [11] H. Craighead, Nature, 2006, 442, 387. [12] K. Sato, et al., Advanced Drug Delivery Reviews, 2003, 55, 379. [13] P. von Lode, Clinical Biochemistry, 2005, 38, 591. [14] M. Toner and D. Irimia, Annu. Rev. Biomed. Eng., 2005, 7, 77. [15] L. J. Kricka, Clinica Chimica Acta, 2001, 307, 219. [16] P. Pavlickova, E. M. Schneider and H. Hug, Clinica Chimica Acta, 2004, 343, 17. [17] K. Sato, K. Mawatari and T. Kitamori, Lab Chip, 2008, 8, 1992. [18] B. Weigl, et al., Lab Chip, 2008, 8, 1999. [19] F. B. Myers and L. P. Lee, Lab Chip, 2008, 8, 2015. [20] C. D. Chin, et al., Lab Chip, 2007, 7, 41. [21] P. S. Doyle, et al., Science, 2002, 295, 2237.

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Portable Diagnostic Technologies for Resource Poor Environments [22] D. Di Carlo, et al., PNAS, 2007, 104, 18892. [23] H. Maenaka, et al., Langmuir, 2008, 24, 4405. [24] JW. Kim, et al., Nano Letters, 2007, 7, 2876. [25] S. Koster, et al., Lab chip, 2008, 8, 1110. [26] S. Sia and L. Kricka, Lab Chip, 2008, 8, 1982.

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Reviewed by Dr. R. Yokokawa, Kyoto University, Japan.

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In: Micro Electro Mechanical Systems ISBN: 978-1-60876-474-7 c 2010 Nova Science Publishers, Inc. Editors: B. Ekwall and M. Cronquist, pp. 317-373

Chapter 12

B ALLISTIC T RANSPORT THROUGH Q UANTUM W IRES AND R INGS Vassilios Vargiamidis1,∗ and Vassilios Fessatidis2,† 1 Department of Physics, Aristotle University, GR-54124 Thessaloniki, Greece 2 Department of Physics, Fordham University, Bronx, NY, USA

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Abstract Phase-coherent electron transport through quasi-one-dimensional systems has developed into a very active and fascinating subfield of mesoscopic physics. We present a review of this development focusing on ballistic conduction through quantum wires (or constrictions) and one-dimensional open rings. In quantum wires the electron conductance versus Fermi energy is quantized as a consequence of the reduced dimensionality and the subsequent quantization of transverse momentum. The presence of scatterers in otherwise “clean” wires can strongly suppress the quantum conductance, and can generate sharp resonances (which are due to quasibound states) if the scattering potential is attractive. These resonances can be of the Fano or Breit-Wigner type, depending on the size or/and strength of the scattering potential. Thermal effects are also considered. The scattering approach is briefly discussed in order to derive the Landauer formula, which is the basic tool for calculating the conductance of a mesoscopic sample. Scattering theory in ballistic quantum wires is formulated in terms of the Lippmann-Schwinger equation while the Feshbach coupled-channel theory is employed in order to treat Fano resonances. The occurrence of Fano resonances in strictly one-dimensional mesoscopic open rings is discussed in the last part of this review.

PACS number: 72.10.Fk; 73.63.Nm

1.

Introduction

Since the 1980s advances in the growth techniques and new electronic materials developed therefrom have provided almost defect-free electronic devices, which have dimensions ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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in one or more directions on the quantum scale [1]. New quantum regimes governing such systems of lower dimensionality have led to novel electronic properties with potential applications. Quantum wells, wires, and dots, which have been implemented in the terminology of condensed-matter physics, indicate not only different dimensionality but also exhibit dramatically different electronic properties. In particular, the electronic transport properties in lower dimensionality have several important features, which have attracted a great deal of both experimental and theoretical interest. One example of this is the progress made for the resonant electronic transport in such systems. In electronic transport studies, if the size of the sample (or device) is smaller than the phase breaking length, electrons have a well-defined phase throughout the device, even though they may experience elastic scattering. The small size of these devices allows the observation of important quantum interference effects, such as the Aharonov-Bohm effect [2]. Numerous publications on this type of transport have appeared, thus contributing to a field called mesoscopic physics, a term that indicates a new length scale for physical events in the borderline between macroscopic and microscopic. As early as 1957, Landauer [3] proposed that electronic conduction in the mesoscopic regime can be viewed as a scattering process, and he related the transmission probability with the quantum unit of conduction e2 /h. His views made a great impact on the physics of mesoscopic systems. There have been efforts to obtain the Landauer formula from linearresponse theory or the Kubo formula [4, 5]. Even though these theories have completely different appearances, in some cases they give identical results. In Sec. II, we briefly review the scattering approach and derive Landauer’s formula. The scattering approach is also suitable for discussing other phenomena, such as current correlations and shot noise in mesoscopic conductors [6]. In 1988, the experiments performed independently by van Wees et al. [7] and Wharam et al. [8] were a breakthrough in the field of quantum ballistic transport in a quantum point contact (QPC) in a two-dimensional electron gas (2DEG). Using high-mobility GaAsAl1−xGax As heterojunctions and the split-gate technique, they imposed a small constriction on the sample. A channel was obtained from this constriction by applying a negative bias to the split gate, and thus by causing the depletion of electrons beneath the gate. In these experiments the width, W , of the constriction is in the range of the Fermi wavelength λF , whereby quantum size effects become relevant, while the length of the constriction is smaller than the electron mean free path. The conductance G as a function of the gate voltage (or, equivalently, as a function of the width of the constriction) was found to rise in a long series of rather sharp steps of magnitude e2 /h (per spin). This observation was interpreted as the quantization of conductance, which occurred at very low temperatures (∼ 0.6K). The observed step structure in the conductance curve can easily be visualized in terms of a new conduction channel opened by a subband dipping into the Fermi level [7]. However, questions as how the temperature and the geometry of the constriction affect the quantization of the conductance required further analysis [9, 10, 11]. For instance, it was shown [9] that (for long constrictions), a small temperature averages the narrow resonances producing steps. The resonance structure, which was not observed in the experimental data, was thus attributed to the finite temperature effects. Furthermore, the role of the impurities is also crucial. In particular, it was shown that a repulsive impurity can strongly reduce the

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conductance [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], while an attractive impurity generates sharp Breit-Wigner or Fano resonances [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] (i.e., dips and peaks). These resonances correspond to resonant reflection and transmission due to quasibound states in the impurity potential. As mentioned above, the presence of an attractive impurity in a constriction may generate Fano resonances in the transmission probability. A bound state in the attractive impurity interacts with the continuum and becomes a quasibound (resonant) state. The interference between direct (nonresonant) transmission and transmission via a quasibound state gives rise to an asymmetric Fano resonance. The Fano effect [37] is a ubiquitous phenomenon and has been observed in a large variety of experiments including atomic photoionization [38], neutron scattering [39], Raman scattering [40], optical absorption [41], and transport through mesoscopic systems with embedded quantum dots [42, 43, 44, 45, 46]. In particular, the Fano effect in transport through mesoscopic systems is of great interest both as a basis for the creation of new resonant nanoelectronic devices and for revealing the quantum-mechanical wave nature of the charge carriers. The Fano effect occurs also in some particular 1D systems; namely, in electronic transport through 1D Aharonov-Bohm rings connected to current leads [47, 48, 49, 50, 51, 52]. In these systems, the electron motion between the junctions is treated as purely onedimensional, i.e., no subband structure is taken into account and therefore no intersubband interaction occurs. Thus the usual interpretation of the Fano effect as being due to the coupling of a bound state, splitting off from a nonpropagating subband, with the continuum of states, does not apply in these systems. Our intention here is to present a review on electronic ballistic transport through quantum wires with scatterers and through 1D open rings. Certain issues were left untouched deliberately; for example, the effect of an external magnetic field, conductance fluctuations, and transmission through quantum waveguides with resonantly coupled cavities are not discussed at all. An attempt to explain these effects would double the size of this Review. On the other hand, since many important ideas are present here, this Review might be useful to those who wish to enter the field as well as to experts who wish to look for the results concerning some particular phenomenon. For completeness, we begin with a discussion of the scattering approach (Sec. II) and the derivation of Landauer’s formula. This is a fully quantum-mechanical theory which applies to phase-coherent transport. In Sec. III, the scattering theory is formulated by means of the Lippmann-Schwinger equation [53] in the case of a general static scattering potential in a quantum wire. This provides an efficient method to calculate transmission amplitudes. In Sec. IV, the conductance through various types of scattering potentials is obtained via Landauer’s formula [3, 54, 55]. In particular, short- and finite-range scattering potentials of either repulsive or attractive type are considered and their effects on the conductance are discussed. The lateral confinement of electrons is taken into account by considering either hard- or soft-wall confining potentials. Various important features of electron transport as well as the differences in the behavior of the conductance through short- or finite-range scatterers are discussed. A detailed account of the transmission resonance structure, in particular the Fano resonance structure, is given in Sec. V by means of the Feshbach coupledchannel approach [56, 57]. The Feshbach theory provides microscopic expressions for all line shape parameters and is therefore particularly suitable for treating resonance phenom-

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ena. The most important aspects of the resonant structure are discussed to some extent; namely, the transformation of a Fano resonance into a Breit-Wigner antiresonance, the reversal of the Fano line shape, and the temperature dependence. In Sec. VI, the resonant behavior of the transmission probability through a 1D mesoscopic ring connected to current leads is discussed. Some concluding remarks are presented in Sec. VII.

2.

The Scattering Approach

The scattering approach (also referred to as Landauer approach) relates transport properties of the system to its scattering properties, which are assumed to be known from a quantum-mechanical calculation. The theory applies to phase-coherent transport in the stationary regime and is based on second quantization [58]. Using this approach we derive the Landauer formula. We consider a sample (i.e., a scattering region) connected by ideal leads to two reservoirs, which will be refereed to as “left” (l) and “right” (r). The reservoirs are assumed to be very large and form free electron gases (see Fig. 1). They are characterized by a temperature Tl,r and a chemical potential µl,r . The distribution functions of electrons in the reservoirs are then Fermi distribution functions 1 (α = l, r). (1) fα (E) = exp [(E − µα ) /kB Tα] + 1

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In the leads, no interactions or scattering by impurities take place. It is also assumed that: i) no inelastic scattering occurs in the sample, and ii) the reservoirs are so large that their chemical potentials do not change if electrons are transferred from one to the other.

Figure 1. Example of a two-terminal scattering problem for one propagating channel. The left and right reservoirs are free-electron gases and characterized by temperatures and chemical potentials Tl, µl and Tr , µr , respectively. The reservoirs are connected to the sample through the leads. In the leads the longitudinal motion of electrons is free and characterized by the continuous wave vector kx . Therefore, we introduce the longitudinal energy Ex = ~2 kx2 /2m as a quantum number. The transverse motion however is confined, and therefore energy in the transverse dimension is quantized and described by the discrete index n (which corresponds to transverse energies En ). It is natural to describe the states by the quantum numbers kx and n. These states are referred to as transverse (quantum) channels. For a given total energy E = Ex + En , the longitudinal energy Ex must be positive and therefore only a finite number of transport channels exists.

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Within the scattering region (i.e., within the sample), the scattering process can be described by a scattering matrix S,   r− t+ S= . (2) t− r+ The scattering matrix relates outgoing current amplitudes to incoming amplitudes. Its matrix elements depend on the total energy E. Here, r− and r+ describe electron reflection back to the left and right reservoirs, respectively. The off-diagonal elements t− and t+ , describe electron transmission to the right and left of the sample, respectively. Here, the signs − and + refer to an incident wave from the left or from the right of the sample. We use second quantization and introduce creation and annihilation operators [58] of electrons in the scattering states. For simplicity we restrict our discussion to the case of ˆ†l (E) and a one transport channel. We introduce the operators a ˆl (E), which create and annihilate electrons with total energy E in the left lead which are incident upon the sample. We also introduce operators ˆb†l (E) and ˆbl (E), which create and annihilate electrons in the outgoing states. These operators obey anticommutation relations n
o
al E 0 = δ E − E 0 , (3) a ˆ†l (E) , ˆ
 al E 0 = 0, a ˆl (E) , ˆ n
o † † 0 al E = 0. a ˆl (E) , ˆ

(4) (5)

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†

Similarly, we introduce operators ˆ ar (E) and ˆ ar (E) which create and annihilate electrons in † ˆ the incident states in the right lead, and br (E), ˆbr (E) for outgoing states. The annihilation operators for the outgoing states are related linearly to the annihilation operators for the incoming states via the scattering matrix. Thus, in lead α, where α = l, r X ˆbα (E) = Sαβ (E) a ˆβ (E) . (6) β

The creation operators a ˆ† and ˆb† obey the same relation but with S †. Flux conservation in the scattering process implies that the matrix S is unitary, and provides a unitary transformation of the a ˆ operators into the ˆb operators. In particular, unitarity of the scattering † matrix, S S = SS † = 1, yields the following relations for the reflection and transmission coefficients, (7) |r− |2 + |t− |2 = 1, |r+ |2 + |t+ |2 = 1,

(8)

∗ t− = 0. t∗+ r− + r+

(9)

Eqs. (7) and (8) express the flux conservation. The current density operator in the left lead is expressed in a standard way, h i ˆjl (x, t) = ~ ψˆ† (x, t) ∇ψˆl (x, t) − ψˆl (x, t) ∇ψˆ† (x, t) , l 2mi l

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(10)

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where the field operators ψˆl and ψˆl† are defined as ψˆl (x, t) = and ψˆl† (x, t) =

Z

dE [2π~υl (E)]1/2

Z

dE [2π~υl (E)]1/2

  e−iEt/~ φl (y) a ˆl eikl x + ˆbl e−ikl x ,

(11)

  eiEt/~φ∗l (y) a ˆ†l e−ikl x + ˆb†l eikl x .

(12)

Here y is the transverse coordinate and x is the coordinate along the leads. Also, φl (y) is the transverse wave function, and we have introduced the wave vector, kl = [2m (E − El )]1/2 /~, and the velocity of electrons υl (E) = ~kl /m in the channel. Note also that 1/2π~υl (E) is just the 1D density of states of the channel. The total current operator in the left lead is obtained by integrating the current density, given in Eq. (10), over the cross section,     Z ~e ∂ ˆ ∂ ˆ† † ˆ ˆ ˆ Il (x, t) = dy ψl (x, t) ψl (x, t) − ψ (x, t) ψl (x, t) . (13) 2mi ∂x ∂x l

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Inserting Eqs. (11) and (12) into Eq. (13) results in a cumbersome expression, which can be considerably simplified [58] if we note that the velocities vary with energy slowly, and therefore one can neglect their energy dependence. Then, Eq. (13) can be expressed in a simpler form as Z h

i e 0 ˆ dEdE 0ei(E−E )t/~ ˆ a†l (E) a ˆl E 0 − ˆb†l (E) ˆbl E 0 . (14) Il (t) = 2π~ ˆl (E) = n ˆ+ Note that a ˆ†l (E) a l (E) is the operator of the occupation number of the incident ˆ− electrons in lead l. Similarly, ˆb†l (E) ˆbl (E) = n l (E) is the operator of the occupation number of the outgoing electrons in lead l. Using Eq. (6) we can express the current in † ˆl alone, terms of the operators a ˆl and a e X Iˆl (t) = 2π~

Z


†
0 dEdE 0ei(E−E )t/~Aαβ l, E, E 0 a ˆα (E) a ˆβ E 0 .

(15)

αβ

The indices α and β in Eq. (15) label the reservoirs and may assume values l or r. The matrix A is defined as

∗ (E) Slβ E 0 . Aαβ l, E, E 0 = δαlδβl − Slα

(16)

ˆα (E). where Slα (E) is the element of the scattering matrix relating ˆbl (E) to a In order to evaluate the average current from Eq. (15), we first note that for a system at thermal equilibrium the quantum statistical average of the product of an electron creation operator and annihilation operator of a Fermi gas is D
E
ˆβ E 0 = δαβ δ E − E 0 fα (E) . (17) a ˆ†α (E) a

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Using Eqs. (15) and (17) and taking into account the unitarity of the scattering matrix S, the average current can be expressed as Z   e X ∗ hIl i = dE δβl − Slβ (E) Slβ (E) fβ (E) 2π~ β Z e dE [(1 − R) fl (E) − T fr (E)] = 2π~ Z e = dE T (E) [fl (E) − fr (E)] , (18) 2π~ ∗ r and T = S ∗ S = t∗ t are the reflection and transmission where R = Sll∗ Sll = r− − − − lr lr probabilities, respectively. If the chemical potentials differ only by a small amount, we can expand the distribution functions as fl (E) − fr (E) ' − (∂f /∂E) (µl − µr ). Then, for a small applied voltage V = (µl − µr ) /e, the conductance is obtained as   Z hIl i e hIl i e2 ∂f = = dE − G= T (E) . (19) V µl − µr 2π~ ∂E

Eq. (19) has been successfully applied to a wide range of problems from ballistic transport to the quantum Hall effect. In the zero-temperature limit, −∂f /∂E = δ (E − EF ), and Eq. (19) reduces to e2 T (EF ) , (20) G= 2π~ which is the one-channel Landauer formula [54, 55, 59] establishing a relation between the scattering matrix and the conductance. The above discussion can be generalized to treat more than one transport channels. The multi-channel generalization of the Landauer formula reads

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nc e2 X Tnm , G= 2π~

(21)

n,m=1

where Tnm is the transmission probability for a carrier incident in channel n in the left lead to transmit to channel m in the right lead. In the case of ballistic transport, i.e., in the case of no channel interaction, Tnm = δnm , and Eq. (21) reduces to e2 , (22) 2π~ which is the total conductance (due to all occupied subbands up to the Fermi level), and nc is the total number of transport channels. Note that Eq. (22) expresses the conductance per spin. The quantity e2 /π~ = 2e2/h (2 for spin) has been known as the quantum unit of conduction. In the next section we discuss the scattering processes in a Q1D wire by employing the Lippmann-Schwinger equation, which provides a nice method for calculating the scattering amplitudes. The scattering amplitudes are then obtained for several examples of model scatterers in Sec. IV, where we also present the conductance through these scattering potentials. As will be shown, the quantization of conductance is distorted owing to scattering by a single impurity in the Q1D wire. The extent of deviation from quantized values depends on the strength and size of the scattering potential. G = nc

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3.

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Scattering Theory in Ballistic Q1D Wires

One of the differences between Q1D and three-dimensional (3D) scattering is the subband structure that exists in a quantum wire due to the lateral confinement. Motivated by the process of multi-channel scattering in 3D, we can also consider the quantum wire as a multi-channel system if we treat the different subbands as “channels”. The quantummechanical calculation of the scattering amplitudes in a Q1D wire is formulated below via the Lippmann-Schwinger equation [53], which makes use of the Green’s function of the “clean” (unperturbed) wire.

3.1.

Scattering and the Lippmann-Schwinger Equation

We consider a uniform quantum wire connected to reservoirs. Electrons in the wire are confined along the y-direction (transverse direction) but are free to propagate along the xdirection, as shown in Fig. 2. The cross section is uniform along the wire. The Hamiltonian can be written as H = H0 + V, (23)

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where V = V (x, y) is the scattering potential of any defects or impurities in the wire. The unperturbed Hamiltonian H0 is the sum of the kinetic energy of the electron plus the confining potential Vc (y), p2 H0 = + Vc (y) , (24) 2m where m is the effective mass of the electron.

Figure 2. Schematic illustration of a uniform quantum wire with a scatterer. We assume infinite leads and hard-walls at y = 0 and y = W . The incident wave is partly transmitted and partly reflected by the scattering potential V (x, y). If the wire were “clean” (i.e., if the scattering potential E were absent, E V = 0), an energy (0) (0) eigenstate would satisfy the eigenvalue equation H0 ψp = E ψp with corresponding unperturbed eigenstates given by 1 (0) eipx x/~ φn (y) , ψp (x, y) = √ 2π~

(25)

where φn (y) are the confinement modes (or quantum channels) of the wire. The corresponding energy eigenvalues of H0 are given as E=

p2x + En , 2m

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where px = ~kx has a continuous spectrum, n represents the subband index, and En is the subband energy. The spectrum of H0 is degenerate, which means that there are more than one eigenstates all with the same energy eigenvalue. In particular, for a given total energy E, a number of channels have energies smaller than E (i.e., En < E). The number of those channels is the degeneracy of the energy eigenvalue E. These channels are called “open” because they can propagate along the wire. All higher channels with En > E will be referred to as “closed” because they cannot propagate (see Fig. 3).

Figure 3. The energy levels of a uniform quantum wire of width W and hard-wall confining potential. For a given total energy, E = ~2kn2 /2m + En , only a finite number of transport channels exists. Here, En = n2 E1 where E1 = ~2π 2 /2mW 2. Higher evanescent modes (kn = iκn ) can serve as intermediate states for electrons. E (0) In the presence of the scattering potential, V (x, y), the energy eigenstate ψp of the “clean” wire is modified. However, since we are dealing with elastic scattering, the energy eigenvalue E will remain also energy eigenvalue of the full scattering Hamiltonian, i.e., Eq. (23). The Schr¨odinger equation we have to solve in the presence of V can be written as (E − H0 ) |ψpi = V |ψpi ,

(27)

which is an inhomogeneous equation for |ψp i. Eq. (27) can be solved with the retarded Green’s function defined by

~2 (3) δ x − x0 . (E − H0 ) G(0) x, x0; E = 2m The solution to the inhomogeneous equation, Eq. (27), can then be written as E E E (+) (0) (+) = ψp + G(0)V ψp , ψp

(28)

(29)

which is the Lippmann-Schwinger equation (LSE) of scattering theory. The above solution corresponds to an incident wave plus an outgoing wave traveling away from the scatterer. This is taken into account, in the usual way, by inserting +iε in the Green’s

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(0) function operator which = (E − H0 + iε)−1 . In Eq. (29) note that if appears E E as G (0) (+) → ψp . Multiplying by hx| and using the completeness relation, V → 0 then ψp R 2 0 0 d x |x i hx0| = 1, of the position eigenkets we can rewrite Eq. (29) as Z ED D E (+) (0) (+) ψp (x) = ψp (x) + d2x0 x G(0) x0 x0 |V | ψp . (30)

Here, the matrix element of G(0) is the Green’s function, which can be expressed as 
~2 G(0) x, x0; E = hx| (E − H0 + iε)−1 x0 . 2m

(31)

The matrix element can be evaluated by first using twice the completeness relation, R 2 last d p0 |p0 i hp0| = 1, of the energy eigenkets in order to rewrite it as G(0) (x, x0; E) =

~2 2m

Z

d2p0

Z

−1

d2p00hx|p0i hp0 | (E − H0 + iε) |p00i hp00|x0 i !−1 Z Z 2 p0 ~2 2 0 2 00 0 d p + iε d p hx|p i E − hp0|p00ihp00|x0 i = 2m 2m !−1 Z Z 2 p0 ~2 (0) 2 0 2 00 (0) d p + iε d p ψp0 (x) E − δ (p0x − p00x ) δn0 n00 ψp00 (x0 ) , = 2m 2m (32)

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where we made use of the orthonormality relation of the energy eigenkets, hp0 |p00i = δ (2) (p0 − p00) = δ (p0x − p00x) δn0 n00 . Inserting the explicit form of the unperturbed eigen(0) states, ψp (x), and using the method of residues [20] we finally obtain [60] ∞
X
eikn0 |x−x0 | φn0 (y) φ∗n0 y 0 . G(0) x, y; x0, y 0; E = 2ikn0 0

(33)

n =1

The wave vectors in Eq. (33) are given as kn0 = [2m (E − En0 )]1/2 /~. Assuming a local potential, which satisfies hx0| V (x) |x00i = V (x0) δ (2) (x0 − x00), allows us to write the second matrix element in Eq. (30) as E Z D


(+)
(+) (+) 0 = d2x00 x0 V x00 hx00|ψp i = V x0 ψp x0 . (34) x |V | ψp The integral equation Eq. (30) can now be expressed as Z Z

(+)
2m ∞ 0 ∞ 0 (0) (+) (0) dx dy G x, y; x0, y 0; E V x0 , y 0 ψp x0 , y 0 . ψp (x, y) = ψp (x, y)+ 2 ~ −∞ −∞ (35) (+) In Eq. (35) we note that the wave function ψp (x, y) in the presence of the scatterer is (0) written as the sum of the wave function for the incident wave ψp (x, y) plus a term that represents the effect of scattering. Eq. (35) is valid throughout the whole wire. The physical interpretation of the Green’s function is that it represents the probability amplitude that a

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particle after interacting at (x0, y 0) propagates to (x, y). Substituting Eq. (33) for the Green’s function into Eq. (35) will allow us to obtain the scattering amplitudes for x → ∞ as (+) ψp (x, y)

∞ X 1 ikn x 1 √ eikn0 x φn0 (y) ft (kn0 , kn ) , =√ e φn (y) + 2π 2π n0 =1

where ft (kn0 , kn) are the scattering amplitudes, which are given as Z

(+) 0 0
π 2m 2 0 1 −ikn0 x0 ∗ 0 0 0 0 √ e ft (kn , kn) = d x φ V x , y ψp x , y 0 y n ikn0 ~2 2π E π 2m D (+) 0 |V | ψp = k . n ikn0 ~2

(36)

(37)

In Eqs. (36) and (37) the indices n and n0 characterize the incident and scattered channels. Similarly, in order to obtain the scattering amplitudes in the asymptotic region x → −∞, one can write |x − x0 | = − (x − x0) in the Green’s function of Eq. (33) and then substitute into Eq. (35). In this region, the scattering amplitudes will be denoted as fr (kn0 , kn). In the above, |kn i stands for the state described by the two quantum numbers kx and n, i.e., |kn i = |kx, ni.

3.2.

Conservation of Energy and Scattering Processes

Since we consider elastic scattering, conservation of energy requires that ~2 kn2 0 ~2kn2 + En = + En0 = E, 2m 2m

(38)

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where E is the given total energy and n, n0 label the incident and scattered channel modes, respectively. From Eq. (38), it can be seen that the wave vector of the scattered channel, kn2 0 =

2m (E − En0 ) , ~2

(39)

can be real if En0 < E or imaginary if En0 > E (see Fig. 3). If kn0 is real the scattered channel wave function is of the form eikn0 |x|φn0 (y), which means that the electron travels away from the scattering potential; namely, it is either reflected or transmitted. If kn0 is imaginary (i.e., kn0 = iκn0 ) the scattered channel is of the form e−κn0 |x|φn0 (y), which means that the electron has finite probability to scatter into an evanescent mode and, therefore, be localized near the scatterer. At this point we note that in 3D scattering the electron can scatter elastically only into a traveling spherical wave, which propagates away from the scattering center. That is, in 3D scattering there are no evanescent waves into which the electron can scatter. To state it another way, the existence of evanescent channel modes is a consequence of the confinement subbands in the Q1D wire. According to Eq. (38), the incident electron can scatter into any state, |kn0 i, as long as the energy of this state is equal to the energy of the state |kn i of the incident electron. If the electron scatters into a channel with En0 < E, the kinetic energy, ~2kn2 0 /2m, of the scattered electron should be greater than the kinetic energy ~2kn2 /2m of the incident electron, so that the total energy remains constant during the scattering process. Thus, we distinguish

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two scattering processes: i) intrasubband scattering, in which the scattered subband is the same as the incident one, and therefore the kinetic energy remains the same, and ii) intersubband scattering, in which the electron scatters into a subband which is different from the incident one, and therefore the kinetic energy is modified. In the next section, where we treat specific examples of scattering potentials, we will discuss the effect of evanescent modes on the scattering properties and the conductance of quantum wires. However, we first discuss the Born approximation scheme in Q1D scattering.

3.3.

Born Approximation in Q1D Scattering (+)

In Eq. (35) or (37) the unknown wave function ψp (x, y) appears also under the integral sign. Analytical solution of Eq. (35) is possible only for relatively simple scattering potentials while, for more complicated potentials, the numerical solution may be computationally demanding. Therefore, it is useful to resort to some type of approximation method. One such method is the Born approximation which, despite the fact that it is valid in a restricted domain of the parameters of the potential, it allows us to obtain easily a first qualitative estimate of the scattering solution. We can obtain an iterative solution of Eq. (35), which leads to a perturbation series (or Born series). The wave function can be expressed in a standard way as a perturbation series (+)

(0)

(1)

(2)

ψp (x) = ψp (x) + ψp (x) + ψp (x) + . . . ,

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(0)

(40) (k)

where ψp (x) is the incident wave (i.e., the unperturbed wave function), and ψp (x) is the kth order correction. Thus, Z

(0)
2m (+) (0) d2x0 G(0) x, x0 V x0 ψp x0 ψp (x) = ψp (x) + 2 ~  2 Z Z

(0) 0 00

(0) 00
2m 2 0 2 00 (0) 0 0 00 d x d x G x, x V x G x , x V x ψp x + ~2 + . . .. (41) (1)

(2)

In Eq. (40) ψp (x) is the first-order Born approximation, ψp (x) is the second-order Born approximation, and so on. Then we can express the scattering amplitude given in Eq. (37) as (1) (2) (3) (42) ft (kn0 , kn ) = ft (kn0 , kn) + ft (kn0 , kn) + ft (kn0 , kn ) . . . , where

π 2m hkn0 | V |kn i , ikn0 ~2 π 2m (2) hkn0 | V G(0)V |kn i , ft (kn0 , kn ) = ikn0 ~2 π 2m (3) hkn0 | V G(0)V G(0)V |kn i , ft (kn0 , kn) = ikn0 ~2 .. . (1)

ft

(kn0 , kn) =

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(43) (44) (45)

Ballistic Transport through Quantum Wires and Rings 329 √
ik x and hx|kni = 1/ 2π e n φn (y). Then the amplitude of the wave function for x → ∞ can be written as (1)

tnn0 = δnn0 + ft

(2)

(kn0 , kn ) + ft

(kn0 , kn) + . . . .

(46)

Note that when the energy E approaches the bottom of the subband, En , then kn0 → 0 and (+) |ft (kn0 , kn)| → ∞. The first-order Born approximation is valid as long as ψp (x) is not (0) too different from ψp (x) at the center of the scattering potential. Usually, this implies that the scattering the Born approximation is valid precisely, potential should be weak. More (+) (1) (0) whenever ψp − ψp  1, i.e., whenever ft  1.

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4.

Conductance through Quantum Wires with Scatterers

The discovery of conductance quantization [7, 8] in narrow channels has motivated a great deal of interest in the description of elastic scattering in such systems. Some of the calculations employed the idealized model of the 2D Dirac δ-function [12, 13, 19, 20, 21, 22, 25, 26, 27]. This model potential is used mainly for two reasons. First, it allows in most cases for an analytical solution of the scattering problem with a relatively small amount of effort, and second, it captures the basic physics of the problem under consideration. This short-range scattering potential has two special features. First, there is perfect transmission whenever the electron energy crosses a channel threshold (i.e., at the bottom of each subband the conductance through the scatterer is equivalent to the ballistic conductance). Second, in the case that the 2D δ-function is attractive, exactly one quasibound state splits off below each confinement subband giving rise to a single resonance in each subband. Both of these features are consequences of the presence of evanescent modes in the wire [12, 13, 19, 20]. A finite-range scatterer modeled by a rectangular barrier or well [13, 29] reveals rather different features, but the general behavior of the conductance qualitatively resembles that for a 2D δ-function. The differences can be summarized as follows. First, there are two sets of subbands; namely, one in the region of the scatterer and one outside this region (i.e., in the clean part of the wire). Second, the conductance is no longer perfect at a subband minimum. Third, in the case of a rectangular barrier, the conductance exhibits geometrical resonances (analogous to the Ramsauer resonances). Fourth, in the case of a rectangular well, multiple quasibound states are formed in each subband giving rise to multiple BreitWigner and Fano resonances coexisting in the same subband. More realistic finite-range scattering potentials with smooth profiles, such as a Gaussian [23, 24], double-Gaussian (or quantum dot) [24], and P¨oschl-Teller [23, 32, 36] potentials have been investigated. In these cases, multiple quasibound states are also formed in each subband, but the resonances are either Fano-type or Breit-Wigner. Below we review some of the most important effects of these model scatterers on electron transport. The calculation of the scattering amplitudes proceeds with the help of the LSE, which was discussed in Sec. III, while the conductance is calculated with the Landauer formula. In the rest of this section we assume an infinite square-well confining potential along the y direction of width W = 30nm. The electron mass is taken to be the effective

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mass for GaAs, which is 0.067 of the free electron mass. The subband energies are then given as En = n2 E1 , where E1 = ~2π 2/2mW 2 = 6.236meV , and n = 1, 2, 3, . . ..

4.1.

2D δ-function Scattering Potential

We consider electron scattering by a δ-function potential described by V (x, y) = γδ (x) δ (y − yi ) .

(47)

The magnitude of γ sets the strength of the potential, which can be either positive or negative, and yi is the position of the scatterer in the transverse dimension of the wire. The integral equation Eq. (35) can be solved analytically [20, 21] for this scattering potential. From Eq. (35) the wave function becomes (+)

(0)

ψp (x, y) = ψp (x, y) +

2mγ (0) (+) G (x, y; 0, yi; E) ψp (0, yi) . ~2

(48)

(+)

Setting x = 0 and y = yi in the last expression we can determine ψp (0, yi). The wave function can then be expressed as (+)

(0)

ψp (x, y) = ψp (x, y) + where D = 1−

 nF  X Sn0 n0 n0 =1

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1 2mγ (0) (0) G (x, y; 0, yi; E) ψp (0, yi) , D ~2

2ikn0

+

nc X n0 =nF +1



Sn0 n0 2κn0



.

(49)

(50)

In Eq. (50), nF denotes the highest occupied subband that lies below the Fermi energy, while the first summation is over the propagating modes and the second one is over the evanescent modes. Truncating the problem to a finite number of modes, nc , has some important consequences on the size of the impurity as we will discuss later. The quantities Slm are given as 2mγ Slm = 2 φl (yi ) φm (yi ) , (51) ~ and represent the coupling strengths between the channel modes l and m. Substituting G(0) (x, y; 0, yi; E) from Eq. (33) into Eq. (49) allows us to extract the scattering amplitudes (see, for example, Refs. [19, 20, 21]). The wave function transmission amplitudes are given as Snn0 , (52) tnn0 (E) = δnn0 + 2ikn0 D where n denotes the incident mode while n0 the transmitted one. By letting kn0 = iκn0 in Eq. (52) we obtain the amplitude in the case that n0 is an evanescent mode. The wave function reflection amplitudes are given as rnn0 (E) =

Snn0 . 2ikn0 D

(53)

The amplitudes given in Eqs. (52) and (53) are the main results in the case of a 2D δfunction scatterer in a Q1D wire. They have also been obtained via the mode-matching

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method (see, for example, Refs. [13, 19]) and also by solving the Dyson equation [61]. In Eqs. (52) and (53) note that rnn0 (E) = tnn0 (E) if n 6= n0 or if n0 is an evanescent mode. Also, rnn0 (E) = tnn0 (E) − 1 for n = n0 , which is a statement of current conservation. The intrasubband transmission amplitudes, tnn (E), represent the result of leftover particles which did not scatter. The current transmission amplitude is obtained as t˜nn0 (E) = (kn0 /kn )1/2 tnn0 , which represents the probability amplitude that an incident electron in mode n from the left will be transmitted in mode n0 to the right of the scatterer. The transmission probabilities Tnn0 are then obtained as [54, 59] Tnn0 =

kn0 tnn0 t∗nn0 . kn

(54)

Having obtained the transmission probabilities, the conductance is calculated via the Landauer formula [54, 55, 59] as G=

e2 X kn0 e2 X Tnn0 = tnn0 t∗nn0 , π~ 0 π~ 0 kn n,n

(55)

n,n

where n and n0 run only over the propagating channel modes of the wire.

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4.1.1. Conductance The conductance of a ballistic quantum wire plotted versus the electron energy is shown in Fig. 4 for a repulsive (solid line) and an attractive (red dashed line) 2D δ-function scatterer of strength γ = 16f eV cm2 and −8f eV cm2 , respectively. Expressed more usefully, γ/W 2 = 1.78meV and −0.89meV respectively. The scatterer’s position is taken at yi = (5/12) W , and we included 100 modes in the calculation. The two most important features of a 2D δ-function scatterer are as follows. First, when the electron energy aligns with the energy of a subband minimum, the conductance becomes unity (i.e., the channel modes decouple). This “perfect transmission” effect, first noted in Ref. [12], holds for either a repulsive or an attractive scatterer, and is independent of the shape of the confining potential. Second, in the case that the 2D δ-function potential is attractive, a quasibound state forms below each confinement subband, which gives rise to resonant reflection. Electrons at the quasibound-state energies spend enhanced periods of time in the region of the scattering potential, and destructive interference of these resonant states with the continuum causes the conductance to drop. 4.1.2. Effect of the Number of Modes Even though the model of a 2D δ-function potential in a quantum wire is useful, it may cause some problems. One such problem is that the location of the poles, and thus the conductance properties are not converging with increasing the number of modes [19, 62]. The poles of the transmission amplitude are given as solutions of 1/tnn = 0 which, according to Eq. (52), takes the form nc X 2mγ |φn (yi )|2 = 0. 1− ~2 2ikn n=1

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Figure 4. Conductance (G, in units of 2e2 /h) through a 2D δ-function scatterer in a quantum wire versus electron energy (E, in units of E1 ). The solid line corresponds to a repulsive scatterer while the dashed line corresponds to an attractive scatterer of strength γ = 16f eV cm2 and −8f eV cm2, respectively. The first few levels, En = n2 E1, of the wire are given in the horizontal axis. A new subband opens every time the electron energy crosses from below a level En . It can be seen that the pole locations depend sensitively on the number nc of channel modes included in the calculation. The behavior of the first pole (below the second subband) was investigated in Ref. [19] as a function of the number of modes. It was shown that the poles tend to move deeper into the complex energy plane with increasing the number of modes. This means that the resonances become broader. This is illustrated in Fig. 5 where the conductance of a quantum wire with an attractive scatterer of strength γ = −8f eV cm2 is plotted versus the electron energy, for nc = 9 and 100 modes. It is seen that both resonances become broader when nc = 100. The above-mentioned convergence problem can be avoided if we modify the δ-function in the transverse dimension to a finite-range Gaussian-type potential [62]. When using such a potential (i.e., a δ-function in the propagation direction and Gaussian in the transverse direction), the result for the location of the poles converges to its exact solution when using a finite number of modes. Furthermore, it has been shown that [20, 22, 63, 64] if the number of modes is extended towards infinity, a 2D δ-function potential in a uniform Q1D wire does not scatter electrons at all. This follows from Eq. (50) in which, if nc becomes very large, we can approximately write κnc ' πnc /W . Then in the limit nc → ∞ one gets D → ∞ and consequently, from (+) (0) Eq. (49), ψp (x, y) = ψp (x, y), i.e., the scattered and incident wave function are equal to each other. One is then forced to keep only a finite number of modes in the scattering problem. However, truncating the number of modes to a finite number, nc , establishes an effective impurity size of W/nc . Thus, in general, the scattering properties and conductance of a ballistic Q1D wire with a 2D δ-function scattering potential must always be discussed for a specific number of modes. This model potential can then be interpreted as an s-like scatterer with finite width W/nc in the transverse dimension instead of a true 2D δ-function.

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Figure 5. Conductance (G, in units of 2e2 /h) through an attractive 2D δ-function scatterer in a quantum wire versus the electron energy (E, in units of E1). Increasing the number of modes affects the position and width of a resonance. 4.1.3. Born Approximation Even though exact analytical solution exists for the 2D δ-function potential, further insight is gained by considering the Born approximation. For this particular model potential, the whole Born series can be summed as a geometric series to the exact result, which is reminiscent of the analogous situation in 1D scattering. Therefore the 2D δ-function provides a nice example for applying the Born approximation. (1) From Eq. (43) the first-order Born amplitude, ft (kn0 , kn ), is obtained as

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(1) ft

Z Z π 2m 2 0 d x d2 x00hkn0 |x0ihx0 |V |x00ihx00 |kni (kn0 , kn ) = ikn0 ~2 Z Z 0 00 π 2m 2 0 d d2 x00e−ikn0 x φ∗n0 (y0 ) V (x0) δ (2) (x0 − x00) eikn x φn (y00 ) x = 2 ikn0 (2π) ~ Snn0 = . (57) 2ikn0

The first-order Born amplitude of the reflected wave turns out to be equal to that of the (1) (1) transmitted wave, i.e., fr (kn0 , kn) = ft (kn0 , kn). If we consider intrasubband scatter(1) ing (i.e., n0 = n) the reflection probability to first-order, Rnn (kn ), is given as R(1) nn (kn ) =

2 Snn , 4kn2

(58)

which is seen to decrease as (E − En )−1 . The second-order Born amplitude can also be evaluated in similar fashion yielding Z Z

π 2m (2) 2 0 x d d2x00hkn0 |x0iV x0 hx0|G(0)|x00iV x00 hx00|kn i ft (kn0 , kn) = 2 ikn0 ~ Snn0 2mγ (0) G (0, yi; 0, yi; E) . (59) = 2ikn0 ~2

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From Eq. (46), the amplitude of the transmitted wave can be expressed as Snn0 Snn0 2mγ (0) tnn0 (E) = δnn0 + + G (0, yi; 0, yi; E) 2ikn0 2ikn0 ~2  2 Snn0 2mγ (0) + G (0, yi ; 0, yi; E) + . . . 2ikn0 ~2  −1 Snn0 2mγ (0) 1 − 2 G (0, yi ; 0, yi; E) , = δnn0 + 2ikn0 ~

(60)

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which is the exact result that we
obtained by solving the LSE exactly [see Eq. (52)] if we recognize that 1 − 2mγ/~2 G(0) (0, yi ; 0, yi; E) = D. As mentioned in Sec. IIIC, (1) the first-order Born approximation is valid whenever ψp  1 (or, equivalently, (1) f (kn0 , kn)  1) at the center of the scattering potential. For the 2D δ-function this condition implies Snn0 (61) 2k 0 D  1. n This limit can be attained whenever the strength of the scatterer is small enough or the parameter D (which depends on the number of modes) is large. Note also that the firstorder Born amplitude does not take into account the sign of the potential, i.e., it gives the same result for both repulsive and attractive potentials. It should also be mentioned that the Born approximation to any finite order fails whenever the electron energy approaches a subband minimum. However, this approximation method is certainly useful for discussing more complex finite-range scattering potentials. In these cases, before employing numerical calculations, it may prove advantageous to understand the basic aspects of the problem by employing an approximation method.

4.2.

Rectangular Scattering Potential

Electron scattering by a finite-range potential has a more rich structure and the conductance exhibits interesting features. Consider a potential described by     L Ws ~2 γ Θ − |x| Θ − |y − yi | , (62) V (x, y) = 2m 2 2 where Θ (x) is the unit step function. The potential described by this function is a rectangular barrier (γ > 0) or well (γ < 0) of length L in the propagation direction and width Ws in the transverse direction (centered at yi ). For simplicity, the size Ws of the potential in the transverse direction is assumed to be very small ( Ws  L,) and thus, in this direction only, the scatterer can be considered as short-ranged and can be approximated by δ (y − yi ). Then, integrating over y, one can replace the
wave function under the integral 2 in the LSE by its value at yi . We also let γ = 2m/~ V0Ws , where V0 has energy units. For the potential given in Eq. (62), the LSE does not lead to a useful analytical expression and therefore a numerical approach is required. A numerical method for solving the LSE in the presence of an arbitrarily shaped scattering potential in a Q1D wire was developed in Ref. [23]. This method can also be employed for investigating multiple scattering centers of any shape.

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4.2.1. Numerical Method Instead of the potential given in Eq. (62), let us consider for the moment a general finite-range scattering potential V (x, y) =

~2γ υ (x) δ (y − yi ) , 2m

(63)

where υ (x) is an arbitrary function of the coordinate x,
and yi is the transversal position of the scatterer. Also, as mentioned above, γ = 2m/~2 V0Ws . Now, since the unknown (+)

wave function ψp (x, y) appears also under the integral in Eq. (35), the starting point for (+) solving the LSE is to find ψp (x, y) in the region of the scattering potential. Once we (+) know ψp (x, y) in the scattering region, we can then substitute this solution under the integral in Eq. (35) and obtain the wave function in the asymptotic region x → ±∞. Before proceeding further with the numerical method, we first note that, in the case of a finite-range scatterer, there are two sets of subbands; namely, one set in the “clean” part of the wire, and one in the scattering region. The subband structure of the “clean” wire was described in Sec. IVA (see also Fig. 3). However, in the region where the scatterer is located, the transverse energy levels (which define the bottoms of the subbands) are obtained numerically from

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sin (αW ) =

γ sin (αyi ) sin [α (yi − W )] , α

(64)

as derived from the Schr¨odinger equation in the transverse dimension. In Eq. (64) we have defined α = (2mE)1/2 /~. The resulting energy levels will be denoted by En,s in order to distinguish them from the levels En of the “clean” part of the wire. Note that in the limit γ → 0 we recover the energy levels of the “clean” wire. Now, for the scattering potential given in Eq. (63), we can write the LSE Eq. (35) as Z ∞

(+)
(+) (0) dx0G(0) x, y; x0, yi υ x0 ψp x0 , yi . (65) ψp (x, y) = ψp (x, y) + γ −∞ (+)

(+)

In order to find ψp (x, y) for x → ∞, we must first find ψp (x0 , yi ) in the region of the scatterer [i.e., in the region where υ (x0) 6= 0]. We assume that υ (x0 ) 6= 0 for |x0| ≤ x0 , and υ (x0 ) = 0 for |x0 | > x0 . In the case of the rectangular barrier (or well) given in Eq. (62), we let x0 = L/2. The interval [−x0 , x0] is then divided into s equal subintervals of width b = 2 |x0 | /s. The coordinates x0 and x in Eq. (65) are discretized as x0 = −x0 + qb and x = −x0 +rb, where q = 0, 1, 2, . . ., s and r = 0, 1, 2, . . ., s. The number s of subintervals can be chosen sufficiently large so that the results converge. Replacing the integral by a sum and setting y = yi in Eq. (65), we obtain a set of s equations (one for each value of r) for (+) the s unknown values of the wave function ψp , which can be expressed in matrix notation as s X 1 Mrq ψq(+) = − ψr(0), (66) γb q=0

where Mrq = G(0) (−x0 + rb, yi; −x0 + qb, yi) υ (−x0 + qb) −

1 δrq , γb

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(67)

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are the entries of an s × s matrix. In Eq. (66) ψq represents the unknown values of (+) the wave function, ψp (−x0 + qb, yi), in the scattering region and is a column vector. (0) Similarly, ψr is a column vector and represents the known values of the wave function (0) ψp (−x0 + rb, yi) of the incident wave in the region of the scattering potential. Inverting (+) the matrix Mrq , allows us to solve Eq. (66) for ψq and therefore find the wave function in (+) the region of the scattering potential. With these values of ψp (x, yi), we can subsequently perform the integral in Eq. (65) by discretization and determine the wave function far to the right (or left) of the scatterer, i.e., for x → ∞ (or −∞). From the resulting wave function we can then extract the transmission (or reflection) amplitude. 4.2.2. Conductance

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We return now to the rectangular potential given in Eq. (62). In the numerical calculations we have taken yi = (3/8) W and Ws = 0.1W . Six modes were included in the calculation. In the following, all distances are expressed in units of the wire’s width W , and all energies in units of E1. The conductance plotted versus the electron energy over the first subband is shown in Fig. 6, for V0 = 8E1 and for three values of the barrier length L. For this value of V0 , solving numerically Eq. (64) gives E1,s = 1.82E1, E2,s = 4.8E1, and E3,s = 9.23E1 (i.e., the energy subbands in the scattering region are shifted upwards). Note that for very small L the conductance resembles that of the 2D δ-function scatterer. In fact, the conductance becomes unity at the bottom of the second subband, a feature that appears in the case of the 2D δ-function (compare with Fig. 4).

Figure 6. Conductance (G, in units of 2e2 /h) through a finite-range rectangular repulsive scatterer in a quantum wire versus electron energy (E, in units of E1 ), for various values of the ratio L, where L is the barrier length and W the wire’s width. The height of the scatterer is V0 = 8E1, while yi = 3/8. Note that the conductance exhibits oscillatory behavior as L increases whereas, for sufficiently small L, it resembles that for the 2D δ-function scatterer. However, the conductance through the barrier exhibits two new features. First, there is electron transport via tunneling in the energy range E1 < E < E1,s , which is due to

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the presence of evanescent modes below E1,s. That is, transmission prior to the opening of the subband E1,s occurs via the evanescent modes in the region of the barrier. However, a contribution from the evanescent mode n + 1 decreases exponentially with increasing the barrier length, becoming negligible when r 2m κn+1,s (E = En,s ) L = (En+1,s − En,s ) L > 1. (68) ~2

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In Fig. 6, note that for L = 2.5W the transmission below E1,s actually vanishes. This also justifies the use of a relatively small number of modes whenever a finite-range scatterer is considered. The second feature is the resonant structure in the conductance which appears as soon as the barrier becomes long enough to damp out the evanescent modes, so that interference effects become important. In fact, when r 2m kn,s (E = En+1,s ) L = (En+1,s − En,s ) L > 1. (69) ~2 the conductance at the nth plateau should start exhibiting oscillations, which gradually evolve into sharp resonances with increasing L. We also note that, as the length of the barrier increases, the conductance is no longer perfect at the bottom of the second subband. The conductance plotted versus the electron energy over the first two subbands is shown in Fig. 7, for two values of the ratio L. The parameters of the potential have the same values as those used in Fig. 6, and we have included again six modes in the calculations. Note that the conductance oscillations evolve into sharp resonances as the barrier length increases. These geometrical resonances appear in the conductance only for a rectangular barrier. They are also more pronounced immediately after the opening of a new subband En,s while, for higher energies, their amplitudes decrease and they become broader. Note that, for a sufficiently long barrier, the rises in the conductance do not occur at the subband energies En of the “clean” wire, but rather at the subband energies En,s .

Figure 7. Conductance (G, in units of 2e2 /h) through a finite-range rectangular repulsive barrier in a quantum wire over the first two subbands, for V0 = 8E1. In the case of a rectangular well (V0 < 0), a new effect occurs; namely, the occurrence of multiple asymmetric Fano resonances in the conductance. In Fig. 8, the conductance

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plotted versus the electron energy is shown for V0 = −11E1. All other parameters are the same as in Fig. 7. For this value of V0, the subband energies in the scattering region are E1,s = −0.51E1, E2,s = 3.11E1, and E3,s = 7.8E1. It is well-known that the Fano line shapes are the result of the coupling between quasibound states formed in the scattering potential with the continuum of states. Depending on the range and strength of the scattering potential, there may be one or more quasibound states which will give rise to one or more Fano line shapes. For L = 1.9W there are two quasibound states in the first subband, which give rise to two Fano resonances. Note that the first Fano resonance is of the 1 → 0 type (i.e., the transmission zero follows the transmission one), while the second is of the 0 → 1 type (i.e., the transmission one follows the transmission zero). For the 1 → 0 type, the resonance energy occurs after the energy of the transmission one, while for the 0 → 1 type, the resonance energy occurs before the energy of the transmission one. Note also that the number of Fano resonances in a given subband increases with increasing the length of the rectangular well; for L = 3.8W there are four resonances, which are due to four quasibound states.

Figure 8. Conductance (G, in units of 2e2/h) through a finite-range square-well in a quantum wire versus electron energy (E, in units of E1), for two values of L, where L is the length of the scatterer and W the wire’s width. The depth of the scatterer is V0 = −11E1. Note that the conductance exhibits Fano resonances due to quasibound states. The number of these resonances increases with increasing L.

Furthermore, when L = 3.8W , there is a broad Breit-Wigner resonance in the first subband while in the second subband there are two. The simultaneous occurrence of multiple Breit-Wigner and Fano-type resonances in the same energy subband was discussed in Ref. [29]. An interesting effect was also shown [29] to occur; namely, the collapse of a Fano resonance for a certain values of the size of the square-well impurity. This is associated with the fact that, for certain critical sizes of the impurity, the corresponding pole of the transmission amplitude moves to the real energy axis and, consequently, the resonance width shrinks to zero. In Sec. V we discuss this and other relevant effects in more detail.

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4.2.3. Born Approximation The first-order Born approximation is expected to be valid when the height of the rectangular scatterer is small compared to the subband separation. However, in this limit, the oscillatory structure of the conductance may not be observable. The first-order Born approximation to the reflection amplitude is obtained as [see discussion below Eq. (37)] fr(1) (kn0 , kn )

Z

(0)
π 2m 1 0 = d2x0 √ eikn0 x φ∗n0 y 0 V x0 , y 0 ψp x0 , y 0 2 ikn0 ~ 2π    2   Z L/2 2m 1 ~ γ π 0 ∗ φn0 (yi ) φn (yi ) = ei(kn +kn0 )x dx0 2 ikn0 ~ 2π 2m −L/2   2  Snn0 ~ 2 sin [(kn + kn0 ) L/2] = . (70) 2ikn0 2m (kn + kn0 )

Then the first-order Born approximation for the intrasubband reflection probability is found as 2  ~2 2  S 2  sin2 (k L) (1) n nn (1) , (71) Rnn (kn ) = fr (kn0 , kn) = 4 2m 4 kn

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which is seen to decrease as (E − En )−2 for large energies. This is in contrast to the corresponding result for the 2D δ-function potential [see Eq. (58)], for which the reflection probability in the first-order Born approximation decreases slower, i.e., as (E − En )−1 . Note that if L becomes small such that kn L  1 one gets sin (kn L) ' kn L, and the result of Eq. (71) reduces to that for the 2D δ-function potential. However, we should note that the first-order Born approximation fails for lower energies, and diverges in the limit E → E1.

4.3.

Gaussian Scattering Potential

A more realistic scatterer should have a “smooth” potential profile with some decay length along the propagation direction. Two model potentials having such “smooth” profiles are the Gaussian [23, 24] and P¨oschl-Teller [23, 32] potentials. Let us consider for the moment scattering by a potential V (x, y) =

 2   x Ws ~2γ exp − 2 Θ − |y − yi | , 2m ρ 2

(72)

which represents a Gaussian barrier (γ > 0) or well (γ < 0) with a decay length ρ in the propagation direction. In the transverse dimension the scatterer has again a rectangular shape, which is modeled by the unit step function (centered at
y = yi ), but with very small size Ws , where Ws  ρ. As in Sec. IVB, we let γ = 2m/~2 V0Ws , where V0 has energy units. For this potential the LSE cannot be solved analytically and therefore we employ the numerical method discussed above. In the following calculations the values of the impurity parameters are the same as those for the rectangular scatterer, i.e., yi = (3/8) W and Ws = 0.1W . Six modes were included in the calculation.

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4.3.1. Conductance

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The conductance plotted versus the electron energy over the first subband is shown in Fig. 9 for V0 = 8E1, and for several values of the ratio ρ. Note that as ρ increases the rise in the conductance becomes gradually sharper. This occurs at the opening of the new subband energy E1,s = 1.82E1, as obtained by solving Eq. (64). Since the first channel is not propagating for E1 < E < E1,s , electron transport for this energy range occurs via tunneling. As in the case of the rectangular barrier, this is due to the presence of evanescent modes (in the scattering region) that lie above the Fermi level. These modes can serve as intermediate states for electrons. However, as ρ increases the contribution of the evanescent modes decreases exponentially [23] and thus, prior to the opening of the first subband E1,s , the conductance is suppressed and becomes negligible.

Figure 9. Conductance (G, in units of 2e2/h) through a finite-range Gaussian barrier in a quantum wire versus electron energy (E, in units of E1 ), for various values of the ratio ρ, where ρ is the decay length of the Gaussian scatterer and W the wire’s width. The height of the scatterer is V0 = 8E1. Note that as ρ increases the conductance approaches 2e2 /h progressively faster while, below E1,s , it is suppressed. This suppression is due to the gradually smaller contribution of evanescent modes as ρ increases. After the opening of the subband E1,s, the conductance approaches the quantum unit 2e2 /h progressively faster as ρ increases. This is due to the suppression of backscattering as the Gaussian potential becomes “smoother”. This is in contrast to the case of the rectangular barrier for which the conductance exhibits resonant structure as the barrier length increases. We should mention here that at finite temperature there would be further smearing of the conductance step [9, 65, 66]. The conductance plotted versus the electron energy over the first two subbands is shown in Fig. 10 for V0 = 11E1, and for two values of ρ. The well-known smearing of the conductance steps, similar to that observed in experiments [7], is present in both steps. We mention here that, if the scatterer was placed on the central axis of the wire (i.e., at yi = W/2), the first mode would be maximally scattered, while the second one would not be scattered at all. In this case, there would be almost perfect quantization of the second conductance step. To this end, we point out that recently the effect of impurity scattering

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on the quantized conductance of a Q1D wire has been investigated experimentally [67].

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Figure 10. Conductance (G, in units of 2e2 /h) through a finite-range Gaussian barrier in a quantum wire versus electron energy (E, in units of E1), over the first two subbands. The height of the scatterer is V0 = 11E1. In the case of a Gaussian well, the conductance may exhibit multiple Breit-Wigner dips. For V0 = −15E1, the conductance plotted versus the electron energy is shown in Fig. 11, for ρ = 0.4W and 0.6W . Note that the number of the resonant dips increases with increasing ρ. As mentioned previously, these symmetric antiresonances in the conductance are explained in terms of the formation of quasibound states - at special energies - in the impurity region [23, 24, 29, 32, 33]. The destructive interference of these states with the continuum causes the conductance to drop. In addition, increasing ρ causes shifting of these resonances toward lower energies. This behavior originates from the gradually larger integrated strength of the potential ( ∼ γρ). This effect occurs regardless the type of scattering potential [13, 20, 22, 29]. Experimental observation of conductance dips induced by an impurity in narrow channel was reported in Ref. [68]. It is also worth noting that the width of a resonance decreases with increasing ρ, but in an oscillatory manner [23]. However, numerical calculations indicate [23] that the resonance width does not vanish as ρ varies. This is in contrast to the case of the square-well impurity [29], in which the width of a Fano resonance shrinks to zero for certain critical sizes of the impurity. To this end, we point out that, besides the symmetric Breit-Wigner dips, the conductance through a “smooth” Gaussian-type scatterer may also exhibit asymmetric Fano resonances. This will further be discussed in the next section by employing the coupled-channel Feshbach theory, which provides a more suitable formalism for treating resonance phenomena. The absence of Fano line shapes from the conductance in Fig. 11 is due to the particular shape of the scatterer in the transverse dimension, which was assumed to be short-ranged. In fact, if the scatterer was made to have a lateral extent then Fano resonances would emerge in the conductance (the reader is referred to Fig. 11 of Ref. [24]). In passing, we also mention that an analytical treatment of scattering by spherically symmetric potentials in a Q1D wire was performed in Ref. [69], in the presence of a general cylindrical confinement. This was done in terms of Green’s functions and by expressing the

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Figure 11. Conductance (G, in units of 2e2/h) through a finite-range Gaussian well in a quantum wire versus electron energy (E, in units of E1), for two values of ρ, where ρ is the decay length of the Gaussian well, and W the wire’s width. The depth of the scatterer is V0 = −15E1. Note that the conductance exhibits symmetric resonant dips due to quasibound states. The number of these dips increases with increasing ρ.

scattering amplitudes in terms of the phase shifts (beyond s waves), and taking into account the couplings of orbital angular momenta. The effect of confinement-induced resonances [70] was also discussed.

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4.3.2. Born Approximation The first-order Born approximation to the reflection amplitude is obtained similarly to the case of the rectangular scatterer, fr(1) (kn0 , kn)



   2  Z ∞ π 2m 1 ~ γ 0 02 2 ∗ = φn0 (yi ) φn (yi ) ei(kn +kn0 )x e−x /ρ dx0 2 ikn0 ~ 2π 2m −∞   2  √ 2 2 Snn0 ~ = π ρ e−ρ (kn +kn0 ) /4 . (73) 0 2ikn 2m

The first-order Born approximation for the intrasubband reflection probability is obtained as 2  ~2 2  S 2 2 2 (1) nn (1) πρ2e−2ρ kn . (74) Rnn (kn ) = fr (kn0 , kn) = 2m 2ikn Note that the reflection probability for the Gaussian potential decreases exponentially for large energies, whereas in the case of the rectangular barrier it decreased only as (E − En )−2 [see Eq. (71)]. This is not accidental. When the scattering potential is “smooth” the backscattering effects are suppressed, and therefore Rnn → 0 faster.

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5.

343

Fano Resonances in Transport through Quantum Wires

When a discrete level interacts with a continuum of states, a quasibound (resonant) state is created around the discrete level giving rise to asymmetric Fano line shapes. Such asymmetric resonances appeared in the conductance of a quantum wire with an attractive square-well impurity (Sec. IVB, Fig. 8). The Fano effect [37] has been investigated theoretically in various condensed matter systems including electronic ballistic transport through Q1D systems with attractive impurities [23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 71, 64] and quantum waveguides with resonantly coupled cavities [71, 64, 72]. Fano resonances have also been observed experimentally in transport through mesoscopic systems with embedded quantum dots [42, 43, 44, 45, 46]. The Fano effect, a result of quantum interference, is of great interest both as a basis for the creation of new resonant nanoelectronic devices and for revealing the quantum-mechanical wave nature of the charge carriers. Resonance line shapes in narrow channels (or constrictions, defined by a split gate [73]) with impurities can also lead to a connection between the line shape and the parameters of the impurity [32]. Let us briefly summarize Fano’s theory [37]. Consider a system with a discrete (localized) energy state embedded in a continuum of states, which can be described by the Hamiltonian  X X Ek cˆ† cˆk + Vk cˆ† ˆb + V ∗ ˆb†cˆk . (75) H = Ec ˆb†ˆb + k

k

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k

k

k

The above Hamiltonian describes a discrete state with energy Ec , while the operators ˆb† and ˆb create and annihilate electrons respectively in this state. This discrete state may be called the impurity. The operators cˆ†k and cˆk create and annihilate electrons in the continuum of states of energy Ek . The continuum may have a finite bandwidth, which occurs in tight binding models, or it can be a free-particle model. In the last term of Eq. (75), Vk is the coupling strength between the discrete state and the continuum, which is the result of mixing between these two kinds of states. In this term there are two types of processes. In one process, represented by cˆ†kˆb, the particle hops off the impurity into the continuum while, in the second process, represented by ˆb† cˆk , the particle in the continuum hops onto the impurity. The nature of the solution of the above model depends critically on whether the energy Ec falls within the continuum band of states Ek , i.e., in the energy range E1 < Ek < E2 . Due to the interactions with the continuous band of states, the energy of the discrete state is modified to a new renormalized energy Ec0 . If this modified energy Ec0 is still within the continuum band of states, i.e., if E1 < Ec0 < E2 , then the solution exhibits a very interesting feature; namely, there would be no more any localized state in the system. This means that a particle in the continuum may hop onto the impurity, spend some time there, and then it may hop off again. Therefore, this is not a well-defined eigenstate, and the impurity state has become a scattering resonance. As mentioned in Sec. III, in a Q1D wire scattered particles at a given energy can have different momenta in the asymptotic region, depending on their subband index. Thus, a bound state in one subband can coexist with an unbound state in another subband. The interaction between these two states may give rise to a Fano line shape in the transmission

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probability. Fano resonances have a line shape of the form

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2 ( + q)2 , T () = tbg 2  +1

(76)

where T () is the transmission probability,  = (E − ER ) /Γ is the dimensionless energy 2 from resonance, Γ is the resonance width, q is the (Fano) asymmetry parameter, and tbg is the nonresonant transmission. The asymmetry parameter is particularly useful for the complete description of the line shape and has been investigated experimentally and theoretically in the context of mesoscopic systems [30, 32, 35, 42, 43, 44, 74]. A theoretical investigation of the case in which the Fano parameter is complex was reported in Ref. [75]. As discussed below, an important feature of the asymmetry parameter is that it depends only on the background transmission and not on the strength of the bound state-continuum interaction. In addition, it is of interest to have a connection between the line shape and the physical parameters that give rise to resonance (such as, the matrix elements of the scattering potential). A suitable approach to treat resonances in electron transmission through Q1D systems is provided by the Feshbach theory [56, 57]. Usually, the Fano line shapes are investigated by employing the two-channel approximation; namely, by considering coupling between the (first) propagating channel and the bound state of the second channel. However, below we also discuss the case of one open and two closed channels (i.e., the three-channel approximation). Feshbach’s approach has been employed in the case of magnetic-field-induced coupling [28], and an equivalent approach [76] was used for the description of symmetric resonances. Recently, this approach was successfully applied to the case of a quantum wire with a local spin-orbit (Rashba) coupling [77], which acts as an attractive impurity. The “reversal” of the Fano line shape, the transformation to Breit-Wigner antiresonance, and the collapsing behavior are discussed (or at least mentioned) below. Here, by “reversal” we mean the change of the sign of q. In addition, we discuss the range of validity of the two-channel model, and comparison with the three-channel model is made.

5.1.

Feshbach Theory in Q1D Systems

As in Sec. IV, we consider a ballistic uniform quantum wire in which electrons are confined along the y-direction (transverse direction) but are free to propagate along the xdirection (see Fig. 2). In the presence of a scattering potential, and an electric field in the transverse direction, the Schr¨odinger equation describing the electron motion in the wire can be written as   2 p + Vc (y) + U (y) + V (x, y) Ψ (x, y) = EΨ (x, y) , (77) 2m where Vc (y) is the confining potential chosen to be parabolic with force constant mω02 , Vc (y) =

1 mω02 y 2, 2

(78)

and U (y) = −eF y represents the potential energy due to the electric field F (e is the charge of the electron, e < 0). Also, V (x, y) is the scattering potential in the wire. The

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total transverse potential V (y)+U (y), providing confinement of the electron motion along the y direction, gives rise to channel modes φn (y), # " p2y + Vc(y) + U (y) φn (y) = En φn (y), (79) 2m where En is the threshold energy for mode n. We expand the wave function of Eq. (77) in terms of the channel modes Ψ(x, y) =

∞ X

ψn (x)φn (y).

(80)

n=0

Substituting Eq. (80) into Eq. (77) we obtain coupled-channel equations for ψn (x): ˆ n(x) = (E − En − K)ψ

∞ X

Vnl (x)ψl(x),

(81)

l=0

ˆ = −(~2/2m)d2/dx2 and Vnl (x) are the coupling matrix elements given as where K Z (82) Vnl (x) = dy φ∗n (y)V (x, y)φl(y).

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Now, in the decoupling limit (Vnl = 0 for n 6= l), only the first channel mode (n = 0) can be found in some scattering state provided that the diagonal matrix elements, Vnn , vanish far away from the scattering region. From Eq. (81), the scattering states for the first mode can be obtained as solutions of the equation i h ˆ + V00(x) + E0 χ± (x) = Eχ± (x), (83) K k k − where χ+ k (x) and χk (x) correspond to scattering states for which the incident wave comes from −∞ and +∞, respectively. These states describe the background (nonresonant) scattering, which is the scattering in a hypothetical system in which the channel coupling is switched off [28, 56, 57]. In Eq. (83), k is the wave vector for the propagating mode, i.e., k = [2m(E − E0 )]1/2/~. The scattering states have the asymptotic form  bg ±ikx (x → ±∞)  t e ± , (84) χk (x) =  ±ikx bg ∓ikx + r± e (x → ∓∞) e bg where the upper signs correspond to an incident wave coming from −∞. Also, tbg and r± denote the background transmission and reflection amplitudes in the wire. In addition to the open channel (n = 0), we consider two closed ones (n = 1 and 2), which are dominated by their bound states Φ01 (x) and Φ02 (x), respectively. From Eq. (81), in the decoupling limit, we obtain the equations for the bound states of the uncoupled channels n = 1 and 2 as   ˆ Φ0j (x) = Vjj (x)Φ0j (x) ej − Ej − K (j = 1, 2), (85) E

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ej are the bound state energies. We now make the approximation of truncating the where E sum in Eq. (81) at n = 2 and obtain the following system of three equations, h i ˆ − V00(x) ψ0 (x) = V01(x)ψ1(x) + V02(x)ψ2(x), E − E0 − K (86) i ˆ − V11(x) ψ1 (x) = V10(x)ψ0(x) + V12(x)ψ2(x), E − E1 − K h i ˆ − V22(x) ψ2 (x) = V20(x)ψ0(x) + V21(x)ψ1(x). E − E2 − K h

(87) (88)

The above system of equations can be simplified by assuming that ψ1(x) and ψ2 (x) are multiples of the bound states of the uncoupled channels [57], i.e., ψ1 (x) = A1Φ01 (x),

(89)

ψ2 (x) = A2Φ02 (x).

(90)

Inserting Eqs. (89) and (90) into Eq. (86) we get an inhomogeneous equation for |ψ0i, which can be solved with the retarded Green’s function operator defined by h i ˆ − V00(x) G ˆ 0(x, x0) = δ(x − x0 ). E − E0 − K (91) ˆ 0 as obtained from Eq. (91), the general solution of that inhomogeneous equation With G can be written as

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ˆ ˆ |ψ0i = |χ+ k i + A1 G0 V01|Φ01 i + A2 G0 V02 |Φ02i.

(92)

To find the constants A1 and A2, we first insert Eqs. (89) and (90) into Eqs. (87) and (88), and then employ Eq. (85) to get the system of equations,   e1 |Φ01i − A2 V12|Φ02i = V10|χ+ i + A1 V10G ˆ 0V01|Φ01i + A2 V10G ˆ 0V02|Φ02i, A1 E − E k (93)   + e ˆ ˆ −A1 V21|Φ01i+A2 E − E2 |Φ02i = V20|χk i+A1V20G0V01|Φ01i+A2V20G0V02|Φ02i. (94) Multiplying Eq. (93) with the bra hΦ01 | and Eq. (94) with the bra hΦ02| we arrive at the system of equations, (95) (E − ε1 ) A1 − W12A2 = W10 , −W21 A1 + (E − ε2 ) A2 = W20,

(96)

where we have introduced the energies εj by ej + hΦ0j |Vj0G ˆ 0 V0j |Φ0j i εj = E

(j = 1, 2).

(97)

The matrix element in Eq. (97) is a self-energy term, which is due to the coupling of the jth bound state with the continuum and has in general both a real and an imaginary part. Here we point out that, if the third channel were not included, the real part of the above e1 acquires, while its matrix element for j = 1 would give the shift that the bound state E imaginary part would give the width of the resulting quasibound state. However, in the

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three-channel model that we consider here there is an additional contribution originating from the coupling W12 between the two closed channels. We have also introduced the matrix elements Wj0 = hΦ0j |Vj0 |χ+ (j = 1, 2), (98) ki ˆ 0 V02|Φ02i = W21. W12 = hΦ01|V12|Φ02i + hΦ01|V10G

(99)

In Eq. (99) the first matrix element represents the direct coupling of the two closed channels, while the second matrix element represents the indirect coupling of the closed channels via the open channel. After solving the system of Eqs. (95) and (96) for A1 and A2 , we insert the resulting expressions into Eq. (92), which is finally written as ψ0(x) = χ+ k (x) + +

∗ h(χ− m + k ) |V01 |Φ01 i [(E − ε2 ) W10 + W12 W20 ] χ (x) 2 i~2ktbg k (E − ε1 ) (E − ε2 ) − W12

∗ h(χ− m + k ) |V02|Φ02 i [(E − ε1 ) W20 + W10 W21 ] χ (x) , 2 i~2ktbg k (E − ε1 ) (E − ε2 ) − W12

(100)

where we have used the explicit form of the retarded Green’s function in 1D [76] given in Eq. (141) of Appendix A. Note that in the absence of the third channel we can set W20 = W12 = ε2 = 0, and the above expression for ψ0 (x) reduces to that of the two-channel case [28]. In Appendix B, we discuss the effect of the interaction, W12 , between the two closed channels and show that the strength of this interaction determines the extent to which the two-channel model is valid.

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5.2.

Short-Range Scattering Potential with Lateral Extent

We consider an attractive scattering potential described by [78, 79, 80, 81, 82]   (y − yi )2 ~2 β δ(x) exp − , V (x, y) = − 2m d2

(101)

which decays in the transverse direction with decay length d, and has its center at y = yi . A Gaussian shape for the scattering potential in the transverse direction is employed here in order to introduce the lateral extent of the impurity, quantified by d, which may provide an extra parameter for fitting experimental data. Here the magnitude of β sets the strength of the scattering potential (β > 0). A scattering potential of this type can be used to model, for example, the negative electrostatic influence of a scanning probe microscope (SPM) tip in experiments studying the imaging of coherent electron flow through a narrow constriction in a 2D electron gas [83, 84]. In fact, a similar potential (i.e., a 2D δ-function) has been used previously [85] in order to approximate the potential induced by the SPM tip, and the obtained results were consistent with the experimental ones. Then, using Eq. (101), the matrix elements of Eq. (82) take the form Vnl (x) = − where υnl =

Z

~2 β δ(x)υnl, 2m

  (y − yi )2 φ∗n (y) exp − φl (y) dy. d2

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(102)

(103)

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The mode eigenfunctions φn (y) are found by solving Eq. (79), which can be written as " # p2y 1 e2 F 2 2 2 + mω0 (y − l) − (104) φn (y) = En φn (y) , 2m 2 2mω02 where l = eF/mω02. The last equation is the Schr¨odinger equation for a shifted 1D harmonic oscillator plus a constant energy term in the Hamiltonian. Therefore, φn (y) =



1 √ n 2 n! π

r

mω0 ~

1/2

r  h mω i mω0 0 2 exp − (y − l) Hn (y − l) , 2~ ~

(105)

where Hn (y) denote the Hermite polynomials, and the eigenenergies are given as En = ~ω0



1 n+ 2



−

e2F 2 , 2mω02

(106)

where n = 0, 1, 2, . . . , which are the Stark-shifted harmonic oscillator eigenenergies. In order to find the scattering states χ± k (x), which are needed for the evaluawe must solve Eq. (83) for the effective 1D potential V00(x) = tion of ψ0(x),
− ~2β/2m δ(x)υ00. The solution proceeds in a standard way and the background transbg bg mission and reflection amplitudes are found as tbg = [1 + (βυ00/2ik)]−1 and r+ = r− = − [1 + (2ik/βυ00)]−1 . The bound states in the effective potentials V11(x) and V22(x), which are also needed for the evaluation of ψ0 (x), are found by solving Eq. (85) and they √ are given as Φ0j (x) = κ0j exp (−κ0j |x|), where κ0j = βυjj /2, j = 1, 2. The corre  ej = Ej − ~2 β 2υ 2 /8m . sponding eigenenergies are E

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jj

Having obtained the scattering states of the open (uncoupled) channel n = 0, and the bound states Φ01 and Φ02 of the closed channels n = 1 and 2, the matrix elements that occur in ψ0 (x) of Eq. (100) can finally be evaluated (see Appendix A). The resulting transmission amplitude can then be extracted from ψ0 (x) = teikx as   
e1 E − E e2 − ~2/2m 2 β 2 υ 2 κ01 κ02 E−E 12 ih i , t = tbg h e1 − (~2 /4ikm) β 2 υ 2 κ01tbg E − E e2 − (~2/4ikm) β 2 υ 2 κ02tbg − W 2 E−E 01 02 12 (107) where W12 is the sum of Eqs. (150) and (151). The transmission probability is obtained as T = |t|2 . In the coupled-channel approach as described above, keeping only two or three modes is legitimate in the weak-coupling regime [28, 29, 56, 57, 76] defined by |Vij |  |Ei − Ej |

(i 6= j),

(108)

where |Ei −Ej | is the distance in energy between two subbands. In this regime, the coupling between different channels is small and becomes progressively smaller if i is kept fixed and j becomes large enough. Therefore, only a small number of channels is involved for the calculation of transmission resonances.

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5.2.1. Fano Resonances in the Transmission Probability In the following we set ~2 /2m = 1 and take the energy unit [86] as ε0 = 17.7meV . Then the length unit is L0 = 5.7nm, and the magnitude β of the scattering potential has dimension of inverse length. We take the center of the potential at yi = 0.29L0, while d = 0.66L0. The transmission probability through the wire plotted versus the incident electron energy is shown in Fig. 12(a) for β = 3 and 3.7 in units of L−1 0 . The transmission exhibits two Fano resonances which are both of the 0 → 1 type; namely, the Fano parameter q is positive for both resonances, which means that the resonance energies occur after the energies of the transmission zeros. In order to compare the three- with the two-channel model, we also plotted the first resonance calculated with the two-channel model by the black dashed line for β = 3. The inclusion of the third channel (black solid line) appears to have a negligible effect on the resonance characteristics. Furthermore, increasing the strength of the impurity causes shifting of the transmission zeros toward lower energies while, at the same time, both resonances are broadened as reflected by an increase in the widths (denoted by γ1 and γ2) as shown in Fig. 12(b). The downward shift of the transmission zeros is a consequence of the progressively larger binding energies Eb , where

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ej − Ej = − Eb = E

2 ~2β 2 υjj 8m

(j = 1, 2),

(109)

as the impurity strength increases. In Fig. 12(b) note that the width γ2 of the second Fano resonance grows much faster than γ1 with increasing β. The physical origin of such a behavior lies in the stronger coupling of the second bound state with the continuum, resulting in a faster decay rate γ2/~ of the quasibound state into a propagating state. Note also that making the impurity potential sharper in the transverse dimension (i.e., decreasing ρ) causes suppression of both γ1 and γ2, since the bound state-continuum interactions become weaker. The black dashed line shows γ1 (for d = 0.66) calculated with the two-channel model. In Fig. 12(c) the asymmetry parameters q1 and q2 of the two resonances are plotted versus the impurity strength. They have been obtained by numerically evaluating (j) ej ER − E qj = γj

(j = 1, 2),

(j)

(110) (j)

where ER is the jth resonance energy, which is a shifted quasibound-state energy ER = ej + δj . Both q’s are positive, but q1 increases faster than q2 . This is due to the fact E that for Hamiltonians that possess inversion symmetry the Fano parameter depends only on the background transmission and not on the strength of the coupling to the quasibound level [28]. In this case, stronger background scattering leads to a larger magnitude of the asymmetry parameter. Close to the first resonance energy the background scattering is stronger, which leads to a larger asymmetry of the corresponding Fano profile. 5.2.2. Temperature Dependence of Fano Resonances In the above we discussed the Fano resonances at zero temperature. To consider thermal effects, we employ the finite-temperature conductance formula given in Eq. (19). We de-

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Figure 12. (a) Transmission probability T through the wire versus incident electron energy (E, in units of ε0 ), plotted for various values of the impurity strength ( β, in units of L−1 0 ), and for fixed values of the impurity position yi and decay length d (i.e., yi = 0.29L0 and d = 0.66L0). (b) Resonance widths (γ1 and γ2, in units of ε0 ) vs impurity strength, for yi = 0.29L0 and for two values of d. (c) Fano asymmetry parameter q vs impurity strength, for yi = 0.29L0 and for two values of d. The black dashed lines in (a), (b), and (c) illustrate the two-channel case (see text) for β = 3L−1 0 , d = 0.66L0, and yi = 0.29L0. fine a dimensionless parameter Td = kB T /ε0 which is a measure of temperature. With this definition, Td = 0.001 corresponds to T = 206mK. The temperature dependence is shown in Fig. 13. The scatterer is located at yi = 0.3L0, while we have taken β = 2.9L−1 0 and d = 0.64L0. Increasing the temperature causes rapid smearing of the second (narrow) resonance, while its amplitude decreases. The effect of temperature on the first (broader) Fano resonance is much weaker. Thus, the Fano resonance in the second subband may diminish

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for temperature ∼ 7K, while that in the first subband may persist at higher energies.

Figure 13. Fano resonances in the first and second subbands plotted for various values of the dimensionless temperature Td , where Td = kB T /ε0. Here we used yi = 0.3L0, β = 2.9L−1 0 , and d = 0.64L0. Note that the first (broader) Fano resonance persists even though higher temperatures were used.

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The origin of the above-mentioned smearing of the resonance structure is the thermal broadening, via the smooth peak in ∂f /∂µ, which obscures the resonance as kB T becomes comparable to the resonance width. Thus, depending on the resonance width, the Fano effect may almost disappear in one subband for relatively low temperatures, while it may persist in another subband even at higher temperatures. Experimental observation of thermal effects on Fano resonances in the conductance through a single-electron transistor has been reported in Ref. [42], and in the conductance through an Aharonov-Bohm ring with embedded quantum dot in Refs. [43, 44]. 5.2.3. Effect of a Transverse Electric Field It has been shown both experimentally [88] and theoretically [89, 90, 17, 20] that shifting the position of the impurity with respect to the “walls” of a constriction causes drastic change of the conductance. For instance, if a point attractive impurity falls on the central axis of the channel, the conductance exhibits no resonance structure in the odd subbands; that is, the resonances strongly depend on the impurity position. The shifting of the impurity can be achieved by, for example, applying different gate voltage to the two parts of a split gate [88], which can be thought of as an applied electric field in the transverse direction of the constriction. This causes shifting of the confining potential, which is equivalent to a “shifting” of the impurity in the opposite direction. Hence an external electric field may be used as a means for controlling the resonance structure. It is desirable therefore to give a detailed account of the effects of an electric field on the Fano resonance. In the following we define a dimensionless parameter ξ, as ξ = eF/mω02 L0. Then, for ξ > 0 the electric field points in the negative direction, and for ξ < 0 it points in the positive direction. In the following we discuss the case in which the electric field points in the negative direction, and show that varying the field strength may cause collapse of the Fano line shape.

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We consider an impurity located at yi = 0.38L0, having strength β = 2.9L−1 0 and decay length d = 0.62L0. The transmission probability through the wire plotted versus the incident electron energy is shown in Fig. 14, for various values of the parameter ξ. The influence of the electric field on the two Fano resonances is rather different. Increasing the field strength causes a systematic downward shifting of the second transmission zero while, at the same time, the resonance width γ2 first increases and then decreases. On the other hand, the energy of the first transmission zero initially shifts upward while the resonance width γ1 gradually decreases. When the field strength becomes such that ξ = 0.38, γ1 shrinks to zero resulting in the collapse of the Fano resonance in the first subband. Further increasing the field strength to ξ = 0.57 the Fano profile is recovered but is down-shifted. The vanishing of γ1 for ξ = 0.38 implies that the coupling υ01 between the first bound state and the continuum of states also vanishes. In fact, this coupling becomes zero as soon as l = yi (or ξ = yi /L0), that is as soon as the field strength becomes such that the center of the confining potential and, consequently, the node of the second channel mode coincides with the impurity center. This can be seen explicitly from the expression of υ01, r   (yi − l)/d2 2~ −(yi − l)2/d2 υ01 = exp . (111) 2 1 + (~/mω0d2) [1 + (~/mω0d2 )]3/2 πmω0 d

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Similar collapsing behavior of Fano profiles has also been found to occur elsewhere; for example, when varying the size of a square well impurity [29] in a straight quantum waveguide, and also when varying the position of a point impurity placed in one arm of a 1D mesoscopic ring [51, 50]. The upward and the subsequent downward shifting in energy of the first transmission zero is a consequence of the competition between: i) the interaction υ11 of the second channel mode with the impurity, and ii) the Stark shift of the energy threshold E1. This can be seen explicitly from the expression of the bound state energy, 2 2 2 2 2 e1 = 3 ~ω0 − e F − ~ β υ11 . E 2 2mω02 8m

(112)

In the last expression it can be seen that while the Stark shift (second term) continuously increases with increasing field strength, the interaction υ11 (third term) initially decreases until it becomes zero. The decrease of the third term however is faster than the increase of e1 and, therefore, in the upward the Stark shift, resulting in the upward displacement of E shifting of the transmission zero. After the node of the second mode passes through the impurity center (where υ11 → 0), the third term starts increasing. This results in the decrease e1 and, therefore, in the downward shifting of the transmission zero. of E e2 of the second bound state is similar On the other hand, the equation for the energy E in form to Eq. (112) with (3/2) ~ω0 and υ11 replaced by (5/2) ~ω0 and υ22, respectively, i.e., 2 2 2 2 2 e2 = 5 ~ω0 − e F − ~ β υ22 . (113) E 2 2 8m 2mω0 For the range of values of the parameter ξ that we use here, the electric field-induced shift of the third mode leads only to an increase of υ22, which results in a continuous decrease e2. Thus, the second transmission zero shifts only toward lower energy values. of E

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The influence of an electric field on the resonant behavior of the transmission probability can further be described by the pole structure of the transmission amplitude in the complex energy plane. In fact the imaginary part of a pole (which determines the resonance width γj ) is related to the time an electron spends in the quasibound state; namely, γj /~ is the probability per unit time of an electron in a quasibound state to leave this state. The poles are given by the zeros of the denominator of the transmission amplitude Eq. (107), i.e., by the solutions of 

e1 − E−E

2 κ tbg ~2β 2 υ01 01 4ikm

  2 2 2 bg 2 e2 − ~ β υ02κ02t E−E − W12 = 0. 4ikm

(114)

For the trajectories of the poles as a function of ξ, and for the collapsing behavior of the second Fano resonance when the electric field points in the positive direction the interested reader can look at Ref. [91]. Further insight into the resonance structure and the collapsing behavior discussed above is provided by the phase of the transmission amplitude given as

tan θ =

Im[t] . Re[t]

(115)

For an isolated resonance it may be written as

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tan θj = 

γj (j)

ER − E

,

(116)

where j = 1, 2. It is well-known that the transmission phase is almost constant away from a resonance and changes by π as the energy of the incident electron is being scanned through a quasibound level [87]. However, the quasibound states are proportional to the coupling ∗ strengths [through the matrix elements h(χ− k ) |V0j |Φ0j i of Eq. (100)] and therefore vanish whenever these couplings become zero (i.e., when the Fano profiles collapse). When this happens, there is no resonant level to interfere with the background and no π phase change is expected. In Fig. 15 the phase is plotted versus the incident electron energy, for three values of ξ. The values of the impurity parameters are the same as those used in Fig. 14. As expected, away from the resonant levels the phase remains zero, but it increases rapidly to π/2 as the energy approaches a resonance from below. At resonance, the phase changes abruptly to −π/2 and as soon as the energy has crossed the resonance, it rapidly becomes zero again. The phase change is more abrupt for the first (narrower) resonance, while it is smoother for the second (broader) resonance (see also Fig. 14). At ξ = 0.38, for which the collapse of the first resonance occurs, we note that there is no π phase change. Similar behavior of the transmission phase has also been found in the case of a square well impurity [29]. The constancy of the phase with respect to energy is also related to the absence of resonant space charge in the local electron density [92].

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Figure 14. Transmission probability T through the wire vs incident electron energy ( E, in units of ε0 ), plotted for various positive values of the dimensionless parameter ξ, where ξ = eF/mω02 L0. Here we used yi = 0.38L0, β = 2.9L−1 0 , and d = 0.62L0 , respectively. It is seen that for ξ = 0.38 (which corresponds to l = yi , where l = eF/mω02), the Fano structure in the first subband collapses.

Figure 15. Phase of the transmission amplitude vs incident electron energy ( E, in units of ε0 ), plotted for three values of ξ. The values of the impurity parameters are the same as those used in Fig. 14. Note that at ξ = 0.38 (for which the first Fano profile collapses), the phase is constant as the energy crosses the quasibound level. Note also that the π phase change is more abrupt if the resonance is narrower, while it is smoother if the resonance is broader.

5.3.

P¨oschl-Teller Scattering Potential

We consider now scattering of electrons by a “smooth” finite-range attractive potential of the P¨oschl-Teller type, " # 1 (y − yi )2 ~2 β exp − , (117) V (x, y) = − 2m cosh2 (αx) d2

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where α−1 is the decay length of the potential along the propagation direction, while d is the decay length in the transverse direction. As before, β is the strength of the scattering potential (β > 0). We employ the two-channel model in the single-subband regime E0 < E < E1 . In order to find the scattering states χ± k we solve Eq. (83) in the same way as in
a 1D scattering problem [87] with an attractive potential V00 (x) = − ~2β/2m υ00/ cosh2 (αx) . The asymptotic form of the wave function as x → −∞ can be expressed as Γ (ik/α) Γ [1 − (ik/α)] Γ (−ik/α) Γ [1 − (ik/α)] + eikx . Γ (−v) Γ (1 + v) Γ [(−ik/α) − v] Γ [(−ik/α) + v + 1] (118) i h p Here Γ (z) is the standard Gamma function, and υ = (1/2) −1 + 1 + (4βυ00/α2) . −ikx χ+ k (x) ∼ e

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The reflection amplitude is the ratio of coefficients in the function χ+ k . The bound state energies in the potential V11 (x) are found as " # r 2 2 ~ α 4βυ 11 ej = E1 − − (1 + 2j) + 1 + , (119) E 8m α2 where j = 0, 1, 2, . . .. Here we consider only one bound state (see Ref. [32] for the necessary conditions such that only one bound state exists). The normalized bound state wave function that corresponds to the first energy level is Φ01 (x) = (α/2)1/2 cosh−1 (αx). The calculation of the matrix elements which are necessary for the wave function ψ0 (x) has been done in Ref. [23]. The resulting transmission coefficient  2 e0 2 E − E , (120) T = tbg  2 e0 − δ0 + (Γ0 )2 E−E is of the Fano form. In Eq. (120), δ0 and Γ0 are the shift and width that the original bound e0 acquires. It can be transformed to the asymmetric Fano function if we state energy E   e0 /Γ0 = δ0 /Γ0 , where define reduced variables  = (E − ER) /Γ0 and q = ER − E e0 + δ0 is the resonance energy, E e0 is the energy of the transmission zero, and q is ER = E the asymmetry parameter. For the P¨oschl-Teller potential the Fano parameter is given as [32] h i p 2) cos (π/2) 1 + (4βυ /α 00 δ0 . (121) =− q= Γ0 sinh (πk/α) Note that varying the size α or the strength β of the potential, q oscillates between positive and negative values. This oscillatory behavior causes sequential reversals of the Fano line shape. As this happens, the resonance energy occurs before,h after, or equals i the energy 2 2 of the transmission zero. In particular, if βυ00/α = (1/4) (2n + 1) − 1 where n = e0 . In this case the transmission exhibits symmetric 1, 2, 3, . . ., we have q = 0, and ER = E 2 e0. In this case Breit-Wigner dips. For 0 < βυ00/α < 2, we have q > 0 and ER > E

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the transmission peak follows the transmission dip (i.e., it is a 0 → 1 Fano resonance). For e0 (i.e., it is of the 1 → 0 type). In this case 2 < βυ00/α2 < 6, we have q < 0 and ER < E the location of the pole is switched with the zero energy and the dip follows the peak (i.e., there is a reversal of the Fano line shape). The reversal of the Fano line shape is shown in Fig. 16(a), where the transmission probability is plotted versus the electron energy for three values of α. We have used β = 7.5L−1 0 , d = 0.63L0, and yi = 0.06L0. Note the transformation of the 0 → 1 type Fano resonance (α = 1.8) to a symmetric Breit-Wigner antiresonance (α = 1.23), and subsequently to a 1 → 0 type Fano resonance (α = 0.98). The reversal of the Fano line shape indicates that the roles of destructive and constructive interference between resonant and nonresonant transmission paths have been reversed.

Figure 16. (a) Transmission probability T through the wire versus incident electron energy (E, in units of ε0 ), plotted for three values of α, where α is the inverse decay length of the P¨oschl-Teller scatterer. Note the reversal of the Fano line shape for α = 0.98. (b) Fano asymmetry parameter q versus α, for various values of d, where d is the decay length of the scatterer in the transverse dimension. Note that q changes sign as α varies.

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The Fano parameter q plotted versus α is shown in Fig. 16(b) for three values of d. We have used β = 7.5L−1 0 and yi = 0.06L0 . Note that q oscillates more frequently as α decreases (i.e., as the size of the P¨oschl-Teller potential increases). These oscillations become gradually weaker as d decreases, which means that the Fano resonances tend to become symmetric as the size of the potential increases in the propagation direction.

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6.

Resonances in Mesoscopic Open Rings

As mentioned in the Introduction, a particular class of mesoscopic systems in which the Fano effect shows up is the 1D ring connected to current leads with [50, 51] or without [52] a scatterer in one of its arms. In these systems the electron motion between the junctions is purely one-dimensional (i.e., no interaction between channels occurs). Thus the usual interpretation of the Fano effect as being due to the interference of two transmission paths does not apply in these systems. Fano resonances have also been investigated in the case of a ring with direct coupling between the leads [49] and discussed in the case of an ideal 1D double ring [48]. Further interesting effects in mesoscopic open rings (see Fig. 17) occur in the presence of Aharonov-Bohm flux. In this case the transmission probability and, in fact, all transport properties are periodic functions of the magnetic flux with period Φ0, where Φ0 = hc/e is the flux quantum. Furthermore, if there is no scatterer in the branches the transmission probability of the ring exhibits Breit-Wigner resonances near the energies of the electronic states of the closed ring [47]. The sharpness of these resonances depends on the strength of the coupling between the leads and the ring. In the weak coupling regime (i.e., when the transmission probability is small), an electron entering the ring spends a long time there before being reflected or transmitted. Electrons which spend a long time in the ring must be in an eigenstate of the closed ring. In this case the poles of the transmission amplitude have small imaginary parts, which means that the resonances are narrow. In the strong coupling regime, the electrons traverse the ring without much scattering, and the resonances transform into broad oscillations. If there is scattering in one (or both) branches the transmission probability in the strong coupling regime exhibits Fano resonances, while in the weak coupling regime it exhibits Breit-Wigner resonances. In addition, an Aharonov-Bohm flux between the two electronic paths causes interference of the wave functions, which gives rise to a circulating (or persistent) current. The persistent current is also periodic in the magnetic flux with period Φ0. Since the prediction [93] and experimental observation [94] of the persistent current in isolated rings, much effort has been devoted to the description of this effect in various open ring geometries [52, 49, 48, 95, 96, 97, 98, 99, 100, 101, 102]. Interesting results were obtained in these investigations, such as persistent current magnification that occurs under certain circumstances [48, 96, 98, 100, 102], the divergence of persistent current [48, 96], the creation of long-living states inside a ring coupled to a reservoir [99], to name a few. In this section we discuss the most important effects related to resonant electronic transport through 1D Aharonov-Bohm open rings connected to current leads. First, we describe the coupling of the ring to the leads via an S matrix, which can be used to derive the transmission probability of the ring. The case of scattering in one branch is considered, as well

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Figure 17. Geometry of a ring connected to current leads and threaded by a flux Φ. There is a scattering potential in the upper arm at point D. as the effect of the magnetic flux on the transmission. We use the simple model of the Dirac δ function as a scatterer, which reveals the basic features of the problem.

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6.1.

Transmission Probability of the Ring

We consider electron transport through a ring connected to current leads in the presence of an Aharonov-Bohm flux, as shown in Fig. 17. For simplicity the leads and the ring are taken to be strictly 1D, i.e., no subband structure is taken into account. The current flow is considered to be from left to right. At a junction of a lead with the ring, the three outgoing waves with amplitudes (α0, β 0, γ 0) are related by an S matrix to the three incoming waves with amplitudes (α, β, γ), O = SI, (122) where O represents the outgoing and I the incoming waves, and  √ √  ε ε − (a + b) √  ε a b . S= √ ε b a

(123)

Current conservation implies that S is unitary and therefore we obtain the relations (a + b)2 + 2ε = 1,

(124)

a2 + b2 + ε = 1.

(125)

The coefficients a and b can be expressed as functions of ε by using Eqs. (124) and (125),
1 √ 1 − 2ε − 1 , 2
1 √ 1 − 2ε + 1 . b= 2

a=

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(126) (127)

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The coupling parameter ε can be in the range 0 ≤ ε ≤ 1/2. In the limit ε = 1/2, the junction is completely transparent for incoming electrons and the current lead is strongly coupled to the ring. In the other limit, ε = 0, incoming electrons from the current lead are totally reflected, which means that the current leads and the ring are completely decoupled. In the presence of a scattering potential in one arm and an Aharonov-Bohm flux Φ, the Schr¨odinger equation describing the electron motion in the ring can be written as # "   2π Φ 2 ~2 ∂ −i + Vi (y) ψ(y) = E ψ(y), (128) − 2m ∂y L Φ0 where y is the coordinate along the ring, Φ0 = hc/e is the flux quantum associated with a single charge e, L is the circumference of the ring, and Vi (y) is the scattering potential in the upper arm. The lengths of the two arms, L1 and L2, are taken to be equal, i.e., L1 = L2 = L/2. The scatterer is described by a Dirac δ function Vi(y) = γδ(y − yi ),

(129)

where yi is the position of the scatterer, and the magnitude of γ sets the strength of the potential, which we take to be repulsive ( γ > 0). The wave functions in the leads are of the form (130) ψj (x) = αj eikx + α0j e−ikx ,

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where j = 1 or 2 for the left or right lead respectively, x is the coordinate along the leads, and k = (2mE)1/2/~. The wave functions in the ring are expressed as   ψ2(y) = e2πi(Φ/Φ0 )(y/L) β10 eiky + β1e−iky , (131)   ψ3(y) = e2πi(Φ/Φ0 )(y/L) c1eiky + c2e−iky ,

(132)

  ψ4(y) = e−2πi(Φ/Φ0 )(y/L) γ10 eiky + γ1e−iky ,

(133)

where ψ2(y) and ψ3(y) are the wave functions before and after the impurity potential respectively, while ψ4(y) is the wave function in the lower arm (see Fig. 17). The amplitudes β2 and β20 to the right of the scatterer are expressed in terms of the amplitudes β10 and β1 to the left of the scatterer with the help of a transfer matrix. The transfer matrix is found using a standard procedure, i.e., by applying the boundary conditions at the impurity and then expressing the coefficients c1, c2 in the resulting equations in terms of β2 and β20 . Inserting y = L1 into ψ3(y) we get two terms, the first representing β2 and the second β20 . We finally get   0   β1 β2 iπ(Φ/Φ0 ) Ti =e , (134) β20 β1 where the transfer matrix Ti is given as  (1/t∗) eiq Ti = ∗ (r /t∗ ) e−iq e2iqsi

(r/t) eiq e−2iqsi (1/t) e−iq



.

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(135)

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In Eq. (135) we have used a dimensionless wave vector defined as q = kL1 and we have also set si = yi /L1. The transmission and reflection amplitudes of the scattering potential are given as 2iq t= , (136) 2iq − V and r=

V , 2iq − V

(137)

where we have used the dimensionless parameter V = 2mγL1/~2 to characterize the strength of the scatterer. Here we point out that a scatterer with a small spatial extension d and a magnitude of γ ¯ can be approximated by a δ function potential with γ = d¯ γ and thus V = 2m¯ γdL1 /~2. Choosing an experimentally realizable [43, 103, 104] arm length L1 = 1µm, the effective mass for GaAs (m = 0.067m0), and d = 0.01µm, the unit of the scatterer’s parameter V = 1 corresponds to γ ¯ = 0.12meV . We consider a wave of unit amplitude α1 = 1, incident from the left. In order to find the transmission probability of the ring T = |α02|2 we have to determine the amplitudes β10 , β20 , γ10 , γ20 under the condition α2 = 0. For this reason we use the S matrix at the two junctions, in accordance to Ref. [47], in order to find these amplitudes. We briefly outline this procedure in Appendix C. The transmission amplitude can finally be obtained from Eq. (173) as ε p , (138) α02 = − 2 eiπ(Φ/Φ0 ) b det (Π) where −1

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p = det (Π) [( 1 0 ) + ( 0 1 )] Ti Π



1 −1



,

(139)

with Π given by Eq. (172). The poles of the transmission amplitude, i.e., the solutions of det(Π) = 0, determine the resonant behavior of the transmission probability. The transmission probability of the ring can therefore be written as T = |α02 |2 =

ε2 |p|2 . b4 | det(Π)|2

(140)

For ε = 0 the current leads and the ring are completely decoupled. In this case [47] det(Π) = 0 is just the eigenvalue equation of the closed ring (in the absence of scatterer), and the poles of the transmission amplitude are the eigenvalues of the closed ring. 6.1.1. Aharonov-Bohm Effect The Aharonov-Bohm (AB) effect tests the sensitivity to an applied magnetic flux Φ of electrons in a path which enclose this flux. In the vacuum AB effect the electron travels in a field-free region and therefore does not “feel” the magnetic field. It is a pure quantum effect, which is a direct consequence of the wave nature of electrons and the gauge invariance of the velocity. As mentioned above, the AB effect also shows up in electron transport through a 1D ring coupled to leads and threaded by a magnetic flux, as shown in Fig. 17. Then, the AB effect is manifest in a periodic flux dependence of all the transport properties.

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In particular, the transmission coefficient (and, consequently, the conductance) of the ring, is a periodic function of the external magnetic flux, with period Φ0 = hc/e. This is illustrated in Fig. 18, for ε = 0.4 and q = π/2. We have defined the dimensionless flux F = Φ/Φ0. Curve (1) shows the case in which there is no scattering in the arm, and curve (2) shows the case in which there is a scattering potential in the upper arm with strength V = 3 (see discussion below Eq. (137)).

Figure 18. Transmission probability T through the ring as a function of the AB flux F , where F = Φ/Φ0 for ε = 0.4 and q = π/2. The solid line corresponds to the case of no scatterer in the arm, while the dashed line corresponds to the case in which there is a scatterer in the upper arm.

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6.1.2. Ring without Scattering in the Branches We first discuss the resonant behavior of the transmission probability through a ring with no scattering in the arms. In Fig. 19 the transition from weak to strong coupling is illustrated, in the presence of a flux Φ = 0.1Φ0 (solid lines) and in the absence of flux (dashed lines), for ε = 1/14, 1/5, and 1/2 in Fig. 19(a), (b), and (c) respectively. The BreitWigner resonances in the absence of magnetic flux are due to constructive interference of the electron waves in the two arms and they occur at q = nπ, where n = 1, 2, 3, . . .. The flux Φ introduces phase shifts eiθ1 and e−iθ2 in the two branches of the wave function along the ring where θ1 + θ2 = 2πΦ/Φ0. This modifies the interference of the electron waves at the point of connection to the leads. The Breit-Wigner resonances have been related to the eigenvalues of the closed ring [47]. In the closed ring ( ε = 0), the poles of the transmission amplitude lie on the real axis. However, as the ring-lead coupling ε increases (i.e., the transmissivity of the junction increases), these poles move away from the real axis into the complex energy plane, resulting in broader resonances. In fact, γn /~ (where γn is the resonance width) is the probability per unit time of an electron in an eigenstate of the closed ring to leave this state (i.e., to leave the ring). It has been shown that γn ∝ ε. The interference is constructive only in the weak coupling limit and the probability to find the electron in the ring is large. Note also that, in the absence of external flux, the transmission is unity in the strong coupling limit.

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Figure 19. Transmission probability T through a symmetric ring (L1 = L2 ) versus dimensionless wave vector q/2π, where q = kL1, for three values of the coupling parameter ε, and no scattering in the upper arm. Solid lines correspond to Φ = 0.1Φ0, while the dashed lines correspond to Φ = 0.

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Figure 20. Transmission probability T through a symmetric ring (L1 = L2 ) versus dimensionless wave vector q/2π, where q = kL1, for ε = 1/2 and Φ = 0. The scattering potential, which is present in the upper arm, has strength V = 8, and is located at si = 0.42, where si = yi /L1. The transmission probability exhibits Fano resonances. We also mention that, in the weak coupling regime, there are two poles associated with a resonance pair. As the coupling strength ε reaches a critical value ε∗ , the two poles coalesce. For ε > ε∗ , one pole moves deeper into the complex energy plane, whereas the other one moves toward the real axis as ε increases.

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6.1.3. Ring with Scattering in the Branches In the case there is additional scattering originating from an impurity in one arm, the transmission probability may exhibit Fano resonances. We illustrate this in Fig. 20 for Φ = 0 in the strong coupling limit ε = 0.5. For the scatterer we used V = 8, si = 0.42 (where si = yi /L1), and L1 = 1µm. In the strong coupling limit the junctions are transparent to incident electrons. However, they scatter in the upper arm, make a round trip and scatter few more times before they exit the ring. This is similar to the case in which electrons exist in a quasibound state. There is interference of these electrons with the continuum in the leads, resulting in the Fano line shapes in the transmission probability. It is also seen that all Fano resonances are 1 → 0 type. That is, the resonance energies occur before the energies of the transmission zeros.

7.

Concluding Remarks

In the past two decades advances in microfabrication have made possible the confinement of electrons in a conductor with lateral extent of few nanometers, leading to narrow channels (or constrictions). The ultrasmall size of these structures allows the phase coherence length of the electron to become larger than the system’s dimensions. As a result, the system becomes essentially an electron waveguide, in which the transport properties are mainly determined by the impurity configuration, the geometrical characteristics of the

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conductor, and the principles of quantum mechanics. In this article we have presented a review of quantum ballistic transport and the effect of elastic scattering in narrow wires. In contrast to other methods used in the past, we based our discussion of electron scattering in the Lippmann-Schwinger equation. The most important quantum effect of the transmission, the quantization of conductance, was reviewed and the effects of various types of scattering potentials were discussed. The type and size of the scattering potential becomes important in the detailed structure of conductance. The conductance through a Gaussian-type scatterer most closely resembles the recent experimental results [67]. We also discussed the general properties of resonance line shapes in Q1D systems where the asymptotic motion of the scattered particle is confined. The Fano effect in electron transport through quantum wires was treated here by means of the Feshbach approach, which provides microscopic expressions for all line shape parameters. The most general resonant line shape in Q1D systems (which includes the Breit-Wigner line shape as a special case) is described by the asymmetric Fano function. It has been suggested that the Fano resonance can be used as a means for probing the phase coherence of electrons [30]. Fano resonances can also provide us with information about the parameters of the scattering potential. An interesting future experiment would be the determination of the impurity parameters from knowledge of the resonance line shape. However, not all resonances can be described in terms of quasibound states interfering with the continuum. An example of a system which does not support quasibound states, but exhibits Fano line shapes, is the 1D symmetric ring coupled to current leads. In this system, even though the motion between the junctions is purely one-dimensional, one finds asymmetric Fano resonances in the transmission probability. The presence of a magnetic flux piercing the ring introduces phase shifts eiθ1 and e−iθ2 in the two branches of the wave function along the ring, where θ1 + θ2 = 2πΦ/Φ0. This modifies the interference of the electron waves at the junction and leads to Aharonov-Bohm oscillations in the transmission probability through the ring.

Appendix A. Evaluation of Matrix Elements In this appendix we present the evaluation of the matrix elements that occur in the wave function of Eq. (100) for the impurity potential given by Eq. (101). First, the retarded Green’s function in 1D can be expressed in terms of the scattering states as  + −  χk (x)χk (x0)

(x > x0)

0 − χ+ k (x )χk (x)

(x < x0)

m G0(x, x0) = 2 bg ×  i~ kt

,

(141)

0 where χ± k (x) are the solutions of Eq. (83). Using this representation of G0 (x, x ), the explicit expression of the potentials V0j given by Eq. (102), and the bound states Φ0j we

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then get

Z ∞    ~2 β √ bg ikx ∗ −ikx (χ− ) |V |Φ = − υ dx e + r e δ(x) κ0j e−κ0j |x| 0j 0j 0j − k 2m −∞ ~2 β √ =− υ0j κ0j tbg , (142) 2m

bg where j = 1, 2. We have also used the fact that 1 + r− = tbg . For Wj0 we obtain

Wj0 =

hΦ0j |Vj0|χ+ ki

=−

~2 β 2m

~2β =− υj0 2m

Z

∞

√ dx κ0j e−κ0j |x| δ(x)tbg eikx

−∞

√ υj0 κ0j tbg ,

(143)

where j = 1, 2. For the evaluation of the self-energy term in εj , we use G0 (x, x0) from Eq. (141) to rewrite this term in the following form, D

Z ∞ Z ∞ m + 0 dx dx0Φ0j (x)Vj0(x)Φ0j (x0)Vj0(x0)χ− k (x)χk (x ) i~2ktbg −∞ −∞ Z ∞ Z x m + 2 bg dx dx0Φ0j (x)Vj0(x)Φ0j (x0)Vj0(x0) i~ kt −∞ −∞   − 0 − + 0 (x)χ (x ) − χ × χ+ k k k (x)χk (x )

E ˆ 0V0j |Φ0j = Φ0j |Vj0G

= I1 + I2 .

(144)

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Inserting the expressions for the bound states Φ0j , the coupling potentials Vj0 , and the scattering states into I1, we get  2 2 m ~ β 2 υ0j κ0j I1 = 2 bg i~ kt 2m Z ∞  Z bg ikx −κ0j |x| −ikx × dxe δ(x) e + r− e −∞

=

m i~2k



~2β 2m

2

∞


0 0 dx0e−κ0j |x | δ x0 tbg eikx

−∞ 2 υ0j κ0j tbg .

(145)

The expression in brackets in the second double integral of Eq. (144) can be represented [28, 76] as follows − 0 − + 0 χ+ k (x)χk (x ) − χk (x)χk (x ) =

1 ∗ (tbg )

 +

 0 ∗ ∗ + 0 χk (x)χ+ − χ+ k x k (x) χk x

 + ∗ + 0
 2i Im χk (x) χk x ∗ (tbg ) 
 = −2i tbg sin k x − x0 . =

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(146)

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With the help of Eq. (146), the second double integral on the right hand side of Eq. (144) can then be shown to vanish,  2 2 ~ β m 2 I2 = 2 bg υ0j κ0j i~ kt 2m Z ∞ Z x

 0 × dxe−κ0j |x| δ(x) dx0 e−κ0j |x | δ x0 (−2i) tbg sin k x − x0 −∞ −∞   2  Z ∞ 1 ~ 2 2 =− β υ0j κ0j dxe−κ0j |x| δ(x) sin (kx) Θ(x). k 2m 0 = 0,

(147)

(148)

where Θ(x) is the unit step function. The self-energy term Eq. (144) is therefore given by the resulting expression Eq. (145). The energies εj in Eq. (97) can then be written as ej + m εj = E i~2k



~2β 2m

2

2 υ0j κ0j tbg .

(149)

Now, the evaluation of the first matrix element in Eq. (99) can easily be performed with the result √ √ ~2 β υ12 κ01 κ02 . (150) hΦ01|V12|Φ02i = − 2m Furthermore, the evaluation of the second matrix element in Eq. (99) is performed in a similar fashion as in Eq. (144) with the final result

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ˆ 0 V02 |Φ02i = m hΦ01| V10G i~2k



~2β 2m

2

√ √ υ01υ02 κ01 κ02 tbg .

(151)

Adding the last two equations will give us the result for the indirect coupling W12 of the closed channels via the open channel. Substituting the expressions of the above matrix elements in the wave function of Eq. (100) we finally get the transmission amplitude given by Eq. (107).

Appendix B. Range of Validity of the Two-Channel Approach In this appendix we further discuss the effect of the third (closed) channel on the Fano resonance that occurs in the first subband. As mentioned in Sec. VB, the matrix element W12 of Eq. (99) represents i) the direct coupling of the two closed channels (which are dominated by their bound states), and ii) the indirect coupling of these states via the continuum. As shown below, it turns out that the smallness of |W12| with respect to |ε1 − ε2 | determines the extent to which the two-channel model is valid. First, let us recall from Eq. (97) that ε1 and ε2 are the resonant energies as if there were no interaction between the two closed channels. These can be written as e1 + δ1 − iγ1 = E (1) − iγ1, ε1 = E R

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(152)

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e2 + δ2 − iγ2 = E (2) − iγ2, ε2 = E R

(153)

e1 , where δ1 , δ2 and γ1, γ2 are the shifts and widths that the original bound state energies E e2 acquire. Now, in the presence of interaction W12 between the two closed channels, E the resonant energies are modified. The degree of modification depends, of course, on the strength of this interaction. The modified resonant energies are determined by the poles of the transmission amplitude, i.e., by the solutions of 1/t = 0. The transmission amplitude can be extracted from Eq. (100) as ψ0(x) = teikx , and therefore the poles are obtained as solutions of 2 (E − ε1 ) (E − ε2 ) − W12 = 0. (154) The solutions of Eq. (154) are given as Ep(±)

ε1 + ε2 = ± 2

r

(ε1 − ε2 )2 2. + W12 4

(155)

(±)

The poles, Ep , represent the resonant energies in the presence of the interaction W12. Let us suppose that |W12| is small compared with the relevant energy scale [i.e., the difference |ε1 − ε2 | between the resonance energies of the “unperturbed” (noninteracting) case], |W12|  |ε1 − ε2 |. (±)

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We can then expand the energies Ep

(156)

to first-order as

Ep(+) = ε1 +

2 W12 , ε1 − ε2

(157)

Ep(−) = ε2 −

2 W12 . ε1 − ε2

(158)

2 /(ε − ε ), the bound state energies E e1 and E e2 Due to the first-order corrections, ±W12 1 2 0 0 (0) (0) (0) (0) acquire additional (small) energy shifts δ1 , δ2 and widths γ1 , γ2 , so that (δ1 , δ2) = 0 0 (0) (0) (0) (0) (δ1 + δ1 , δ2 + δ2 ) and (γ1, γ2) = (γ1 + γ1 , γ2 + γ2 ). (+) Let us consider Ep , which corresponds to the first Fano resonance. From the condition given in Eq. (156), it is seen that

|W12|2  |W12|. |ε1 − ε2 |

(159)

This implies that the magnitude of the first-order correction in Eq. (157) is even smaller (0) than |W12| and, consequently, it is even much smaller than |ε1 − ε2 |. In this case, δ1  δ1 0 (0) (+) and γ1  γ1, and one can assume that Ep ' ε1 (that is, one can assume that the shift δ1 0 and width γ1 in the presence of the interaction are approximately equal to the shift δ1 and width γ1 in the absence of interaction). This means that the position and width of the Fano resonance are unaffected by the presence of the third (closed) channel. Thus, we conclude that under the condition given in Eq. (156) one can use the two-channel model to describe a Fano resonance in the first subband.

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In particular, as Eq. (99) suggests, the interaction |W12| of the two closed channels depends on the impurity strength and position through the coupling potentials Vnl , i.e., |W12| ∼ |V12 + [V10V02/(E1 − E0 )]|. Obviously, as the impurity strength decreases, the interaction |W12| becomes gradually weaker (since Vnl decrease), thus validating use of the two-channel approach. For the model impurity of Eq. (101) this is illustrated in Figs. 13(b) and (c), where the two- and three-channel models yield essentially identical results for small impurity strengths. Furthermore, as the impurity position shifts toward the symmetry axis of the wire, the potentials V12 and V01 are gradually reduced. Consequently, |W12| is also reduced, thus validating again use of the two-channel model. The inclusion of even more channels can be done in a similar way. However, it is expected that the two-channel model will still remain valid at least in the regime of small impurity strengths and when the impurity is close to the central axis of the wire.

Appendix C. Calculation of Wave Function Amplitudes in the Ring At the junction to the right we find from Eq. (122), √ √ α02 = εβ2 + εγ2 ,

(160)

β20 = aβ2 + bγ2,

(161)

γ20 = bβ2 + aγ2.

(162)

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We use Eqs. (161) and (162) in order to express the γ 0s in terms of the β 0 s gives  0    γ2 β2 = Tj , γ2 β20

(163)

where Tj is a matrix given as 1 Tj = b



b2 − a2 −a




a 1



.

At the junction to the left we find with the help of Eq. (122) and using α1 = 1, √ √ α01 = −(a + b) + εβ1 + εγ1, √ β10 = ε + aβ1 + bγ1, √ γ10 = ε + bβ1 + aγ1. Using Eqs. (166) and (167) we express the β 0 s in terms of the γ 0s. We then get   0  √    ε b−a β1 γ1 + Tj = , β1 γ10 −1 b

(164)

(165) (166) (167)

(168)

While the amplitudes in the upper arm are transferred according to Eqs.(134) and (135), for the lower arm we insert y = L2 into ψ4(y) of Eq. (133) which then yields two terms, the

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first being γ2 and the second γ20 . Eliminating γ2 and γ20 from these two terms by employing Eq. (163) yields     γ1 β2 iπ(Φ/Φ0 ) 0 =e Tj , (169) γ10 β20 where Tj0 is given as Tj0

1 = b




 b2 − a2 eiq aeiq , −ae−iq e−iq

(170)

Using Eqs. (168), (169) and (134) yields an equation for β10 , β1 which can be written as  0   √  ε b−a β1 =− Π , (171) β1 −1 b where Π = Tj eiπ(Φ/Φ0 )Tj0 eiπ(Φ/Φ0 ) Ti − 1.

(172)

Eliminating γ2 in Eq. (160) with the help of Eq. (161) yields an expression for the transmission amplitude in terms of β2 , β20 √
ε 0 α2 = β2 + β20 . (173) b In Eq. (173) we have also employed the relation b − a = 1. The amplitudes β2, β20 in Eq. (173) are determined by first solving Eq. (171) for β1 , β10 and using Eq. (134).

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In: Micro Electro Mechanical Systems ISBN: 978-1-60876-474-7 c 2010 Nova Science Publishers, Inc. Editors: B. Ekwall and M. Cronquist, pp. 375-381

Chapter 13

L ATTICE B OLTZMANN M ODEL AS A N I NNOVATIVE M ETHOD FOR M ICROFLUIDICS Lajos Szalm´as Department of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, 38344 Volos, Greece

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Abstract The lattice Boltzmann model is an innovative method to simulate gaseous or liquid flows. It successfully fills the gap between the macroscopic description, the NavierStokes equation, and particle based methods, such as the direct simulation of MonteCarlo. In the last years, the lattice Boltzmann model has attracted increasing attention in modeling flows in microfluidic devices. This area has developed rapidly. Microelectromechanical systems (MEMS), gaseous sensors, lab-on-chips produce significant grow in various areas of engineering and technology. In this chapter, we give an overview of the lattice Boltzmann method and its key concepts. We review recent developments related to the field of microflows and microfluidics. Particular attention is paid for non-continuum behaviours observed at the micro-scale, such as the slip and jump of macroscopic variables in a micro-device. Finally, we provide further ideas and directions which can serve as targets of future developments. Beyond the lattice Boltzmann model, these ideas can stimulate progress and investment in applications and technology.

1.

Introduction

In the last few decades, the lattice Boltzmann method (LBM) has matured into a flexible computational platform in modeling complex hydrodynamic problems. The essence of the method is that the description of the media starts at the kinetic scale instead of the usual hydrodynamics which operates with macroscopic quantities. Due to this kinetic nature, the model is well-suited for modeling microflows. In recent years, the field of microflows has undergone a rapid development [1]. These flows show special physical behaviors due to the increased role of the molecular interactions. The flow pattern and the profile of the fluid dynamic variables in a microfluidic device

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basically differ from the ones observed at the macro-scale. Understanding and modeling flows under such circumstances are challenging and important. There is a huge interest in microflows in engineering and technology. In this chapter, I give an introduction into the LBM and overview its main concepts. Then, I discuss recent developments in the LBM for modelling microflows. Finally, some challenges for further research in the LBM are presented.

2.

Concepts of the LBM

Historically, the LBM was derived from its predecessor the lattice gas automate (LGA). McNamara and Zanetti replaced the Boolean particles travelling on lattice links by continuous distribution functions [2]. The new-born model was free from some practical holdback factors of the LGA, noise and the so-called spurious invariants. Later, subsequent developments were performed. The collision operator was linearized, and the BhatnagarGross-Krook (BGK) approximation was introduced. It was realized that the LBM has large capabilities in computational fluid dynamics. The theoretical background of the method was established. The method was successfully married with computational fluid dynamic techniques. An important result was that the discretization of the velocity and the coordinate spaces can be chosen independently [3]. This gave rise to the development of different realization techniques, such as the finite-difference based LBM (FDLBM). The starting point of the LBM is a discrete kinetic equation of the form

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∂fi (r, t) ∂fi (r, t) + ci = Qi , ∂t ∂r

(1)

where fi (r, t) is the one-particle distribution function, ci is a set of discrete speed vectors and Qi denotes the collision operator encoding the molecular interactions between the particles. The macroscopic fluid dynamic quantities, the particle density n, the velocity P u, the pressure tensor p are defined as the moments of the distribution function n = ab i fi , P P u = i ci fi , pab = i cia cib fi . The essential features of the model depend on the speed vectors and the collision operator. The speed vectors can be constructed in different ways. Here, we mention the DnQm family which is based on the roots of the Hermite polynomials. n, m denote the dimension and the number of speed vectors, respectively. In several applications, the collision term is usually defined as the BGK operator. Note that the resulting dynamic system on the basis of the BGK model contains one-relaxation time. As a consequence, only one transport coefficient can be chosen freely, which is usually the viscosity. However, much can be gained by applying the so-called multi-relaxation-time (MRT) collision operator. This allow us choosing independently the relaxation times for different kinetic fields. We mention that the LBM was born in MRT form. However, it is crucial to realize that the idea of MRT is more general. One can show that MRT collision operators with variable Prandtl-number in the LBM are basically equivalent with the Shakov (S) model in rarefied gas dynamics. Up to this point, our kinetic model, Eq. (1), accounts for a set of partial differential equations. This immediately raises the following question. How we can develop a real computational method on the basis of Eq. (1)? This can be achieved by the so-called realization procedure. The coordinate space is discretized by choosing a coordinate grid

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Lattice Boltzmann Model as An Innovative Method for Microfluidics

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comprising fluid nodes. In this chapter, I show two basic realization procedures, the socalled top-down (or bottom-up) and the finite-difference based approaches. In the top-down approach, the grid is constructed from the speed vectors. If one fluid node r resides on the grid, its shifted neighborhoods r ′ = r + ci δt also determine fluid nodes on the grid for all ci . Here, δt is the time step. This construction results in a lattice structure. This approach is usually applied for generating cubic lattices, and typical examples are the D2Q9 or the D3Q15, D3Q27 lattices. For a given lattice, one can also define the space step δx . After the lattice is constructed, Eq. (1) is integrated by using the so-called trapezoid integration scheme. The resulting dynamic system can be written by

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fi′ (r + ci δt , t + δt ) − fi′ (r, t) = Q′i ,

(2)

where fi′ (r, t) and Q′i are transformed quantities. Eq. (2) is called as the lattice Boltzmann equation. I established a general procedure for deriving the top-down LBM for the MRT operator. In the finite-difference based LBM, the coordinate grid is independent of the speed vectors. The grid can be chosen freely, and one can use body-fitted, curvilinear grids as well. The time and space derivatives in Eq. (1) are replaced by finite-difference forms. In my works, I frequently used the combination of the Runge-Kutta time marching scheme and midpoint or upwind, downwind differential forms for the space derivatives. This choice ensures second-order accuracy in both time and space. The time and space discretizations introduce also the time and space steps δt , δx as in the top-down approach. The next question is how to connect the LBM to real physical experiments. In our kinetic model, we encounter four important parameters, the magnitude of the speed vectors c, a main relaxation time τ which is usually connected to the viscosity and the time and space steps δt , δx . Having these quantities, we can dimensionally connect the model to real, physical processes. Furthermore, we can define the mean free path l0 , the average distance travelling a molecule between two subsequent collisions, and the Knudsen number Kn = l0 /L, the ratio of the mean free path and a relevant macroscopic size L of the problem, such as the channel width for microflows. Finally, let us discuss the macro-dynamic behavior of the method. The governing equations of the fluid dynamic quantities are obtained from Eq. (1) as its moments. The derivation of the hydrodynamic equations is usually based on the Chapman-Enskog expansion. With an appropriately chosen collision term, the equations of continuum fluid mechanics can be recovered by the LBM in the small Kn number, hydrodynamic limit.

3.

LBM for Microflows

The lattice Boltzmann model is especially well-suited for modelling microflows. This is because the method is based on micro-physical concepts instead of the continuum description. Microflows can significantly depart from macro-hydrodynamics usually characterized by the Navier-Stokes equations. In recent years, it has been demonstrated that the LBM succeeds in simulating microgas flows. In this case, the main challenge is to capture the so-called rarefaction effects. In contrast to the simple macro-hydrodynamics, the Knudsen number is significantly larger

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for micro-gas systems and mainly lies in the range of Kn = 0.001 − 10. Micro-gas flows can be basically classified into two regions. The slip flow assumption is used in the range of 0.001 < Kn < 0.1. In this situation, the Navier-Stokes equation can be considered valid, but an appropriate slip or jump condition needs to be applied at the solid walls. The so-called first-order slip model characterizes the slip or jump of the macroscopic quantities, velocity or temperature, as a linear function of the Kn number. The slip-flow assumption can be extended for larger Kn numbers by using the second-order slip model, which applies a second-order expansion in the Kn number. The transient region lies in the range of 0.1 < Kn < 10. In this case, the Navier-Stokes hydrodynamics loses its validity. The so-called Knudsen layer, a highly non-equilibrium gas region near the solid-walls, starts to overlap in the micro-device. There is clear evidence that the LBM as a kinetic model describes well micro-gases in the slip region. The LBM with Navier-Stokes level velocity models, such as the D1Q3 class, captures the slip and jump conditions of the macroscopic quantities on the solid walls. In addition, the LBM permits to use realistic boundary conditions for the distribution function, as derivatives from continuous kinetic theory [6, 4, 5]. These LBM boundary conditions can apply accommodation coefficients, which are used to describe a variable momentum transfer at the solid walls. They allow a sophisticated and better description of the solid-gas interface. In this direction an important result was that the LBM with the multi-relaxation-time collision operator captures the second slip model [7]. I showed that the relaxation times in the MRT operator can be chosen to recover tunable second-order slip coefficient in the slip model. This approach is especially useful for modelling pressure driven flows well beyond the slip flow assumption. For micro-channels, the mass flow rate is usually an important quantity from either theoretical or technological standpoints. In this case, one encounters the so-called Knudsen minimum phenomenon, that is the mass flow rate has a minimum point as a function of the Kn number in the transient region. In Fig. 1 we show the mass flow rate provided by the MRT LBM versus the Kn number. It is demonstrated that an excellent agreement can be achieved with the second-order slip model. The remarkable results provided by the LBM in the slip flow region suggest that the method is extend-able into the transient region. Since the LBM is a derivative model from the Boltzmann transport equation, we can increase the number of velocity vectors used in the approach to get a better approximation. In this way, we obtain the so-called higher-order lattice Boltzmann models. These models formally recover some higher-order macroscopic moment equations than those provided by the Navier-Stokes description. As it has been shown, the higher-order LBM can be successfully used to capture the rarefaction effects in the transient region. A cardinal landmark was in the development the so-called Knudsen layer theory of the higher-order LBM [9]. The Knudsen layer characteristic of the method was elaborated. It was shown that the additional velocity vectors used in the higher-order LBM provides additional contribution to the hydrodynamic velocity profiles. These modes are in close relationships with the Knudsen layer functions appearing in rarefied gas theory. I also showed the advantageous features of the MRT collision operator in modelling the Knudsen layer. The MRT operator provides a boundary layer with a variable layer width. This finding is extremely useful for modeling micro-gas flows in the transient region. In order

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LBM I LBM II LBM III Cercignani Hadjiconstantinou No slip

MP

10

1

0.01

0.1

1 Kn

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Figure 1. Normalized mass flow rate for Poiseuille flow. LBM I,II denote the results of the LBM MRT with kinetic boundary conditions. LBM III represents the result of the LBM BGK with bounce back boundary condition. The flow rate from Cercignani and Hadjiconstantinou second slip models [8] are presented by lines. The result of the NavierStokes equation with no slip boundary condition is also plotted. to demonstrate the promising results of the higher-order MRT LBM, we show the velocity profiles provided by the LBM for Couette flow in comparison with the accurate solution of the Boltzmann equation in Fig. 2. Very good agreement can be achieved between the two approaches. Other results also suggested that the LBM models for transient micro-gas flows should be based on higher-order velocity sets. Another approach has been also appeared to capture the Knudsen layer characteristics, the so-called wall-function concept [11]. In this case, the relaxation time is redefined in the boundary layer near the wall. This is a phenomenological approach and based on the idea that the effective mean free path may be smaller near the wall than in the bulk region due to the wall collisions. The method has attracted some interest in modelling microflows. The approach has been also used with the higher-order LBM.

4.

Outlook and Further Challenges

In the LBM development there are further directions and challenges regard to the microflow domain as targets for research. It was realized that the higher-order LBM provides reliable results for microflows. In principle, this method could be further developed. There are complex problems in microflows of which description requires larger moment systems than those provided by the standard LBM. A typical example is thermal problems, where a larger moment system should be used. Another example is gaseous mixtures, where the higher-order dynamics needs to be recovered in order to capture important rarefaction effects. From methodological viewpoint, the higher-order LBM could be developed by using optimized velocity sets. It needs to be realized that the Knudsen characteristics depend on the used velocity model, hence an optimization strategy could be useful. An important advantage of the LBM is that it is a real time model. As a consequence, it can be generally used for modelling time-dependent gaseous flows. A good research project would be the development of the higher-order LBM with larger velocity sets on parallel computers for

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Kn=0.113 Kn=0.339 Kn=0.677 Kn=2.257 Ohwada et al.

u

0.5

0

-0.5

-1 -1

-0.5

0 x

0.5

1

Figure 2. Normalized velocity profiles for Couette flow between two parallel plates. Symbols represent the results of the higher-order LBM MRT versus the Kn number. The corresponding velocity profiles on the basis of the linearized Boltzmann equation with hardsphere interaction [10] are plotted with solid lines. fast computation. We remind the reader that the LBM can be easily parallelized as a result the parallel-performance can be adopted to the higher-order scheme.

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5.

Conclusion

In this chapter, I have given a brief introduction into the lattice Boltzmann model. I have discussed the aspects of the method in modelling microflows. Recent developments clearly show that the LBM is a viable tool for modelling this area. The model with its kinetic origin has large capabilities to describe important rarefaction effects. I have also presented research topics for further developments in the field. It is straightforward to develop the LBM further along these lines.

References [1] Ho,C.M.; Tai,Y.C. Annu.Rev.Fluid.Mech. 1998, 30, 579 [2] McNamara,G.R.; Zanetti,G. Phys.Rev.Lett. 1988, 61, 2332 [3] Cao,N.; Chen,S.J,; Martinez,D. Phys.Rev.E 1997, 55, R21 [4] Szalm´as,L. Phys.Rev.E 2006, 73, 066710 [5] Szalm´as,L. Int.J.Mod.Phys.C 2007, 18, 15 [6] Tang,G.H.; Tao,W.Q.; He,Y.L. Phys.Fluids 2005, 17, 058101 [7] Szalm´as,L. Phys.A 2007, 379, 401 [8] Hadjiconstantinou,N.G. Phys.Fluids 2003, 15, 2352

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Lattice Boltzmann Model as An Innovative Method for Microfluidics [9] Szalm´as,L. Europhys.Lett. 2007, 80, 24003 [10] Sone,Y.; Takata,S.; Ohwada,T. Eur.J.Mech.B/Fluids 1990, 9, 273

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[11] Zhang,Y.H.; Gu,X.J.; Barber,R.W.; Emerson,D.R. Phys.Rev.E 2006, 74, 046704

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INDEX

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A activation energy, 156 actuation, 78, 83, 84, 88, 89, 90, 93, 118, 119, 120, 124, 159, 226, 230, 231, 234, 276, 286 actuators, 67, 68, 70, 93, 123, 125, 126, 161 adaptation, 286 adhesion, x, 76, 77, 115, 241, 242, 244, 257 adhesion force, x, 241, 242, 244, 257 adjustment, 49, 128, 143 adsorption, 191 aerospace, 143, 162 age, 10, 51, 52, 55, 57 Aharonov-Bohm effect, 318 AIDS, 65, 272 alcohol, 273 alcohol consumption, 273 algorithm, 26, 273, 275, 276, 291 alloys, 107, 154 alternatives, viii, 2, 3, 8, 17, 23, 49, 50, 62, 135 aluminum, 220 amorphous polymers, 190 amplitude, x, 85, 87, 88, 90, 93, 105, 108, 109, 114, 189, 200, 201, 202, 234, 241, 242, 243, 244, 326, 328, 329, 330, 331, 333, 334, 336, 338, 339, 342, 348, 350, 353, 354, 355, 357, 360, 361, 366, 367, 369 annihilation, 321, 322 artificial intelligence, 66 assessment, 4, 5, 14, 29, 56, 60, 64, 269, 273 assignment, 38, 50, 263 assumptions, 78, 277 asymmetry, 344, 349, 350, 355, 356 atomic force, x, 241, 242, 257, 258 atoms, ix, x, 165, 166, 167, 168, 169, 170, 173, 174, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187, 188, 189, 190, 191, 225, 227, 230, 231, 232, 237 automobiles, 161

B

band gap, 128, 129, 130 bandwidth, 343 barriers, 233, 235, 237 batteries, viii, 127, 128, 136 beams, 72, 123, 125 behavior, x, 65, 95, 170, 173, 181, 185, 225, 237, 242, 244, 258, 260, 276, 319, 320, 329, 332, 336, 341, 344, 349, 352, 353, 355, 360, 361, 377 Beijing, 225, 238 bending, 72, 76, 78, 79, 81, 83, 84, 96, 101, 117, 119, 120, 125, 157, 172 bias, 87, 88, 318 binding, 166, 170, 343, 349 binding energies, 166, 349 biomaterials, 313 Boltzmann constant, 168, 235 bonding, 69, 93, 146, 147, 166, 170, 171 bonds, ix, 165, 169, 170, 173, 174, 175, 181, 182, 186, 187, 188, 190, 191, 232 bounds, 15, 29 breakdown, 115, 123 brittleness, 107, 115 broadband, 135 buffer, 200, 201, 202, 204, 208, 213, 218, 219, 222 burning, 156

C capillary, 115 carbon, x, 139, 140, 166, 225, 226, 231, 232, 237 carbon atoms, 232 carbon dioxide, 136, 137, 139, 140, 141, 166 carbon nanotubes, x, 225, 226, 231, 232, 237 carrier, 323 catalyst, 137, 138, 140, 141, 142, 143, 154, 156 catalytic effect, 156 categorization, 56, 57, 58 cell, ix, x, 123, 125, 128, 129, 130, 136, 137, 139, 140, 141, 142, 195, 196, 197, 198, 200, 202, 203, 210, 217, 218, 222, 312 ceramic, 144, 145, 154

backscattering, 340, 342

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Index

channels, 138, 140, 152, 153, 154, 198, 228, 320, 323, 324, 325, 327, 329, 343, 344, 345, 346, 347, 348, 357, 363, 366, 367, 368 chaos, 90 chemical properties, 168 chirality, x, 225, 233, 234, 235, 237 chromium, 77, 258 clarity, 278 classes, 69, 270 classification, 176 cluster analysis, 2 clustering, viii, xi, 2, 3, 8, 9, 13, 17, 21, 22, 32, 33, 34, 48, 49, 50, 51, 62, 63, 66, 259, 260, 264, 269, 270, 271 clusters, viii, ix, 2, 3, 8, 9, 10, 17, 18, 19, 20, 21, 31, 33, 34, 48, 49, 50, 51, 63, 128, 165, 166, 167, 168, 169, 174, 179, 180, 181, 182, 183, 184, 187, 188, 189, 190, 191, 261, 264, 271 coagulation, 168 codes, 52 coherence, 363, 364 collisions, 168, 377, 379 combined effect, 70, 108 combustion, viii, 127, 128, 130, 131, 132, 134, 135, 145, 148, 149, 151, 152, 154, 155, 156, 157, 161 communication, 145 compensation, 143, 176, 186, 191, 278 competition, 352 complement, 26, 27, 32, 263 complexity, 143, 199 compliance, 70, 71, 76, 81, 115 components, x, 68, 71, 72, 91, 100, 106, 111, 112, 129, 137, 231, 241, 297 composition, 70, 168, 169 compounds, 166, 176 compression, 83, 94, 97, 98, 104, 112, 157, 176 computation, viii, xi, 33, 50, 62, 67, 70, 72, 74, 75, 82, 84, 85, 93, 110, 112, 116, 293, 380 computational fluid dynamics, 376 computer simulations, 169 computing, 26, 27, 32, 33, 51, 52, 58, 100 concentration, 71, 116, 138, 139, 141, 142, 168 condensation, 168 conductance, xi, xii, 317, 318, 319, 323, 328, 329, 331, 332, 334, 336, 337, 338, 339, 340, 341, 342, 343, 349, 351, 361, 364 conduction, xi, 152, 226, 228, 229, 230, 317, 318, 323 conductivity, 138, 152, 231 conductor, 231, 363, 364 conductors, 318 confidence, 10, 11, 12, 21, 22 confidence interval, 10, 11, 12, 21, 22 configuration, 74, 78, 83, 85, 96, 145, 154, 155, 156, 171, 173, 174, 185, 186, 363 confinement, 319, 324, 327, 329, 331, 341, 345, 363 conservation, x, 225, 226, 228, 229, 231, 237, 321, 327, 331, 358 constant load, 94, 105

constant rate, 5 construction, x, 128, 136, 225, 377 contamination, 115, 143 contingency, 260, 261, 272 continuity, 229 control, xi, 10, 11, 13, 21, 28, 34, 36, 37, 49, 68, 71, 102, 118, 128, 143, 149, 155, 168, 171, 235, 242, 260, 261, 263, 268, 269, 270, 272, 275, 276, 277, 278, 279, 281, 283, 284, 286, 290, 291, 312, 313 control condition, 260 control group, 10, 11, 13, 21, 261, 268, 269 convergence, 27, 39, 41, 42, 52, 57, 60, 86, 283, 290, 332 conversion, vii, viii, 68, 69, 94, 127, 128, 129, 152, 157, 161 conversion rate, 152 cooling, vii, viii, 104, 106, 127, 128, 129, 131, 157 copper, x, 154, 160, 241, 242, 252, 253, 257, 258 correlation, 31, 32, 49, 65, 71, 74, 264, 271 correlations, 49, 170, 264, 318 costs, vii, 312 couples, 70, 157 coupling, 68, 69, 70, 74, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 100, 107, 118, 119, 152, 232, 275, 276, 285, 286, 319, 330, 338, 343, 344, 345, 346, 347, 348, 349, 352, 353, 357, 359, 361, 362, 363, 365, 366, 368 covalent bond, 181 crack, 93, 101, 104, 108, 109, 113, 114, 115, 120 creep, 70, 93, 94, 96, 98, 100, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 118, 120 critical value, 109, 363 cross-validation, viii, 2, 3, 22, 23, 26, 27, 29, 30, 32, 39, 54, 58, 66, 263 crystal structure, ix, 165 crystalline, ix, 165, 167, 169, 173, 174, 176, 177, 178, 179, 181, 190, 196 crystals, 166, 176, 190 curvilinear grid, 377 cycles, 87, 94, 97, 98, 99, 100, 101, 102, 104, 107, 108, 109, 110, 111, 113, 242, 243, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256 cycling, 123 cytometry, 196, 198

D damping, xi, 70, 71, 76, 88, 90, 115, 116, 119, 122, 125, 230, 275, 276, 277, 285, 286, 288, 290 data analysis, 4, 5, 66 data collection, 272 data set, 23, 28, 32, 35, 36, 37, 38, 39, 41, 44, 45, 46, 48, 49, 50, 51, 56, 57, 59, 61, 62, 63, 64, 263 decay, 76, 339, 340, 342, 347, 349, 350, 352, 355, 356 decoupling, 345 deduction, 214 defects, 100, 107, 120, 222, 229, 324

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Index deformation, xi, 71, 123, 203, 235, 252, 255, 293, 294, 304, 306, 309 deformation distribution, 294 degenerate, 230, 325 degradation, 115, 154 delivery, 136, 137, 139, 312 density, 33, 48, 128, 129, 130, 134, 135, 136, 151, 157, 161, 173, 174, 177, 190, 191, 227, 229, 321, 322, 353, 376 density functional theory, 191 deposition, 160 derivatives, 176, 377, 378 destruction, 128, 151 detection, ix, 73, 116, 195, 196, 197, 198, 200, 201, 202, 203, 210, 217, 218, 219, 222, 312 developing countries, xi, 136, 311, 312, 314 deviation, 210, 235, 323 diaphragm, 144, 145 dielectric constant, 92 dielectric permittivity, 74 dielectrics, 115 differential equations, 173 differentiation, 179, 305 diffraction, ix, 195, 196, 197, 198, 199, 200, 202, 203, 204, 205, 208, 209, 210, 211, 213, 214, 216, 217, 218, 220, 221, 222 diffusion, 77, 105, 152, 160, 165, 168, 169, 188, 189, 191, 226, 231, 312 dimensionality, xi, 317, 318 diodes, 167 direction control, 235, 237 discipline, 238 discontinuity, 85, 88 discreteness, 184 discretization, 84, 336, 376 dislocation, 105, 120 disorder, 152 dispersion, vii, viii, 1, 2, 3, 4, 5, 13, 14, 15, 21, 22, 23, 24, 25, 26, 28, 29, 30, 33, 34, 38, 39, 40, 41, 42, 46, 52, 57, 58, 59, 60, 61, 62, 63, 64, 271 displacement, x, 70, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 92, 93, 94, 96, 118, 231, 241, 242, 243, 244, 245, 251, 252, 352 distribution, 3, 4, 5, 15, 23, 24, 26, 49, 76, 78, 79, 80, 82, 85, 95, 96, 118, 132, 134, 155, 168, 176, 185, 306, 309, 320, 323, 376, 378 distribution function, 320, 323, 376, 378 divergence, 146, 357 DNA, 311 doping, 191 drug abuse, 260, 272 drug abusers, 260 drug treatment, 28, 36, 37 drug use, 260, 261, 273 dualism, 94 ductility, 104, 110, 111, 117 duration, 173 DWNT, 232, 233, 234, 235, 237

E elastic deformation, 70, 72, 75 elasticity, 122, 309 electric charge, 69, 167 electric field, 68, 74, 75, 82, 88, 89, 92, 118, 294, 304, 305, 344, 351, 352, 353 electricity, 128, 129, 130, 157, 313 electrodes, 69, 75, 76, 93, 118, 136, 137, 286 electromagnetic, 69, 83, 94, 123, 159 electromagnetism, 124, 309 electromigration, 231 electron, xi, 167, 231, 251, 317, 318, 319, 320, 321, 322, 324, 327, 328, 329, 330, 331, 332, 333, 334, 336, 337, 338, 340, 341, 342, 344, 345, 347, 349, 350, 352, 353, 354, 356, 357, 358, 359, 360, 361, 363, 364 electronic circuits, 68 electronic materials, 317 electronic structure, 166, 167, 191 electrons, 129, 138, 167, 231, 318, 319, 320, 321, 322, 325, 332, 340, 343, 344, 354, 357, 359, 360, 363, 364 elongation, 70, 74, 82, 92, 93, 94, 96 emission, 143, 152 emitters, 129, 135 encapsulation, 313 encoding, 376 energetic materials, 159, 160 energy density, 127, 132, 136, 150, 231 entropy, 152, 176 environment, ix, 76, 107, 110, 115, 119, 128, 162, 165, 167, 170, 174, 185, 232 environmental conditions, 110 epidemiology, 65 equality, 182 equilibrium, 77, 79, 82, 84, 85, 87, 88, 89, 90, 91, 322 estimating, 23, 122, 169, 261 etching, 147, 200 evanescent waves, 327 evaporation, 147, 169, 179, 183, 184, 189, 191 evolution, 98, 149, 176 excitation, 69, 78, 87, 88, 90, 99, 100, 102, 104, 119, 120, 126, 159, 282, 283, 286, 290 execution, 41 experimental condition, 247, 248, 253 exposure, 108, 138, 139 extinction, 154, 155 extrapolation, 217

F fabrication, vii, x, 65, 95, 128, 130, 135, 143, 146, 154, 157, 161, 196, 198, 199, 200, 222, 225, 273, 278, 286, 312 factor analysis, 27

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Index

failure, viii, 67, 68, 69, 73, 75, 82, 87, 88, 90, 96, 107, 108, 111, 112, 115, 116, 120, 124 fatigue, viii, 67, 70, 76, 78, 84, 87, 88, 89, 90, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 113, 117, 118, 119, 120, 121, 122, 125 fatigue endurance, 102 FEM, 82, 84, 86, 93, 124, 125 Fermi level, 318, 323, 340 fidelity, 203, 208 films, x, 123, 125, 187, 241, 257, 258 filters, 70, 129, 135 financial support, 257 flame, 131, 134, 151, 152, 154, 155, 156 flame propagation, 152 flammability, 152, 154, 156 flammability limit, 152, 154, 156 flood, 271 flooding, 138, 140 flow field, 137, 138, 139, 140 fluctuations, 319 fluid, 71, 144, 157, 227, 228, 229, 231, 375, 376, 377 fluorescence, 196, 198 focusing, xi, 220, 314, 317 force constants, 123 fossil, 128 fractures, 120 fragments, 169, 190 free energy, 169 freezing, 191 friction, x, 106, 107, 118, 231, 232, 241, 242, 244, 250, 257 fuel, viii, 127, 128, 131, 134, 135, 136, 137, 138, 139, 141, 142, 143, 150, 152, 190 fullerene, ix, 165, 166, 173, 174, 175, 177, 178, 179, 180, 182, 185, 186, 187, 188, 189, 190, 191 fusion, 75

G gas diffusion, 137 gases, 152, 155, 166, 320 gene, 59 generalization, 323 generation, vii, viii, x, 45, 120, 127, 128, 151, 152, 241, 242, 244, 247, 257 genomics, 311 germanium, 166, 169 goals, 72, 118, 120, 260 gold, 77, 80, 120, 123, 125, 226 gold nanoparticles, 226 graphite, 136, 138, 166, 170 gravitation, 143 group membership, 262 grouping, 35, 37, 55 groups, xi, 29, 36, 55, 151, 154, 259, 260, 261, 268 growth, xi, 169, 196, 259, 264, 269, 270, 271, 314, 317

growth rate, 169 guidelines, viii, 127, 128

H HAART, 272 Hamiltonian, 324, 325, 343, 348 hardness, x, 241, 242, 249, 253, 257, 258 harvesting, viii, 68, 69, 125, 126, 127, 128, 156 health, vii, xi, 68, 311, 312 health services, 311 heat, 71, 120, 128, 129, 130, 133, 134, 151, 152, 154, 155, 156, 157, 160, 226, 227, 228, 229, 230, 231, 232, 312 heat capacity, 232, 312 heat loss, 128, 151, 152, 154 heat transfer, 120, 154 heating, 151, 152, 167, 181, 182, 184 height, 116, 131, 132, 201, 203, 247, 250, 336, 339, 340, 341 heterogeneity, 196 HIV, 1, 260, 261, 272, 273 HIV infection, 272 HIV/AIDS, 272 humidity, 243, 250, 251 hydrogen, ix, 128, 135, 136, 150, 152, 165, 167, 168, 169, 171, 172, 174, 179, 180, 183, 184, 185, 186, 187, 188, 189, 190, 191 hydrogen atoms, ix, 165, 167, 168, 172, 174, 180, 183, 184, 186, 190, 191 hysteresis, 76, 98, 103, 105, 110, 111, 112, 252, 258 hysteresis loop, 252

I identification, 70, 116, 118, 265, 269, 271, 278, 286, 290 ignition energy, 149, 150 illumination, 199, 210 image, ix, 78, 117, 195, 196, 197, 198, 199, 203, 205, 208, 210, 211, 214, 217, 218, 219, 220, 222, 226, 231 images, 160, 196, 203, 208, 210, 245, 246, 247, 252, 253, 254, 255, 256 implementation, 82, 272 impulsive, 76 impurities, 160, 318, 320, 324, 343 incentives, 261 inclusion, 39, 197, 349, 368 indentation, 253, 254, 255, 256, 258 indication, 152 indicators, 11, 56 indices, 172, 195, 217, 322, 327 induction, 94 industry, 143 inefficiencies, 119 inelastic, 103, 112, 320

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Index inequality, 282 inertia, 74, 81, 90 infectious disease, 311, 312, 314 infinite, 324, 329 inflation, 161 inhibitor, 272 initial state, 173, 174, 178 initiation, x, 93, 113, 241, 242, 252, 254, 257, 313 insight, 167, 333, 353 instability, 75, 88, 166 instruments, 203 insulation, 147 integrated circuits, 124, 125 integration, 143, 145, 149, 161, 200, 297, 312, 314, 377 integrity, 116 intelligence, 66 interaction, 11, 12, 13, 14, 21, 22, 56, 64, 76, 88, 111, 172, 173, 177, 319, 323, 343, 344, 347, 352, 357, 366, 367, 368, 380 interaction effect, 11, 14, 21, 22, 56, 64 interaction effects, 11, 21, 22, 56, 64 interactions, 11, 12, 56, 64, 152, 171, 172, 174, 181, 187, 189, 227, 231, 232, 312, 320, 343, 349, 375, 376 interface, 77, 78, 89, 254, 378 interference, 198, 318, 319, 331, 337, 341, 343, 356, 357, 361, 363, 364 interval, 24, 27, 28, 35, 36, 136, 204, 208, 221, 262, 277, 282, 303, 335 Intervals, 36 intervention, 2, 4, 10, 11, 12, 13, 14, 21, 22, 63, 260, 261, 262, 264, 265, 268, 269, 270, 271, 272 interview, 36 intrusions, 104 invariants, 376 inversion, 349 investment, xii, 375 ionization, 143 ions, 62, 167 iron, 154 isolation, 276, 291 Italy, 67, 69, 75, 117, 121 iterative solution, 93, 328

K kinetic model, 376, 377, 378 kinetics, 137, 140, 141, 142, 143, 188 KOH, 200

L lasers, 167 lattices, 226, 227, 231, 377 leakage, 115, 143, 147 lens, 196

life cycle, 106, 108, 112 line, 40, 70, 74, 79, 83, 84, 96, 97, 126, 179, 196, 243, 291, 309, 319, 320, 331, 332, 338, 341, 343, 344, 349, 351, 355, 356, 361, 363, 364 linear function, 378 linear model, 4, 10, 22, 23, 25, 26, 28, 40, 51, 54, 66 linear modeling, 4, 22, 23 linearity, 78, 93, 119 linkage, 179 links, 376 liquid crystals, 190 liquid phase, 178 liquids, 239, 312 low temperatures, 181, 186, 188, 190, 258, 318, 351 luminescence, 167 luminescent devices, 167 Lyapunov function, 281, 284 lying, 74, 173

M macros, vii, 1, 2, 4, 34, 41, 45, 65, 262 magnet, 313 magnetic field, 69, 319, 360 magnetic particles, 313 magnetic properties, 118 magnetism, 169 maintenance, x, 113, 259, 260, 261 management, 137, 157, 260, 261, 272 manufacturing, 161, 293 mapping, 293, 309 market, 136, 314 mass spectrometry, 191 matrix, 31, 32, 33, 41, 51, 262, 264, 279, 280, 281, 282, 283, 321, 322, 323, 326, 335, 336, 344, 345, 346, 347, 348, 353, 355, 357, 358, 359, 360, 364, 366, 368 meanings, 36, 44, 48 measurement, 78, 115, 118, 149, 152, 223, 260, 275 measures, 3, 27, 30, 45, 66, 260, 269, 271 mechanical properties, 82, 101, 108, 118, 138, 159 mechanical stress, 70, 92, 94, 99, 100, 105, 108 media, ix, 135, 165, 228, 375 median, 269 medication, vii, x, 1, 2, 28, 65, 259, 260, 261, 272, 273 medication adherence, vii, x, 1, 2, 28, 65, 259, 260, 261, 273 melting, ix, 154, 160, 165, 169, 176, 177, 190 melting temperature, 169, 176 membranes, 137, 169 memory, 257, 258 messages, 41, 45 metallurgy, 123 metals, 101, 102, 107, 122, 154, 156 methanol, viii, 127, 128, 136, 137, 138, 139, 140, 141, 142, 143, 169 microelectronics, 160

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388

Index

microfabrication, vii, 70, 77, 78, 80, 93, 95, 96, 100, 107, 116, 118, 146, 312, 363 microgravity, 276, 291 micrometer, vii, 249, 312 microscope, 196, 198, 200, 218, 220, 221, 222, 251, 347 microscopy, x, 123, 196, 198, 241, 242, 251, 257, 258 microspheres, ix, 195, 220, 221, 222 microstructure, 69, 70, 71, 74, 75, 76, 77, 78, 90, 94, 96, 100, 106, 107, 108, 109, 115, 116, 118 microstructures, vii, viii, 68, 70, 72, 77, 83, 113, 122, 123, 124, 127, 128, 312 microturbines, viii, 127, 128 migration, 169 miniaturization, 128, 138, 143, 312, 314 missions, 143, 226 mixing, 132, 343 mobility, 188, 189, 190 modeling, vii, viii, xi, xii, 1, 2, 3, 4, 10, 14, 15, 22, 23, 25, 29, 39, 40, 44, 46, 50, 55, 58, 59, 62, 65, 66, 125, 149, 259, 260, 262, 264, 269, 271, 273, 375, 376, 378 modulus, x, 70, 74, 80, 81, 92, 103, 105, 107, 117, 118, 122, 123, 171, 241, 242, 250, 253, 257, 295, 297, 298, 305 mold, 199, 200 molecular dynamics, ix, x, 165, 169, 171, 173, 175, 176, 190, 191, 225, 226, 237 molecules, 169, 172, 227 momentum, x, xi, 225, 226, 229, 231, 237, 317, 378 monolayer, 187, 189 Monte Carlo method, 178 morphology, 116, 168 motion, x, 68, 72, 83, 87, 89, 90, 93, 116, 117, 120, 123, 173, 225, 226, 227, 228, 229, 230, 231, 232, 233, 235, 236, 237, 276, 278, 286, 304, 308, 309, 319, 320, 344, 345, 357, 359, 364 movement, 120, 136, 233, 234, 286 multiwalled carbon nanotubes, 231

N nanocomposites, 169 nanocrystals, 167 nanodevices, 225, 226, 235 nanofretting, vii, x, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258 nanoindentation, 117, 123, 258 nanometer, x, 236, 241, 242, 249 nanometer scale, x, 241, 242, 249 nanometers, 116, 363 nanoparticles, ix, 165, 166, 167, 168, 169, 173, 175, 176, 177, 178, 189, 191 nanostructures, 169, 369 nanosystems, 68 nanotechnology, x, 124, 136, 225

nanotube, x, 225, 226, 231, 232, 233, 237 nanowires, 160 naphthalene, 166 National Institutes of Health, 65, 272 natural gas, 135 neglect, 322 neodymium, 313 network, 68, 166 nickel, 258 NiTi shape memory, 242 nodes, 377 noise, 260, 275, 318, 376 nonlinear dynamics, 124 normal distribution, 15, 24, 25, 26, 49, 65 nucleation, 101, 108, 109, 120 nuclei, ix, 195, 200, 204, 209, 214, 222 nucleic acid, 190 nucleus, 202, 217 numerical tool, viii, 67

O objectives, 143, 225 observations, 27, 29, 30, 35 oligomerization, 166 one dimension, 229, 230 operating system, 34, 45, 113 operator, 321, 322, 326, 346, 376, 377, 378 optical density, 200 optical properties, 167, 169 optical systems, ix, 195, 312 optimization, 379 optoelectronics, 167 orbit, 128, 143 ores, 371 oscillation, 87, 234, 276 oxidation, 70, 98, 100, 104, 105, 107, 109, 111, 113, 114, 115, 120, 142, 251 oxides, 166 oxygen, 120, 136, 140, 142, 251

P palladium, 156 parallel processing, 312 parameter estimates, 38, 42, 52, 57, 59, 263 particle collisions, 168 particles, ix, 156, 165, 167, 168, 169, 176, 181, 182, 183, 184, 190, 231, 313, 331, 343, 376 partition, 27, 31, 262 passivation, 169 passive, 136 patents, 122 PCR, 312 permeability, 94 permittivity, 118 phase shifts, 342, 361, 364

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Index phase transitions, 176 phonons, 167, 227, 228, 231 photoelectron spectroscopy, 191 photographs, 140 photolithography, 147 photoluminescence, 168 photons, 129, 130, 134, 135, 231 photovoltaic cells, 128 physical properties, 105, 113, 117 physicochemical properties, ix, 165, 167, 177 physics, xi, 229, 231, 238, 317, 318, 329 piezoelectricity, 118, 122 plasma, 143, 168, 200 plastic deformation, 71, 252 plasticity, 106, 111 platinum, 153, 154, 156 polarization, 93, 141, 142 polydimethylsiloxane, 200 polymer, 93, 198, 200, 220 polystyrene, ix, 195, 220, 221, 222 porous media, 156 ports, 145 power, vii, viii, 3, 10, 23, 28, 30, 38, 39, 40, 41, 42, 47, 52, 53, 55, 57, 58, 59, 60, 125, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 142, 144, 145, 149, 150, 151, 155, 156, 157, 159, 161, 169, 210, 214, 218, 226, 263, 312 precipitation, 107 precision engineering, 70 prediction, viii, 15, 17, 28, 48, 67, 70, 75, 76, 87, 88, 93, 94, 100, 107, 108, 109, 110, 112, 113, 116, 118, 124, 125, 232, 357 predictor variables, 23, 28, 30, 38, 39 predictors, viii, 1, 2, 3, 4, 22, 23, 38, 55, 58 pressure, 79, 143, 144, 148, 159, 176, 177, 178, 179, 189, 190, 227, 228, 229, 230, 231, 243, 247, 248, 249, 250, 251, 253, 376, 378 prevention, 115 probability, 23, 24, 55, 191, 318, 319, 320, 323, 326, 327, 331, 333, 339, 342, 344, 348, 349, 350, 352, 353, 354, 356, 357, 360, 361, 362, 363, 364 probe, 196, 242, 257, 258, 347 process control, 34 production, 156, 191 program, 136, 313 proliferation, 150 propagation, x, 93, 108, 109, 113, 115, 120, 241, 242, 252, 254, 257, 332, 334, 339, 355, 357 propane, 152 proportionality, 229, 232 proteins, 190 prototype, 130, 131, 137 psychopharmacology, 273 pumps, 136, 137, 143 purity, 168

Q quantization, xi, 167, 317, 318, 320, 321, 323, 329, 340, 364 quantum dot, 167, 319, 329, 343, 351 quantum dots, 167, 319, 343 quantum Hall effect, 323 quantum mechanics, 168, 364 quantum objects, 167

R radiation, 128, 129, 130, 132, 134, 135, 154, 155, 167, 231 radius, x, 79, 173, 174, 178, 183, 184, 185, 186, 200, 201, 202, 203, 204, 205, 208, 209, 210, 211, 213, 216, 217, 218, 219, 221, 241, 242, 243, 249, 252, 253, 255, 257, 258 random assignment, 27 range, ix, 10, 24, 44, 55, 90, 98, 100, 101, 102, 103, 105, 108, 109, 110, 111, 113, 122, 154, 155, 165, 168, 169, 170, 176, 179, 181, 184, 185, 186, 187, 189, 190, 318, 323, 336, 338, 340, 343, 344, 352, 359, 378 reactant, 138 reactants, 159 reaction zone, 152 reactivity, 168, 191 real time, vii, ix, 195, 379 reason, 3, 231, 250, 252, 360 recall, 366 recognition, 196 reconstruction, 78, 166 recovery, 252 recycling, 143 reference frame, 91 reflection, 115, 319, 321, 323, 330, 331, 333, 336, 339, 342, 345, 348, 355, 360 refractive index, ix, 195, 198, 200, 201, 202, 203, 217, 218, 219, 220, 221, 222 refractive indices, ix, 195, 197, 198, 204, 213, 218, 219, 222 region, 69, 82, 84, 85, 86, 101, 115, 156, 173, 174, 175, 176, 177, 187, 293, 294, 295, 320, 321, 327, 329, 331, 335, 336, 337, 338, 340, 341, 343, 345, 360, 378, 379 regression, vii, viii, 1, 2, 3, 4, 5, 10, 11, 14, 22, 23, 24, 25, 26, 30, 34, 37, 51, 52, 53, 55, 57, 58, 65, 66, 260, 262, 264, 271, 272 regression analysis, 37 regression method, viii, 2, 3, 4, 271 reinforcement, 261, 273 relationship, 168, 170, 201, 227, 269 relative size, 28 relaxation, 120, 166, 376, 377, 378, 379 relaxation times, 376, 378 repetitions, 50, 109

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residuals, 26, 32, 55 residues, 326 resistance, 138, 191, 229, 230, 237 resolution, ix, 116, 195, 196, 198, 251 resonator, 87, 91 resources, xi, 222, 311, 312, 313 rings, xi, xii, 317, 319, 357 robotics, 136 room temperature, 120, 137, 251 rotations, 69, 70, 82, 94

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S saturation, 94, 118 scaling, 117 scatter, 327, 331, 332, 333, 363 scattering, xi, xii, 196, 229, 231, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 338, 339, 340, 341, 342, 343, 344, 345, 347, 348, 349, 354, 355, 357, 358, 359, 360, 361, 362, 363, 364, 365 scavengers, 68, 126 scores, viii, 2, 3, 4, 5, 8, 10, 27, 28, 29, 30, 32, 33, 34, 38, 40, 42, 45, 49, 50, 55, 56, 58, 59, 62, 64, 263, 264 sea level, 148 self-assembly, 314 semiconductor, vii, 167, 169, 312, 369 semiconductors, 190 sensing, 68, 69, 275, 276, 286 sensitivity, 169, 225, 360 sensors, xii, 67, 68, 152, 161, 167, 275, 375 separation, 234, 339 serum, 312 shape, ix, 78, 79, 80, 83, 85, 96, 137, 168, 169, 179, 185, 190, 195, 199, 201, 202, 210, 214, 217, 222, 243, 247, 249, 257, 258, 319, 320, 331, 334, 339, 341, 343, 344, 347, 351, 355, 356, 364 sharing, 121 shear, 79, 92, 252, 304 signals, 283 signs, 321, 345 silane, 167 silicon, ix, x, 120, 129, 138, 144, 145, 146, 147, 154, 157, 160, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 199, 200, 230, 241, 242, 243, 245, 247, 250, 251, 252, 255, 257, 258 simulation, ix, xi, xii, 125, 165, 169, 178, 188, 189, 191, 195, 199, 222, 232, 239, 258, 275, 276, 286, 293, 294, 309, 313, 375 SiO2, x, 147, 241, 242, 243, 244, 257, 258 skeleton, 177 smart materials, 91, 118 snaps, 74 social interests, 312

software, 2, 35, 196, 197, 198, 199, 203, 217, 218, 220, 222 solar cells, 130 solid state, 161, 168, 169, 231, 238 space, ix, 68, 90, 149, 156, 165, 174, 186, 187, 189, 191, 353, 376, 377 special theory of relativity, 227 species, 151, 152 specific heat, 178, 179, 227 spectrum, 135, 325 speed, 40, 76, 90, 108, 113, 134, 157, 196, 200, 203, 220, 227, 234, 376, 377 speed of light, 227 spin, 169, 318, 323 stability, ix, xi, 151, 152, 155, 159, 165, 166, 167, 168, 169, 233, 275, 276, 281, 284 stabilization, 155, 275, 291 standard deviation, 10, 31, 49 statistics, 65 steel, 67, 101, 138, 153, 154, 250, 252, 253, 254 storage, 68, 128, 135, 136, 143, 152 strain, 69, 70, 72, 74, 76, 77, 78, 79, 82, 83, 84, 91, 92, 93, 94, 96, 97, 99, 100, 102, 103, 104, 105, 106, 107, 110, 111, 112, 114, 115, 116, 117, 118, 122, 304, 306, 308, 309 strategies, 261, 276, 291 strength, viii, xi, 67, 69, 71, 82, 84, 90, 94, 99, 101, 120, 123, 154, 170, 317, 323, 330, 331, 332, 334, 338, 341, 343, 344, 347, 349, 350, 351, 352, 355, 357, 359, 360, 361, 363, 367, 368 stress, 69, 70, 71, 72, 76, 77, 78, 80, 81, 82, 83, 84, 87, 92, 94, 95, 96, 97, 98, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 112, 113, 114, 116, 117, 118, 120, 122, 125, 126, 250, 251, 252, 254, 304, 309 stretching, 79, 81, 82, 172 structural transformations, 190 substrates, 157, 231, 258 superimposition, 99 supply, 140, 149 suppression, 169, 340, 349 surface area, 129, 157 surface properties, 169 surface reactions, 153, 156 surface tension, 168, 169, 190, 313 surface treatment, 120, 313 suspensions, 286 switching, 124 symmetry, 23, 166, 167, 299, 349, 368 symptoms, 273

T targets, xii, 375, 379 technological progress, 312 temperature dependence, ix, 165, 166, 177, 179, 180, 181, 184, 189, 226, 320, 350 tensile strength, 101

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Index tension, 112 terminals, 308 textbooks, 97 therapy, 66, 260, 272 thermal energy, x, 120, 154, 225, 226, 228, 230 thermal evaporation, 160 thermal expansion, 77, 82, 95, 100, 105, 118, 120 thermal oxidation, 147, 160 thermal properties, 118, 154 thermal quenching, 147, 151 thermal stability, 191 thermodynamics, 228, 231 thin films, 123 three-way interaction, 11, 56 threshold, 101, 108, 112, 120, 126, 217, 219, 329, 345, 352 time periods, 13 time use, 110 timing, 271 titanium, 258 tones, 41 total energy, 320, 321, 325, 327 TPV, 128, 129, 131, 151, 156 tracking, 276, 278, 280, 286, 287 tracks, 233, 234, 235 training, 32, 50 trajectory, 279, 282, 283, 286 transducer, 69, 159, 258 transformation, 68, 166, 181, 258, 295, 296, 320, 321, 344, 356 transformations, xi, 38, 98, 107, 108, 178, 258, 293 transistor, 351 transition, x, 176, 178, 241, 243, 244, 247, 248, 249, 252, 257, 258, 361 transitions, 176 translation, 294, 303, 304 transmission, 309, 318, 319, 320, 321, 323, 329, 330, 331, 336, 337, 338, 343, 344, 345, 348, 349, 352, 353, 354, 355, 356, 357, 358, 360, 361, 363, 364, 366, 367, 369 transport, x, xi, 135, 136, 137, 138, 141, 225, 226, 227, 228, 229, 230, 231, 232, 317, 318, 319, 320, 321, 323, 325, 329, 336, 340, 343, 357, 358, 360, 363, 364, 369, 376, 378 tungsten, 191 tunneling, 336, 340

U uniform, xi, 27, 95, 96, 155, 202, 259, 271, 296, 324, 325, 332, 344 updating, 84, 93

V vaccine, 312 vacuum, ix, x, 74, 115, 148, 149, 165, 166, 167, 185, 186, 187, 189, 190, 227, 241, 242, 243, 244, 245, 247, 250, 251, 257, 258, 360 validation, 27, 116, 125, 126, 152, 271 vapor, 157 variability, vii, 1, 3, 5, 14, 15, 17, 18, 19, 20, 21, 22, 29, 33, 34, 60, 61, 62, 63, 64, 96, 118, 271 variables, viii, xii, 2, 3, 4, 10, 13, 23, 24, 31, 34, 35, 36, 37, 38, 39, 40, 41, 45, 48, 49, 51, 56, 57, 58, 63, 64, 65, 89, 263, 355, 375 variance, 2, 15, 23, 24, 28, 31, 32, 49, 58, 59, 60, 286 vector, 23, 25, 26, 31, 32, 262, 277, 282, 336 vehicles, 68 velocity, xi, 152, 155, 228, 229, 230, 231, 232, 275, 276, 277, 278, 279, 283, 286, 289, 290, 303, 304, 322, 360, 376, 378, 379, 380 vibration, viii, 68, 76, 87, 88, 90, 119, 122, 124, 125, 126, 127, 128, 156, 159, 172, 226, 227, 242, 276, 291 virtual work, xi, 293, 297, 309 viscosity, 169, 376, 377 voice, 273 vouchers, x, xi, 259, 260, 261, 271

W wall temperature, 134, 155 waste, vii, 129, 157, 312 wave number, 201 wave vector, 320, 322, 326, 327, 345, 360, 362, 363 wavelengths, 130, 209, 210, 214, 217, 219 weak coupling limit, 361 weak interaction, 231 wealth, 260 wear, 76, 91, 108, 115, 242, 244, 247, 251, 257, 258 wells, 167, 318 wind, 231 windows, 45 wires, xi, xii, 317, 318, 319, 328, 364

X X-ray diffraction, 196

Z zeolites, 190

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