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Nano scale based model development for MEMS to NEMS migration

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Nano scale based model development for MEMS to NEMS migration
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Carrasquilla, Andres Lombo
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Nanoelectronics
Quantum mechanics
Integrated modeling
VHDL-AMS
SCORM
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF   ( lcsh )
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bibliography   ( marcgt )
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ABSTRACT: A novel integrated modeling methodology for NEMS is presented. Nano scale device models include typical effects found, at this scale, in various domains. The methodology facilitates the insertion of quantum corrections to nanoscale device models when they are simulated within multi-domain environments, as is performed in the MEMS industry. This methodology includes domain-oriented approximations from ab-initio modeling. In addition, the methodology includes the selection of quantum mechanical compact models that can be integrated with basic electronic circuits or non-electronic lumped element models. Nanoelectronic device modeling integration in mixed signal systems is reported. The modeling results are compatible with standard hardware description language entities and building blocks. This methodology is based on the IEEE VHDL-AMS, which is an industry standard modeling and simulation hardware description language.^ ^The methodology must be object oriented in order to be shared with current and future nanotechnology modeling resources, which are available worldwide. In order to integrate them inside a Learning Management System (LMS), models were formulated and adapted for educational purposes. The electronic nanodevice models were translated to a standardized format for learning objects by following the Shareable Content Object Reference Model (SCORM). The SCORM format not only allows models reusability inside the framework of the LMS, but their applicability to various educational levels as well. The model of a molecular transistor was properly defined, integrated and translated using SCORM rules and reused for educational purposes at various levels. A very popular LMS platform was used to support these tasks. The LMS platform compatibility skills were applied to test the applicability and reusability of the generated learning objects.^ ^^Model usability was successfully tested and measured within an undergraduate nanotechnology course in an electrical engineering program. The model was reused at the graduate level and adapted afterwards to a nanotechnology education program for school teachers. Following known Learning Management Systems, the developed methodology was successfully formulated and adapted for education.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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Includes bibliographical references.
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by Andres Lombo Carrasquilla.
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Includes vita.

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Nano Scale Based Model Developmen t for MEMS to NEMS Migration by Andres Lombo Carrasquilla A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Wilfrido Moreno, Ph.D. James Leffew, Ph.D. Sanjukta Bhanja, Ph.D. Kimon Valavanis, Ph.D. Fernando Falquez, Ph.D. Date of Approval: November 7, 2007 Keywords: nanoelectronics, quantum mechanic s, integrated modeling, vhdl-ams, scorm Copyright 2008, Andres Lombo Carrasquilla

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DEDICATION To my family and everyone who loves knowledge and uses it in peace and harmony.

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ACKNOWLEDGMENTS I wish to thank Dr. Wilfrido Moreno for his continuous guidance during this research. I also thank the Centro de Investigaciones y Desa rrollo Cientifico at Universidad Distrital Francisco Jose de Caldas for its partial support.

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i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT vi CHAPTER 1. INTRODUCTION 1 CHAPTER 2. NANO SCALE MODELING 5 2.1. Mesoscopic Observables in Nanostructures 9 2.1.1. Ballistic Transport 11 2.1.2. Phase Interference 11 2.1.3. Universal Conductance 12 2.1.4. Weak Localization 13 2.1.5. Carrier Heating 13 2.2. Mathematical Description of Transport in Nanodevices 13 2.2.1. Non-Equilibrium Green’s Function Method 16 2.2.2. Density Matrix Method 20 2.2.3. Wigner Transport Equation 23 2.3. Nanodevice Modeling and Simulation 25 CHAPTER 3. OBJECT ORIENTED MODELING 30 3.1. The Nanosystem 34 3.2. Connections 34 3.3. Subsystems 35 3.4. SCORM – Shareable Content Object Reference Model 38 CHAPTER 4. VHDL-AMS CAPABILITI ES TO MODEL AND SIMULATE NANODEVICES 40 4.1. Constructing Models 42 4.2. Molecular Transistor Model 42 4.3. Circuits Includi ng the Proposed Model 59 4.3.1. Analog Circuits Test Bench 60 4.3.2. Digital Circuits Test Bench 63 CHAPTER 5. CONCLUSIONS 68

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ii REFERENCES 70 APPENDICES 75 Appendix A: Matlab Code of Molecular Transistor Model 76 Appendix B: VHDL-AMS Code of Molecular Transistor Model 79 Appendix C: XML Code of Molecular Transistor Model 83 ABOUT THE AUTHOR End Page

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iii LIST OF TABLES Table 1: Parameters for Simulation of the Molecular Transistor Model 45 Table 2: Matlab Code of a Molecular Transistor 46 Table 3: VHDL-AMS Code of a Molecular Transistor 51 Table 4: VHDL-AMS Code In tegrating the Models 61 Table 5: VHDL-AMS Code for a Simple Two-Input NAND Gate 65 Table 6: VHDL-AMS Code for a Two -Input, Four-Transistor NAND Gate 66

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iv LIST OF FIGURES Figure 1: Sketch of Transistor Regions and Relations for MOSFET Modeling and Simulation; Adapted from [9] 17 Figure 2: Entity-Architecture Pairs Us ed as Object Instances for a NanoSystem 31 Figure 3: Component Variable Equi valences among Various Domains; Adapted from Ansoft Corporation’s Simplorer 7.0 VHDL-AMS Tutorial, 2004 33 Figure 4: Hierarchical Organiza tion of Nanodevices for NEMS 36 Figure 5: Schematic View of a Molecular Transistor Model 43 Figure 6: Molecular Transistor I-V Characteristic from the Matlab Code (Left) Compared with Results Obtained in the Arizona State Experiments (Right) [31] 47 Figure 7: I-V Curve Obtained from the Molctoy Tool 48 Figure 8: Plots of the Molecular Tr ansistor Response using Molctoy; a Simplified Quantum Model 49 Figure 9: Conductance and Current Va riations from the VHDL-AMS Model Level-0 52 Figure 10: Current Variations with Charging Coefficient from the VHDLAMS Model Level-0 53 Figure 11: Current Variations with the Molecular Potential Energy Level from the VHDL-AMS Model Level-0 54 Figure 12: Current Variations with the Fermi Energy from the VHDL-AMS Model Level-0 55

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v Figure 13: Current Variations wi th Broadening Coefficient Gamma (Symmetric Case) from th e VHDL-AMS Model Level-0 56 Figure 14: Current Variations wi th Temperature VHDL-AMS Model Level-0 57 Figure 15: Noise Generation Insi de the VHDL-AMS Model Level-0 58 Figure 16: SCORM Translated Mode l viewed from a Web Browser 58 Figure 17: Analog Circuit Test Benc h Including the Original Model 60 Figure 18: Analog Circu it Test Bench Response 63 Figure 19: Basic NAND2 Gate Schematic using Two Molecular Transistors 64 Figure 20: Input Output Beha vior for the NAND2 Gate 64 Figure 21: Two Input NAND Circuit w ith Four Molecular Transistors 66

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vi NANO SCALE BASED MODEL DEVELO PMENT FOR MEMS TO NEMS MIGRATION Andres Lombo Carrasquilla ABSTRACT A novel integrated modeling methodology for NEMS is presented. Nano scale device models include typical effects found, at this scale, in various domains. The methodology facilitates the insertion of quant um corrections to nanoscale device models when they are simulated within multi-domain environments, as is performed in the MEMS industry. This methodology includes dom ain-oriented approximations from abinitio modeling. In addition, the met hodology includes the sel ection of quantum mechanical compact models that can be integr ated with basic electr onic circuits or nonelectronic lumped element models. Nanoelectronic device modeli ng integration in mixed sign al systems is reported. The modeling results are compatible with stan dard hardware descri ption language entities and building blocks. This methodology is based on the IEEE VHDL-AMS, which is an industry standard modeling and simulati on hardware description language. The methodology must be object oriented in order to be shared with current and future nanotechnology modeling resources, which are available worldwide. In order to integrate them inside a L earning Management System (LMS), models

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vii were formulated and adapted for educa tional purposes. The electronic nanodevice models were translated to a standardized format for learning objects by following the Shareable Content Object Reference Mode l (SCORM). The SCORM format not only allows models reusability inside the framew ork of the LMS, but their applicability to various educational levels as well. The m odel of a molecular transistor was properly defined, integrated and translated using SCORM rules and reused for educational purposes at various levels. A very popular LMS platform was used to support these tasks. The LMS platform compatibility skills were applied to test the applicability and reusability of the gene rated learning objects. Model usability was successfully tested and measured within an undergraduate nanotechnology course in an electrical engi neering program. The model was reused at the graduate level and adapte d afterwards to a nanotechnology education program for school teachers. Following known Learning Management Systems, the developed methodology was successfully formulat ed and adapted for education.

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1 CHAPTER 1 INTRODUCTION Nanostructures, nanodevices or devices which are termed mesoscopic, can be built with dimensions smaller than the appropriate mean free path. The prefix “meso” is used to indicate that a device is larger th an atomic scale devices but smaller than the macroscopic scale, where Boltzmann transport th eory has been demonstrated to be valid. The Nano Electro-Mechanical Systems, or NEMS, integrate nano scale electromechanical sensors and actuators in the same way and as well as Micro ElectroMechanical Systems, MEMS, integrate mi cro scale electro-mechanical sensors and actuators. Both systems must be specified, designed, modeled and simulated in an integrated hybrid environment in order to ac hieve reliable and well designed prototypes. These prototypes can be fabricated with exis ting fabrication processes. Furthermore, they can be integrated within commercial products to further economic development. NEMS characterization and simulation i nvolves integrated environments taking into account mechanical, electr ical (analog and digital), a nd optical properties within a single experiment or simulation session. As the dimensions of integrated circuit devices continue to shrink, the finite dimensions of the atoms within the st ructures will lead to statistical variations in critical dimensi ons and thus in device properties. As a consequence, physical properties will deviate from the bulk properties used for MEMS.

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2 Such deviations arise from quantum mechanical and mean free path effects. The effects of statistical variations ma y be even more pronounced in multi-layer structures with components only several atoms thick. Statistical variations require th at intrinsic defects and the quantum mechanical effects of confined structures be accounted for in order to characterize a device and pred ict circuit performance for large-scale integration. With the National Nanotechnology Initiative (NNI), new concepts and design methodologies are needed to create new nanoscale devices, synthesize nano-systems and provide for their integration into architect ures for various operational environments. Design methodologies also require multiple layers of abstraction and various mathematical models to represent component behavior at the different layers. On the other hand, the emergence of new processe s in nanostructures, nanodevices and nanosystems create an urgent need for theoreti cal development, mode ling, simulation and new design tools in order to understand, control and accelerate development at the scale of these regimes. Quantum mechanics and quantum chemistry, multi-particle simulation, molecular simulation, grain and continuum-bas ed models, stochastic methods and nano mechanics must be combined in order to accomplish all of the required development. Nano-scale modeling and simulation proce sses have always been computationally expensive. High performance co mputer clusters have been arranged to run atom by atom models and to obtain exact particle res ponse from each material. These kinds of simulations provide a detailed de scription of carrier relations inside the atom, molecule or interface between atoms or molecules. Depe nding on the number of particles involved and the nature of the respons e to be extracted from the simulation, algorithms for these kinds of tasks have very high complexity orders. On the other hand, current MEMS

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3 modeling and simulation tools can be run on re latively cheap workst ations without high performance requirements. Lumped elemen t models simplify multi-physics experiments and provide fast and complete system response to the designer. This characteristic is enforced by a growing market with high dema nds, which is looking for reliable systems compatible with existing CMOS and Bi-CMOS manufacturing lines. Industry standard hardware descript ion languages such as Verilog and VHDL have demonstrated their a pplicability in top-down desi gn methodologies for MEMS. However, the nanotechnology industry has not us ed this approach due to the absence of affordable models that fit with the existi ng VHDL models of microscale devices. Additional tools are required to complete the whole design methodology. Digital, analog, mechanical and other domains enforce the require ment for the addition of specific tools. Additionally, cross domain verifications must be addressed in orde r to couple results from each design stage. To date, hardware description languages have not been used to describe nanoscale devices or to inte grate such devices in a top-down design methodology. Results from molecular dynami c simulations or quantum mechanics modeling tools have not yet been directly applied to a desi gn flow. The primary reason for the absence of the application of such t ools is their complexity and incompatibility with existing Integrated Circ uits (IC), and MEMS tools, which apply continuous theory rules. The statistical nanodevice nature comp licates the task of expressing it by means of standard hardware description language statements. Therefore, a coupled or approximated solution, which permits system s, subsystems and devices modeling and simulation, has been pursued. Any strategi es for modeling and simulation must be accessible to the scientific community by means of an open language platform, which

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4 ensures their readability and reusability. Th is fact is particularly important when educational processes involve nano scale devices and ot her modeling and simulation tools are also involved. Nanodevice models must also be pres ented as educational objects inside a learning platform. Current learning platform s have been successfully integrated by means of traditional computational environments as well as mobile computing devices. Advances in programming languages, operati ng systems and computing hardware and applications are leveraging the opporunities fo r integration of comp lex computing models into user friendly learning platforms. These platforms commonl y support graduate and undergraduate studies in th e nanotechnology area. Modeli ng and simulation platforms specifically dedicated to th e nanotechnology area are ava ilable on line from various educational institutions around the world.

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5 CHAPTER 2 NANO SCALE MODELING The coupling between quantum models continuum models and the quasicontinuum models as a middle point between the first two is currently an important development area related to the modeling a nd simulation market. Some methods have been proposed to analyze defects in solid s [1]. These methods are intended for application to multiscale (nano and micro scal e), modeling of crystalline materials under mechanical loads. Other methods have propos ed two ways of coup ling using a sequential method and a coupled method between the co ntinuum and the nanoscale domain for microfluidic applications [2]. Yu and colla borators have proposed a quantum mechanical correction method, which is used for simulation of logic circuits and silicon MOS devices operation [3]. Yu’s method uses a one-di mensional solution of the Schrodinger and Poisson equations, which employs a Density Gradient approach for quantum modeling. Most researchers have been working from the typical Natahnsons appr oach of a resonant gate transistor, which was published in 1967. Natahnsons approach employs a “lumped” mass-spring model where small sets of electric circuit elements represent the behavior of devices [4]. This approach has been tested with many microelectromechanical devices. However, it has been shown to be limited when nanodevices co-exist with microdevices. These limitations are mainly encountered wh en no-steady state solutions must be

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6 analyzed, dynamical information must be coll ected or a specific nano-scale region must be solved. S.Ai and J. Pelesko provide an analysis of the viscosity dominated and time harmonically forced mass-spring model [5]. Th ey demonstrate the applicability of this approach when inertial forces are both negligible a nd non-negligible. A more detailed work with respect to modeling at the nanoscale has been performed for electronic devices. Zhiping Yu et al. have discu ssed the inclusion of quantum mechanical corrections to the classi cal transport model for devices and circuits [3]. Initially, the Hansch model, the Van Dort model and a hybrid model were compared. The first model considers the repulsive boundary condition for channel carriers at a Si/SiO2 interface and introduces a shape f unction, which is imposed upon the carrier concentration in the transverse direction. The second model u tilizes the fact that energy quantization increases the bandgap at the subs trate surface under the gate. Both methods have drawbacks that could be solved by a hybr id model, which combines the Hansch and Van Dort models. Probably the most important effort in the modeling and simulation of nanodevices for the non-equilibrium condition has been reported by Professor Mark Lundstrom’s group at Purdue University. J. Rew, fr om the Purdue group, developed a study to understand essential physics of quasi-ballistic transport and it s implications to nanoscale device simulation based on macroscopic transpor t models [9]. R. Venugopal worked the non-equilibrium Green's function (NEGF) form alism into model quantum transport in nanoscale silicon transistors [9]. The objectives of these works have been: Implement the appropriate physics and methodology for nanoscale device modeling,

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7 Develop new TCAD (technology computer ai ded design) tools for quantum scale device simulation, Examine and assess new features of carrier transport in future developments in nanoscale transistors. P. Damle presents an approach to model quantum transport in nanoscale electronic devices [9]. Damle’s approach is based on the non-equilibrium Green's function (NEGF) formalism method. Damle tr eats a few nanoscale devices of current interest. The devices treated by Damle in clude a dual-gate silicon nanotransistor (effective mass model) a three terminal molecular device (semi-empirical atomic orbital model) and two terminal molecular wires (rigo rous ab initio atomic orbital model). Results from these investigations provide useful insights into the underlying physics in these devices. Several important features su ch as charge transfer, self-consistent band lineup, I-V characteristics and voltage drop were analyzed and explained. The NEGF approach is very consistent with a low temperature operation point. However, other techniques are appropriate when higher temper atures and external driving fields are involved in the modeling and simula tion process. The density matrix approach and the Wigner distribution method are ava ilable when those special conditions are encountered. Adequate construction of nanodevice models is necessary to develop the theory of transport in nanostructures. The first a pproach to this phenomenon is the ballistic (coherent or unscattered) transpor t. To characterize ballistic transport, all the dimensions must be comparable to or less than the inelastic electron mean free path. For GaAs at 300

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8 K this factor is approximately 120 nm. This consideration allows for the extraction of the energy relaxation length and the gate lengt h, which are very important dimensions. The energy relaxation length is given by: le = vFe, (1) where vF is the Fermi velocity and e is the energy relaxation ti me. When the gate length is comparable to the inelastic mean free pat h, phase interference effects appear in the transport phenomena, which makes the Boltzmann equations non-applicable. When phase interference effects appe ar, carriers move w ith no scattering along the coherence length. This leng th can be defined as the distance over which the electrons lose their phase memory. Loss of phase me mory yields an average broadening of the energy levels, Ea, which can be related to the numb er of states contributing to the current (I). The current is given by: I = eV/(dn/dE) (2) through the equation: 2 2 d dn dE EL ea, (3) where L is the diffusion length. Each state c ontributes with a current proportional to one electron per second. Variables to be observed and measured fr om nano scale systems are primarily the phase interference, conductan ce fluctuations, resistance, ca rrier heating and scattering time [7]. Other variables related to the th ermal and magnetic behavi or of the device are of secondary importance.

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9 2.1. Mesoscopic Observables in Nanostructures In order to develop an adequate char acterization of nanostr uctures certain key parameters must be consider ed. Most of these paramete rs can be related to their corresponding values in th e continuum domain where Boltzmann’s equations can normally be used. However, in the temperatur e regime, when conditions are restricted to low temperatures, these parameters cannot be related to normal processing and fabrication steps. Under condi tions of low temperature they must be related to normal operation conditions. Most of the parameters look familiar but they must be related to the quantum regime. The parameters to be considered are: Density: In particular, the sheet density of carriers in the quasi 2D electron gas at any interface. Mobility: Specifically at the interfaces. Scattering time: Derived from the corresponding mobility. Fermi wave vector: Determined by the density through: 2 12s Fn k (4) Fermi velocity: Derived from the expression: m k vF F (5) Elastic mean free path: Derived from the expression: sc F ev l (6) where sc is the scattering time. This expression is only valid at low temperatures.

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10 Inelastic mean free path: Derived from the expression: F inv l, (7) where is the inelastic mean free time or phase breaking time. Diffusion constant: Derived from the expression: d v Dsc F/2 (8) Phase coherence length or Thouless le ngth: Derived from the expression: 2 / 1 D l (9) this parameter can be used as the di ffusion length. On the other hand, the inelastic length is the distance trav eled by an electron ballistically in time. Thermal length: Derived from the expression: 2 / 1 T k D lB T. (10) Magnetic length: Derived from the expression: 2 / 1 eB lm. (11) All of these calculations will be affect ed by the potential barriers between each quantum box, quantum dots, na no-wires, or any other device involved. It is also valid when connections between each part of the sy stem are performed by means of an electromagnetical effect, which are very usual when micro-scaled systems are scaled down to the nano-regime. The actual potential shap e and values depend on many-body effects within each box or system. Th ese systems can be classified as non-equilibrium systems.

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11 They are usually formulated using the non-equi librium Green’s function in real time. At times these systems may require classificati on as far-from equilibrium. Formalism has not been formulated for far-from equilibrium systems. An equilibrium approach to this kind of devices can be also acceptable if working conditions matches operationals requirements for system operation. In any cas e this matching must be considered for conserving quantum behavior effects inside and outside of each potential barrier. 2.1.1. Ballistic Transport Some basic definitions must be stated to characterize transport in nanostructures. All of the definitions are related to the phe nomenon called ballistic tr ansport. The term ballistic is used to characterize the trans port condition under which the traveling distance of the carriers is comparable or lesser than the mean distance betw een scattering events. In consequence, many carriers can travel fr om the injection point to the point of extraction without any scattering, which is a be havior similar to proj ectiles or electrons in a vacuum tube [8],[46]. Their associated ch arge flow is known as ballistic transport. Ballistic transport can be formulated from the Landauer formula for conductance. At any part of the material, which can be viewed as a two-port network, the carrier concentration variation is related to conductance variation by: xdK dE dE dn R T e V I G2 (12) 2.1.2. Phase Interference The phase of an electron in the presence of a vector potential can be calculated

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12 using the Peierl’s substitution: ) (eA pK, (13) then r eA p ) (0 (14) Therefore, the phase difference is given by: ringnds B e 02 (15) where e h 0 (16) is the quantum unit of flux, and is the magnetic flux coupled through the ring. According to the Aharonov – Bohm effect, th is result can be used to calculate the resistance or conductance as the periodic oscillation of the flux. 2.1.3. Universal Conductance Time independent oscill ations are periodic in equation (16), in systems whose size scale is of the orde r of the phase breaking length. Th ey occur with variations in the Fermi energy. Therefore, these systems ar e sample-dependent. For a FET device, the gate voltage or magnetic field characterizes the random interference of the trajectories inside a sample, if the sample size is comparable with the phase coherence length. Thus the solution of Landauer’s formula when the ap plied gate voltage is more negative yields step variations of conductance of the order of (25.812,8 ohms)-1, [32]. A similar situation is experienced for the quantum Hall effect.

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13 2.1.4. Weak Localization This phenomenon can be measured when the electron tends to return to its original position, interfering with itself, and possessing a velocity, which is negatively correlated to its original velocity. 2.1.5. Carrier Heating The quantum kinetics, not considered in Boltzman equations, can be calculated using a reduction of the Liouville-von Neumann equati on for the density matrix with boundary conditions, as: ] [ ] [ ] [0 F V H t i (17) In equation (17), the first term on the right side corresponds to the carriers, phonons and impurities Hamiltonian, the sec ond term corresponds to the electronscatterer interaction and the last term corresponds to the electron coupling Hamiltonian. These definitions provide for a proper formulation of the transport processes through mesoscopic devices and yield importa nt conclusions about the behavior of nanodevices, specifically those involved in NEMS. 2.2. Mathematical Description of Transport in Nanodevices In this section, a description of th e mathematical models for nanodevices is presented. In addition, the main software tools available are expl ored and conclusions

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14 are obtained in order to find an adequa te way to model and simulate NEMS. In order to develop an adequate ma thematical formulation for a generic nanodevice, the Hamiltonian operator facilitate s a matrix representation of the effective mass equation. This approach must be used because of the variati ons of each quantity at a particular spatial coordina te. Solutions for the Schrodi nger equation when a periodic potential is considered and the effectiv e mass is included, can be written as: ) ( ) ( ) ( ) ( ) ( 2 ) (2 2t r t r U t r t r E t r m t r t iS C (18) where EC is the band edge energy and US is the scattering potential. H` is the Hamiltonian operator defined as: ) ( U ` t r H HS (19) and ) ( 22 2r E m HC (20) Then the effective mass equation can be written as: ) ( ` ) ( t r H t r t i (21) or can be also written with matrix notati on using a set of orthonormal functions as: ) ( ) ( ) ( t t H t dt d i (22) where ) ( t is a column vector not explicitly dependent on r and a matrix is used instead of a differential operator. This concept is widely used in multi-particle systems where a wavefunction is almost impossible to formulate. The state vector belonging to an N-dimensional state space can be written as:

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15 N i i ir u t t r1) ( ) ( ) (, (23) where the set of unit vectors ui are orthonormal and i i is the probability of an electron being in state i. The elec tron density can be formulated as: e allt r t r_) ( ) ( t) n(r, (24) The summation is performed over all the particles (electrons). For instance, the current density can be calculated as: ) ( * ) ( 2 ) ( m iq t r J. (25) Equation (25) proposes a connection with the co ntinuity equation. However, at this time non-equilibrium conditions enforce independent calculation of the electron density at each port of any nanodevice. Non-equilibrium conditions can arise from many situations such as through shining light, or maintaining a temperature, or potential gradient at the nanodevice. Transport in nanostructures needs to be expressed by means of a non-equilibrium system whose energy is affected by the poten tial barriers between each quantum box (e.g. quantum dots or any other nanodevice), which is known as the many-body effect. Finite element techniques or perturba tion techniques are useful for single particle problems. However, in most practical computations, di fferent techniques must be combined [9]. The non-equilibrium Green’s function (NEGF), method is one of the most popular solution methods for the problem. The Dens ity Matrix method and the Hartree-Fock method are more sophisticated methods for so lution of electron-electron interactions. These methods are usually applied to qua ntum chemistry calculations. The Density

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16 Matrix method needs to work with the atomic orbital coordinates more than the real space coordinates. The Density Matr ix approach can be used to compute the electron density or the current itself. The Wigner distribution method has been studied and applied to the solution of electron-electron interactions by various research groups ar ound the world. With an adequate computing facility, Monte Carlo met hods provide an affordable method to solve the many-body Schrodinger equation. A spectral-domain method demonstrating good accuracy and faster response than other second-order finite-difference methods is described in [10]. Its effectiveness must be tested with 2D and 3D problems. Those methods are reviewed in the next paragraphs. 2.2.1. Non-Equilibrium Green’s Function Method In order to explain this method a typi cal Metal-Oxide-Semiconductor Field-Effect transistor (MOSFET), is modeled. It can be viewed, as presented in Figure 1, as three regions forming two contact region s with a potential difference of 12 [9]. The particles (electrons or holes) coming from the two contacts, from left to right or viceversa, have rates of 1 and 2 respectively. As it can be seen on Figure 1, the net current at left si de is given by: ) (1 1N N q IL (26) and the net current at the right side is given by: ) (2 2N N q IR (27)

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17 Figure 1: Sketch of Transistor Regions and Relations for MOSFET Modeling and Simulation; Adapted from [9] Assuming no other current sources, I=IL=IR, (28) so ) ( ) ( 22 1 2 1 2 1E f E f q I (29) where 1 1 1 1) exp 1 ( 2 ) ( 2 T k E E f NB (30) and 1 2 2 2) exp 1 ( 2 ) ( 2 T k E E f NB (31) are the number of electr ons. The quantities f1 and f2 are the Fermi functions at the contacts. The instantaneous number of part icles responsible of conduction, N, varies from N1 to N2 for non-equilibrium conditions. After some algebraic manipulations [42], Contact 1 Contact 2 1 2 1/ 2/

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18 N is found to be given by: 2 1 2 2 1 1) ( ) ( 2 E f E f N (32) and the total current is given by: ) ( ) ( 22 1 2 1 2 1E f E f q I (33) At this point it is necessary to include the average broadening of energy states, which affects the total current across the device. Using the definitions given in, [9], the self-consistent energy function at each side of the device is found to by given by: USC = U[N-N0] = U[N-2f0(Ef)], (34) which yields the total energy inside as: SCU 0 (35) where 0 is the closest molecular level to Ef. For instance, a single broadened energy level can be described as: 2 2) 2 ( ) ( 2 1 ) ( E E D. (36) Consequently, the new expressions for the am ount of electrons and the total current are given by: 2 1 2 2 1 1) ( ) ( ) ( 2 E f E f E dED N (37) and ) ( ) ( ) ( 22 1 2 1 2 1E f E f E dED q I (38)

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19 The Green function is formulated as: 1 2 12 ) ( i E E G (39) using the above definitions fo r a single broadened energy level. However, it must be generalized for a multiple level condition which commonly happens inside a molecular device according to the all prev ious considerations. A seconda ry spectral function can be defined as: A(E) = -2Im{G(E)} (40) and the broadening as ) ( ) ( E A E D (41) which results in new definitions fo r electron density and current as: ) ( ) ( ) ( ) ( 2 22 2 2 1 1 2E f E G E f E G dE N (42) and ) ( ) ( ) ( 22 1 2 2 1E f E f E G dE q I (43) It is more exact to use a matrix formulation for the Green function: 2 1) ( H ES E G, (44) where terms have been replaced by their corresponding matrices, which contain the variables for each particle-particle interaction. S is the identity matrix. The single energy level 0 is replaced by the Hamiltonian matrix H. In order to be consistent, the broadening terms are replaced by the complex energydependent self-energy matrices named [9].

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20 Clearly, this approach needs a large com putation facility since the complexity of the calculations increases dramatically when many interacting particles are considered. The solution to the problem has to be follo wed through an iterative process to converge when the solutions of the NEGF are applied to the matrix density. These solutions are approximately the same as those obtained fr om Poisson’s equation. The process can be outlined as follows: Depending on the device to be modeled, (e.g. a molecular one or a bigger one such as a MOSFET), select an appropriat e matrix representation. Either an atomic orbital basis or a real sp ace discrete lattice basis, [43]. Write down a suitable Hamiltonian matrix for the device. Calculate the contact self-energy functions, according to the device. Select a value of the self-consistent potential to begin the iterations. Solve the NEGF to obtain the density matrix. Calculate the new self-consistent potentia l from the density matrix and iterate from the previous step until a proper convergence can be confirmed. Use the final density matrix to calculate the electron density, the current and other necessary functions. Depending on whether the device is mol ecular or any other mesoscopic device, the method can vary and other variables can be included. 2.2.2. Density Matrix Method The Density Matrix Method is based prim arily in the Hartree-Fock variational principle. In the variational principle many system wave-functions are a product of

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21 antisymmetrized single particle wave-functions The Hartree-Fock equation states that the single particle potential is given by: ) ( ) ( 2 13r r r n r d v vext HF S, (45) where vext represents the states for the external pot ential applied to the particle. However, equation (45) does not account for any correlati on between the partic les. If particle interaction is to be taken in account, the Dens ity Functional Theory is applied [14]. This theory includes all exchange and correlation effects. There exists, almost always, one external potential that, when doubly occupied by two non-intera cting electrons, yields the exact density of a H2 molecule. The exact form of th e exchange-correla tion functional is unknown. The simplest approximation is th e Local Density Appr oximation (LDA). A local functional provides information about the function at a single point contributing to the final solution. However, this approxima tion needs to be optimized by gradient terms through a process such as the Lagrange me thod or the Generalized Gradient method. Application of the Density Matrix Met hod to an N electron system requires a set of 3N spatial coordinates, (ri), and N spin coordinates, ( i), where: ) (i i ir x (46) The electron probability is written as: 1 ) , (2 1 1 N Nx x dx dx (47) where each integral is applied over all space. The electronic density is obtained as a summation, including both spins and is given by: 2 2 2) , ( ) (N Nx x r dx dx N r n (48)

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22 This procedure is very computationally expensive, since many methods have to be tested in order to solve this system. Even though the original fo rmulation divides the electron density into subsystems, the computational expense is large. Approximations to the density functional lead to linear scaling methods, mainly for metals, but can be generalized to a ny material. Each orbital contains 2fi electrons where 1 0 if and the electronic densit y can be simplified to: i i ir f r n2) ( 2 ) ( (49) and the non-interacting kinetic en ergy functional is given by: ) ( ) 2 1 ( ) ( 2 min ] [2 } { }, {r r f n Ti i i i n f J Si i (50) If all the interactions are included, then the total functional is given by: ) ( ) ( ] [ ] [ ) ( ) 2 1 ( ) ( 22 *r V r n dr n E n E r r dr fext XC J H i i i i ; (51) however, equation (51) needs to be minimized with respect to the occupation numbers, fi, and the orbitals,{| i|}. Using a common Lagrange me thod, a set of Schrodinger-like equations is obtained as: ) ( ) ( ) ( 2 12r r r V f fi i i i i (52) where: i i if (53) and equation (52) simplifies to: ) ( ) ( ) ( 2 12r r r Vi i i (54)

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23 Equation (54) is multiplied by *i(r) and integrated, which yields the expression: i i i ir V r dr r r dr ) ( ) ( ) ( ) 2 1 )( ( *2 2, (55) which is related to the energy levels that any orbital can occupy. This procedure is currently used inside the ABINIT tool to find the total energy, charge density and electronic structure of systems made of electrons and nuclei (molecules and periodic solid s), using pseudopotentials and a planewave basis. ABINIT also includes options to optimize the geomet ry, perform a molecular dynamics simulation or generate dynamical matrices for Born effec tive charges and dielectr ic tensors. Control of the computational costs of this me thod is implemented through the use of computational clusters, which is th e method currently utilized by USF. 2.2.3. Wigner Transport Equation The Wigner Transport Equation method is based in the use of a phase space defined function, f(k,r,t), for electrons, which is given by: du u k i t u r t u r P t r k fj j j j) exp( ) 2 ( ) 2 ( ) , (* (56) This function satisfies the transport equation given by: C Qt f t r k C k f U f m k h t f ) , ( 1 *. (57) The parameter Pj in equation (56) is the occupa ncy probability of any electron at state j. The third term on the left side of equation (57) corresponds to the drift term and

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24 the fourth term corresponds to the quantum corr ection. The term on the right side of the equality in equation (57) is the collision term. Assuming no phonon occurrence [12] the re laxation time must be less than the carrier transition time and the electron drift en ergy much lesser than the thermal energy. These assumptions allow equation (57) to be simplified. The simplified version of equation (57) is given by: Ct f k f m n U f m k t f 12 ) ln( 1 *2 2 (58) A slightly different approach to evalua tion of the Wigner function is presented in [13]. The approach in [13] uses a truncated form of the Wigner potential, which is given by: ) ) 2 ( ) 2 ( ( ) exp( 2 ) ( E qs s r V s r V s ik i ds k r V (59) This form of the Wigner potential assumes positive and negative values. Therefore, it cannot be considered as a proba bility density. This fact makes numerical methods such as the Monte Carlo method, which might be applied to solve the Wigner unstable potential since the particle weight (positiv e or negative) grows exponentially and the variance would also grows exponentially. Ho wever, as in [13], this method has been tested with acceptable results for devices of 1D at room and at low temperatures where scattering is present. However, the coheren ce length must be sufficiently large compared to feature sizes, otherwise the resonant peaks cannot be resolved properly. On this way, a conclusion of most of the methods presented can be yield. Drawbacks reside mainly in the computational complexity that a soluti on for a nanodevice can present. The expense

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25 of computational complexity is huge when la rge-scale integration circuits need to be solved. A point in favor of the NEGF met hod is that it can be applied to the contacts between devices, asumming that nanoscale an alysis can be performed there and that nanoscale effects can be partially ignored for the rest of the device. It is also based in a superposition principle, where partial soluti ons for each part contribute linearly to the solution. A simplified formulation method needs to be found for systems at nano or micro scales. At the scales of NEMS and MEMS numerous contacts are involved inside the analysis. In addition, what the microele ctronics industry is currently applying as modeling tools does not perform these types of considerations. Therefore, an evident requirement is for the development of a solu tion system that will provide for integration between the nano and the micro scales approach to modeling and simulation. At this point, the nanometric system ab straction level arises as a key point to consider for any computational system. A deeper analysis of modeling and simulation approaches is performed in next paragraphs. Th e aim is to facilitate a valid abstraction of quantum mechanics principles applied to syst ems, which include interfaces with micro scale and nano scale devices. 2.3. Nanodevice Modeling and Simulation There are various research initiative s and groups whose pr imary concern is nanodevices modeling and simula tion [33-39]. Most of research is oriented toward finding an adequate computational algorithm, which represents the quantum behavior of structures [45]. Currently available soft ware tools run on relatively onerous and time-

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26 consuming computational platforms. PROPHET, MOSES, SIMON and SETTTRANS represent some of the tools, which have b een created to involve quantum calculations within the micro and nano-electronic desi gn process. NEMO 3-D, from the Jet Propulsion Laboratory of NASA and CalTech, arises as one of the best and complete efforts to simulate semiconductor nanodevices. This group is working on a tool that can be applied to the simulation of optical propert ies of quantum dots. Initial studies have been oriented to metallic quantum dots in various implementations. These implementations include lateral quantu m dots, through electrical gating of heterostructures, vertical quantum dots, th rough wet etching of qua ntum well structures, pyramidal quantum dots, through self-assembled growth, and trench quantum wires. The NEMO 3-D tool excludes band structure effects from the el ectron scattering simulation mainly because of the computational expens e associated with the inclusion of these effects. Benchmarking has been performed using single and multiple CPUs with shared memory. Machines such as the Sun E450 Ultra-Sparc 2 running at 300 MHz, the SGI Origin 2000 running at 200 MHz, the HP V cl ass PA 8000 at 200 MHz, clusters of the HP/Convex SPP-2000 – 256 CPUs at 180 MHz, the SGI Onyx – 4 CPUs at 200 MHz, and the Intel Pentium II – 16 CPUs at 200 MHz running the LINUX operating system, were incorporated in the benchmarking. In Germany, the Walter Schottky Institute and personnel from the University of Rome developed Nanoext3, which is a software suite for 3D nano-device simulation that solves an 8-band momentum and the Schrodinger-Poisson equation using a library of IIIV materials. Nanoext3 can calculate the stru cture of 3D heterostructure quantum devices under bias and its current density close to equilibrium. Nanoext3 uses a mixture of

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27 methods. These methods range from the totall y quantum mechanical solvers, to find the electronic structure, to a semi-c lassical approach of local Fermi levels, to find the current. In order to perform these calculations Nanoe xt3 assumes that the carriers are in local equilibrium. Nanohub is a web-based initiative spear headed by the NSF-funded Network for Computational Nanotechnology (NNC), whic h includes seven universities. Nanohub provides on-line simulation services and uti lizes tools such as 2DS, Schred 2.0, NanoMOS and TBGreen. 2DS is a tool for solving Schrodinger’s equation in a 2D quantum well with infinite potential barriers and an arbitrary, user defined, potential field inside the well. The tool discretizes Schrodinger’s equation through the use of a Finite Cloud Method (FCM). 2DS solves the eigenvalue problem by using ARPACK. Schred 2.0 calculates the envelope wavefunctions and corresponding bound-state energies in a typical MOS structure. Th ese calculations involve solutions of the one-dimensional Poisson and Schrodinger e quations. Schred 2.0 assumes certain conditions for the quantum simulation. Th e Si/SiO2 interface is assumed parallel to the [100] plane. The conduction band is represented by the six equivalent valleys. Then the effective masses are calculated from the valley curvature. The valence band is represented by the h eavy-hole band and the light-hole band and uses the same masses. Schred 2.0 is wr itten in Fortran 77. It takes about 10 seconds per bias point calc ulation in quantum mode on a SPARC-5 workstation. However, only about 2 to 3 minutes ar e required for bulk calculations where subband energies crowds together.

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28 NanoMOS is a 2D simulator for thin body, (less than 5 nm), fully depleted, double-gated n-MOSFETs. NanoMOS implem ents five different transport models. Two of the models, quantum ball istic, [46] and quantum diffusive, are related to quantum transport. TBGreen calculates the transmission and re flection coefficients at any port using a tight-binding Green’s function method. No benchmarks or comparisons have been reported using this tight-binding method. On the other hand, NANOTCAD is a project funded by the European Commission [58]. The project resides w ithin the Nanotechnology Information Devices initiative of the Information Society Tec hnologies (IST) Program. NANOTCAD’s main objectives are the development and validation of a software package for the simulation and the design of a wide spectrum of devices. The devices are based both on semiconductors and on transport through singl e molecules. In addition, NANOTCAD, [54] proposes demonstration of a procedure for the realization of prototype nano-scale devices based on detailed modeling. Other approaches to NEMS modeling a nd simulation have been oriented to a proper representation and inte rchange of nanodevice designs. NanoTITAN Inc., has been a leader in this area when publishing nanoML, which is a data markup language for systematic organization, representation and interchange of nanodevice designs. Nanodevice designs include the molecular comp onents, structure and information about the properties, interoperability operational characteristics, display and legal status of nanodevices. A recent release of this t ool, called nanoXplorer IDE, provides for molecular dynamics simulations.

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29 However, the complexity in simulation environments becomes a problem when the system involves mixed signals designs. Problems also arise with a mixing of nano and micro scale devices where a multiphysics anal ysis is required to be performed. The complexity is also represented by the com putational resources needed by nanoXplorer IDE when it performs nanodevices visualizat ion graphics since graphic acceleration hardware and/or software are required. According to the software tools reviewed, it can be concluded that all the tools are primarily oriented to find an adequate co mputational algorithm, which represents the quantum behavior of structures. However, most of the new tools lack portability and compatibility with existing processing tools. The new tools must be integrated with existing tools in order to perform a fabrication process. In addition, these new tools must be able to accept an existi ng and reliable model of the whole system at micro and nano scales, [55] which solve customer requirement s in a flexible way. An operating system, expensive computational processi ng capabilities, fast processor, and/or processor clusters and large memory capacities are intensively requ ired by all the tools e xplored, [56]. This fact is very important when designers and fabrication facil ities must be in concordance for a particular production process. The indus try wants the fastest a nd most reliable tool to integrate with its production lines. The de signer wants the more accurate tool to be sure that the design is in compliance with what the customer desires. However, accuracy, speed and reliability must meet at one poi nt where the final product quality must be satisfied. In the next chapters an object orie nted solution to this matter is formulated and successfully applied.

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30 CHAPTER 3 OBJECT ORIENTED MODELING This section defines the object oriented approach, which was applied to the modeling and simulation of nanodevices. Em phasis is placed on the results obtained from a typical nanoelectronic device. The re sults have been exporte d as learning objects to be used by engineering students who are users of a Learning Management System. Conclusions with respect to the applicability and limitations of this kind of solution are presented. The benefits for a learning pr ocess in nanotechnology at the undergraduate and graduate level ar e also presented. Nanodevices can be considered primaril y as analog devices inside a VHDL-AMS framework. However, these devices are closel y related to digital systems if the designer is concerned with logical gates. In this case the mixed signal approach proves to be superior. This chapter presents a fram ework for nanosystems definition from the modeling and simulation point of view. This fo rmulation is needed in order to facilitate the description of component re lationships, the definition of simulation domains, and the corresponding influences that a variable from one domain can have over a variable in another domain. From the systems point of view, a na nosystem can be considered as a well organized set of nanodevices and interfaces. The devices and interf aces are sufficiently

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31 variable in number and type to allow a desi gner to form an interp retation of the whole system within one or all of the domains, [ 40], which encompass el ectronic, mechanical, optical, fluidic, thermal or electromagnetic char acteristics. The system must be evaluated from the lowest level, which is the device level, to the highest one, which is the functional level. The framework for a VHDL-AMS formulation is the design entity, which consists of an (entity declaration)-(architecture body declaration) pair, which is more commonly referred to simply as an ent ity-architecture pair or design entity. The design entity represents the instance of an object when applied to the construction of complex nanosystems. This formalism is presented in Figure 2. Figure 2: Entity-Architecture Pairs Used as Object Instances for a Nano-System ENTITY entity_name IS PORT ( interface_dscription) -Summary -Contacts -Digital rights END ENTITY ARCHITECTURE structural_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE first_functional_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE second_functional_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE n_functional_architecture_name IS BEGIN … END ARCHITECTURE Object definition Instances definition ENTITY entity_name IS PORT ( interface_dscription) -Summary -Contacts -Digital rights END ENTITY ARCHITECTURE structural_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE first_functional_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE second_functional_architecture_name IS BEGIN … END ARCHITECTURE ARCHITECTURE n_functional_architecture_name IS BEGIN … END ARCHITECTURE

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32 In order to adopt an object oriented m odeling strategy, systems, subsystems and their connections must be addressed by an object. They must have properties to be inherited by other objects when they take part of a nanode vices object library. Each object, (nanodevice), must be characterized by: Entity Declaration, (Entity): Consis ts of a description containing the nanodevice's name, the set of interfaces that allow nanodevice connections, a device summary, contacts and any digital rights associated with elements of the design. Architecture Body Declarati on, (Architecture): The ar chitecture can represent structure, which enables precise c ontrol over the sele cted nanodevice's elements. The architecture can also be functional, which allows the nanodevice's physical, electromagnetic, chem ical and optical characteristics and properties to be represented as a set of formulas and algorithms. In accordance with the objectives of this research, the preferred architectures must be the functional and the primar y domain evaluated is the electrical domain. It is not complicated to make use of a similar set of functions in other domains. In addition, recognized microelectronics industry standard modeling tools have been stated. A brief equivalence among various domains is clearly formulated in Figure 3, wh ich was extracted from an industries’ modeling tool. It each domain, an equiva lent effort of flow variable complies with the same relationship inside a particular object. With this fact in mind a proper definition of a nanode vice entity can be formulated. Furthermore, the

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33 entity can be translated to other domai ns while maintaining the same set of relations and interpreting them inside a different conceptual framework. Figure 3: Component Variable Equivalenc es among Various Domains; Adapted from Ansoft Corporation’s Simplore r 7.0 VHDL-AMS Tutorial, 2004

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34 This structure must be applied to common electrical nanodevice model formulations for a VHDL-AMS platform [50-53]. Once available VHDL-AMS software packages have been evaluated, proper formulation of the nanodevice is executed. 3.1. The Nanosystem In order to formulate a nanosystem the designer must collect a set of requirements, which must be organized within an abstract system formalism. This formalism allows the designer to propose a collection of component s that fit with the original requirements from the structural a nd functional point of vi ew. The designer must formulate or, in the structural case, organize and adapt a set of primitive components that complies as exactly as possible with the user demand. There must be a formal model, which is applied to specify (write) the requirements of each nanosystem as well as e ach nanodevice. In fact, each nanodevice is a nanosystem by itself. The smallest nanode vice that can be analyzed is the hydrogen atom. However, it is better to create a syst ematic framework in order to accelerate the time to market in the nano-related industry A nanosystem can be decomposed in a number of subsystems according to specific design goals [40]. 3.2. Connections A connection must be esta blished between a pair of systems or subsystems. A connection can be modeled in two ways. Th e two types of connections comprise the rigorous metallic connection and the electrom agnetic coupling, which permits particle transport between two reservoirs These connections can be considered as point-to-point

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35 as well as multipoint depending on their geometry or the kind of molecules used in performing the connection. It is importan t that a designer define what kinds of relationships are described insi de a particular NEMS design. This requirement arises due to the possibility of multiphysics analysis and correlated measurements, which can be performed over the same connecti on between two or more nanodevices. 3.3. Subsystems Molecular systems are the most populat ed group of devices with current applications within the industry. Conseque ntly these systems can be easily modeled and simulated. In addition, the solid state electronic nanodevices are important for the industry. In the solid state electronic group there are one, tw o or many terminal devices. Examples of this group include quantum dots, resonant tunneling devices (diodes and transistors) and single electron transistors. A hierarchical view of th e most common nanodevices is pr esented in Figure 4. It shows that a system is composed of subsyste ms and their connections. At the nano scale those connections can be considered in two wa ys. One way to form a physical contact is between materials of different molecular st ructure, which are usually known as metal contacts. Another connection is formed when there is very close proximity between the regions with different or similar molecular structure. The latter situation performs the connections by electromagnetic coupling. On the other hand, subsystems, which can be considered as a complete system, can be classified depending on their structure. A molecular device is composed by a unique patt ern of molecules that can be behaviorally isolated at the model. The Schrodinger e quation can be solved independently for each

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36 molecule, except at the boundaries [41]. De pending on the specific behavior exhibited, it is possible to classify the device differently in a particular domain. A molecular device might be classified as electrochemical, photoa ctive or electro-mechani cal as a function of the principal behavior exhibite d in a particular domain. So lid state nanodevices are also classified based on its particular behavior inside a system. It is also possible to include more additional classifica tions depending on a specific domain to be modeled and simulated such as microfluidics, thermodynamics, and mechanics [41]. Figure 4: Hierarchical Organi zation of Nanodevices for NEMS Nano Electro MechanicalSystem Subsystem Molecular devices Connections Solid-State Nano ElectronicDevices Hybrid Micro-Nano Electronicdevice Quantum Devices CMOS Devices Nano CMOS CNFETs Other Devices OtherD omains Wires EM Field Coupling Resonant Tunneling D evices Resonant TunnelingDiodes Resonant TunnelingTransistors Quantum Dots Single Electron Transistors Electrochemical Photoactive Electromechanical Other Devices OtherD omains

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37 In order to adopt an object oriented m odeling strategy, systems, subsystems and their connections must be addressed. Object properties must be inherited from/to other objects when they are part of a nanodevices object library. Each object must be characterized by [28]: A description containing the nanodevi ce's name, summary, contacts and any digital rights associated with elements of the design. A structural architecture, which enable s precise control over the selected nanodevice's elements. A functional architecture, which contai ns a set of the nanodevice's physical, electromagnetic, chemical and optic al properties for easy reference. This structure must be applied to common electrical nanodevices model formulation for a VHDL-AMS platform. All na nodevices have been well characterized, in many reports, from the continuum theory point of view. However, probabilistic behavior must be included in common tool s such as the available VHDL-AMS software packages. From the hierarchical view some devices can be modeled as a function of their computational complexity. It can be eval uated from the dimensi onal point of view. However, many other aspects of the model must be considered. Models can be considered as: Zero-dimensional: such as quantum dots, One-dimensional: such as quantum wires, Two-dimensional: such as quantum we lls (inside a molecular transistor, a single-electron transistor or tunneling devices such as diodes or transistors),

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38 Three-dimensional: such as quantum bul ks (nano-cantilevers and nano-tools) [41]. The hierarchical represen tation of nanoscale devices pr esented in Figure 4 shows that a system can collect as many subsystems and/or devices as exist in the abstraction levels where they are related. Fully ela borated VHDL-AMS designs can use XML as an intermediate way of representation [27]. XML can extract complete static semantical information, which is inherent to VHDL-AMS and dynamic simulation related information such as the current values of signal drivers or the dynamic equation sets. 3.4. SCORM – Shareable Content Object Reference Model Further development of the XML stru cture yields standardized knowledge objects, which are written following worldwid e standards such as SCORM, (Shareable Content Object Reference Model). SCORM pr ovides for the descri ption and deliverance of e-learning content in diffe rent software platforms. The importance of SCORM lies in the ability to represent educational contents which can be shared and in the interface between these contents and the e-learning platforms that use them. Multiple platforms, either commercial or open-source, suppor t the SCORM specificat ion called ADL 2004. The main SCORM components are: The CAM (Content Aggregation Mode l), which defines a model for packaging learning content. CAM d eals with Assets, Shareable Content Objects (SCO), and Content Aggrega tion Packages. Assets are single individual objects such as HTML pages. SCOs are collections of assets.

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39 They should be independent of the le arning context and are intended to be subjectively small units such that potential reuse across multiple learning objectives is feasible. Content Aggreg ation Packages comprise one or more SCOs or assets. Therefore SCOs compri se one or more learning objects. SCOs should be structured in such a way that they are ready for delivery to a student. The RTE (Run Time Environment), defines an interface for enabling communications between learning content a nd the system that launches it such as a LMS. The RTE deals with an API (Application Programming Interface) adapter and a RTE service routine. The API adapter enables communications between learning content and the LMS fr om which it is launched. The RTE service routine is provided by the LM S and is responsible for providing the user nterface for the student [57].

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40 CHAPTER 4 VHDL-AMS CAPABILITIES TO MODEL AND SIMULATE NANODEVICES According to the IEEE standard 1076. 1-1999, VHDL-AMS is a superset of the IEEE 1076-1993 standard language with capabil ities for modeling and simulation of analog and mixed-signals designs. This can be accomplished by including a capability for representing and analyzing non-linear ordi nary differential and algebraic equations. The models can follow the energy conservati ve principle, using nodes as a TERMINAL, or non conservative princi ples, using nodes termed a QUANTITY. In the case of QUANTITY, inputs are only mathematically modified and presented at outputs. Additionally, the unknowns can denote any wave form or a time series of values. In order to achieve an adequate mode ling and simulation of nanodevices or any multi-particle device at the nano scale, those quantities have to be written by means of a set of quantum correlated matrices. All iterations to be performed are followed by a “break” statement, which informs the analog so lver to schedule an appropriate solution point and to determine a new initial soluti on for the next continuous functional segment or piece. No analog solver has been fixed an IEEE standard. Therefore, each implementer can choose the appropriate method for the solutions of equations. However, it is not yet clear wh ich method is the best when m odeling and simulating nanodevices. Another factor to be considered is th at software platforms, which include VHDL-

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41 AMS, vary in the way of implementing the language standard. Some platforms exclude certain capabilities formulated by the IEEE or limit the capab ilities of certain primitives to certain types of variables. This fact can severely decrease the capabilities of each nanodevice implementation inside a particular software tool. Multiple experiences have been reported about formulation of VHDL-AMS models for MEMS, [24] –26], [4 0]. None has been reported, which include nano scale devices involving the quantum corrections mentioned previo usly. Describing a partial differential equation (PDE) using VHDL-AMS requires a proper PDE definition. The PDE definition must include all its parame ters, its boundary conditions and a contact interface with the rest of the system [25]. VHDL-AMS does not directly support PDEs. However, the equation can be discretized with respect to spatial variables, which leaves the time derivatives to the language itself. A more complicated situation arises when multiple domains are involved in a systems simulation [26]. Reduced order m odeling of linear systems can be achieved, including non linear systems. However, the interface of analog components may use non-conservative nodes (QUANTITY), which can be connected to conservative nodes (TERMINAL). However, this type of c onnection is not allowed by the language. Therefore, it is necessary to modify th e system, subsystem or component model interfaces in each design. Implementing th is idea, in practice, leads to multiple architecture modeling (MAM) through th e use of TERMINAL nodes instead of QUANTITY nodes when low or high abstra ction design levels are modeled and simulated. This idea must be similarly applied when modeling nanoscale devices and their connections with microscale devices.

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42 The final consideration that needs to be attended is that VHDL-AMS is not an object oriented language. However, the instantiation of VHDL-AMS entities is completely valid as a modeling formalism. This type of formalism uses a hierarch ical fashion to create new devices and relate them to their parent devices while maintaining original properties and adding others. 4.1. Constructing Models A set of preliminary models inside th e electrical domain, was developed using existing simulation tools. The models behave in accordance with the quantum mechanical theory and display the expected resp onse from a circuit th eory point of view. More models, where other domains are involved such as the electromechanical domain, must be proposed and tested. The nano-can tilever would seem to be a likely candidate for investigation. In the following secti ons a description of models developed is presented and explained. Detailed coding of the models is presented in Appendix A. 4.2. Molecular Transistor Model The Molecular Transistor model is base d on quantum mechanical approximations of a molecular transistor. It was pointed out in Chapter 2 that conductance fluctuations are periodic in h/e. The ga te voltage determines those fluctuations. The model was formulated using the theory presented and e xplained in Chapter 2. An exact model was simulated using the University of Purdue nanohub facilities using the approach proposed in [9]. The nanohub simulation used an ex act solution with vari ables in a matrix

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43 notation, which made the solution of high el ectron populated models very expensive but more exact than others. A simplified model was developed wherein scalar quantities are used. The model was initially presented in Matlab and late r translated to VHDL-AMS code. Steady state simplifications were performed according to reported experiences in [31]. Parametric simulations were designed and structured outputs were obtaine d. Four kinds of response can be displayed, depending on the simplification level of the model: One energy level (the lowest ~ the most probable) response, without broadening effects and spin effects. One energy level with broadeni ng but without spin effects. One energy level with spin effects, wh ich is also called the “unrestricted” model. Two energy levels: taking into account the two main energy levels E0 and E1. A schematic view of the mode l is presented in Figure 5. Figure 5: Schematic View of a Molecular Transistor Model Contact 1 Contact 2 1 21/ 2/ Rate (e-/s)

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44 The transistor was modeled as tw o bulk regions with potential energy 1 and 2; this potential can be viewed as the electro chemical potential related with the Fermi function. Consider a typical transistor with grounded source and an applied drain voltage VD. For such a transistor the relationship: DqV 2 1 (60) is valid. 1/ and 2/ are the rate at which any charge d particle (electrons) can escape from or to the bulk regions and depend on c oupling with the gate molecule. Broadened density of states at contacts is modele d using a Lorentzian function centered at The Lorentzian function is given by: 2 2) 2 ( ) ( 2 1 ) ( E E D. (61) A generic molecule forms the channel region. If the contact bulk region is metallic the states distribution is continuous. However, if the material is semiconductor, effects such as negative differential resistan ce and other related effects can be present. The parameters chosen are summarized in Table 1. External parameters are related to the surrounding circuit to be connected. Intern al parameters depend on transistor gate molecular composition. Mixed parameters involve the two previous conditions. Primarily, the broadening effect depends on the modification of molecular energy levels when it makes contact with the source and drain bulk. As a primer approach, Matlab code descri bing the equations used in this model is presented. A complete coding is presented in Appendix A. The code was organized as follows. An initial parameter definition was written. It states molecular and device operating conditions, defines the molecular charging energy, the molecular potential

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45 energy and coupling conditions with the microsca le regions inside the transistor, as well as the condition for equilibrium through the device. Table 1: Parameters for Simulation of the Molecular Transistor Model External parameters: Range Fermi Energy. (depending on contact materials) {-7 eV ,.. -3 eV} Temperature {50 K,.. 1000 K} Internal parameters Molecular conduction energy levels, and charging energy (depending on molecular composition) {-8 eV,.. -2eV} {0,..4 eV/electron} Mixed parameters (internal-external) Broadening factors 1 and 2 {0.025, ..1} Then, an energy grid was defined in or der to initiate simplified Hamiltonian calculations where the parameter NE indicates the density of the grid. NE determines how exact will be the solution obtained. NE also determines the complexity of the calculations and the length of the computation time. The next step was to define biasing conditions. At this point it was important to insure that the real voltage range for transistor operation corresponded with the range expected for th e actual transistor used. Most of the reported experiments deal with a short voltage range (-0.8v to 0.8 v). Therefore, this range was included in the c ode. The parameter IV was increased in value in order to achieve a more exact solution. This parameter also directly determines the computational time. The rest of the code describes the computation of Fermi functions and the corresponding current values. Tabl e 2 shows the complete Matlab code.

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46 Table 2: Matlab Code of a Molecular Transistor. % Parameters definition U0 = 0.25; % charging energy in eV kT = 0.025; %energy in eV %at room temp T=300K mu = 0; ep = 0.2; % in eV N0 = 0; Alphag = 0; % molecular coupling alphad = 0.5; % molecular coupling %Energy grid definitions NE = 501; E = linspace(-1,1,NE); dE = E(2) E(1); g2 = 0.005*ones(1,NE); %gamma 2 g1 = g2; %gamma 1 g = g1 + g2; % absolute broadening factor %Bias definitions IV = 101; VV = linspace(-0.8,0.8,IV) ; % applied voltage for iV = 1:IV Vg = 0; % gate voltage Vd = VV(iV); Vg = VV(iV); mu1 = mu; mu2 = mu1 Vd; UL = -(alphag*Vg) (alphad*Vd); U = 0; %Self-consistent field dU = 1; while dU > 1e-6 f1 = 1./(1 + exp((E mu1)./kT)); f2 = 1./(1 + exp((E mu2)./kT)); D = (g./(2*pi))./(((E ep UL U).^2) + ((g./2).^2)); D = D./(dE*sum(D)); N(iV) = dE*2*sum(D.*((f1.*g1./g) + (f2.*g2./g))); Unew = U0*(N(iV) N0); dU = abs(U Unew); U = U + 0.1*(Unew U); end I(iV) = dE*2*I0*(sum(D.*(f1 f2).*g1.*g2./g)); end hold on h = plot(VV,I); grid on In this model, the larger the vector distance NE, the more exact will be the solution. However, the resulting computati onal time also increases. Results from this initial model are presented in Figure 6. This code was translated into VHDL-AMS entity architecture pairs with appropriate selection of parameters.

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47 Figure 6: Molecular Transistor I-V Char acteristic from the Matlab Code (Left) Compared with Results Obtained in the Arizona State Experiments (Right) [31] These results show a very close agre ement with those reported by Goodnick and Gerousis, from Arizona State University, [31]. However, they use a different simulation package called SIMON 2.0, which is a si ngle-electron circuit simulator based on the Monte Carlo method. SIMON 2.0 includes quantum corrections across multiple junctions. However, it is a very high tim e and hardware consuming tool. While the simplified model simulation time is appr oximately two seconds, running on a PC platform, SIMON requires approximately 100 se conds running on a cluster with parallel computation. In order to complete the results valida tion process, a different tool for modeling the same device, was tested. The tool Mol ctoy runs at the University of Purdue’s nanohub facility. Figure 7 presents a detailed I-V curve obtained from the Molctoy tool. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x 10 -6 Current (A) Applied Voltage (v)

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48 Figure 7: I-V Curve Obtain ed from the Molctoy Tool Molctoy formulates a simplified toy m odel molecular transistor. The Matlab response agrees more with results in, [31], because of the simplific ations already made inside Molctoy. In fact, Mol ctoy also performs more calcula tions. As in Arizona State’s simulations, Molctoy spent approximately 100 seconds to perform the required calculations. Figure 8 presents current, c onductance and number of electrons variations when an external voltage is applied to this model. The discrete quantum response for conductance is affecting the electrical curren t variations. Energy broadening effects can be determined from smooth sh apes in all the three plots.

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49 Figure 8: Plots of the Molecular Transi stor Response using Molctoy; A Simplified Quantum Model Further considerations must be taken in to account when the device is considered as a subsystem inside a more complicated system. Applied voltage ranges are usually shorter than the range chosen at nanohub’s simulations. Re stricting the app lied voltage to (-0.8v, 0.8v) displays a more accurate system behavior. The next step in the validation process was to produce an equivalent model with code rewritten in VHDL-AMS. The VHDL-AMS code was run using the hAMSter Simulation System Version 2.0 from Ansoft Corporation on a PC equipped with an Intel Pentium M processor at 1.5 GHz. Simulation time for most of the hAMSter simulations was 20 milliseconds. VHDL-AMS coding adjust ments must be performed according to

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50 the limitations present of the available version of the hAMSter software. The model developed and simulated applied the ballistic principle. The VHDL-AMS model assumed no scattering, a constant Fermi level, used a grounded contact 1, low bias and a minimum broadening of molecular energy leve ls. A very important simplification was included regarding the Green’s function solution. In the Ma tlab code the function is solved approximately. A value of NE, the energy grid parameter, and dU, the energy differential increment were selected as high as possible in order to obtain more iterations, (in the presented case dU>1e-6). In VHDL-AMS, the Green’s function equation simplification was peformed by means of the fullfilment of an equilibrium condition defined by the statement given by: IL=IR. (62) Partial results computed while this conditi on is not verified ar e not valid solutions for the transistor current. Simulation results show proper results in accordance with the quantum conductance definition. Discrete changes in conductance and its corresponding change in the transistor current were also in accordance with expect ations derived from theory. The VHDL-AMS code is shown in Table 3. Figures 9 through 15 present the results simplified molecular transistor behavi or obtained from simulations of the VHDLAMS code. In Figure 9 the x-axis presents variat ions in the applied energy, which are equivalent to variations in the applied vo ltage as in the previous Matlab and nanohub models.

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51 Table 3: VHDL-AMS Code of a Molecular Transistor -Model name: Molecular Transistor Level-0 -This is a discrete model of a molecular tr ansistor. The molecular resistance is associated -with the interface between the narrow wire and the wide contacts. Ballistic -transport model, zero scattering is assumed. Contact 1 is grounded -Low bias and minimum broadening of the molecular energy levels is assumed -This code is optimized to be simulated with the hAMSter tool by Ansoft Corporation -A constant Fermi level was assumed -The output voltage obtained represents only the positive part LIBRARY IEEE; USE IEEE.MATH_REAL. ALL ; -entity definition including real quantities for scalar simplified model ENTITY molectranslev0 IS QUANTITY mu1, mu2: REAL; QUANTITY eVg: REAL; QUANTITY USC: REAL; QUANTITY N, N0, N1, N2: REAL; QUANTITY ep: REAL; QUANTITY f1, f2: REAL; QUANTITY IL, LR: REAL; QUANTITY I, G: REAL; CONSTANT eta: REAL := 0.5; -charging coefficient: could be 0 molec-potential energy level in equlibrium CONSTANT Ef: REAL := -5.0; - Fermi level CONSTANT hbar: REAL := 1.1356e-15; --Planck's constant CONSTANT g1: REAL := 0.1; -Broadening coefficient gamma1 CONSTANT g2: REAL := 0.1; -Broadening coefficient gamma2 CONSTANT U: REAL := 0.001; -charging constant CONSTANT kT: REAL := 0.025; -Boltzman constant at room temperature CONSTANT q: REAL := 1.602e-19; -electron charge END ENTITY molectranslev0; ARCHITECTURE behav OF molectranslev0 IS BEGIN N0 = = 2.0/(1.0 + exp((ep0 + Ef )/kT)); -electrons in equilibrium state eVg = = now; -(electronvolts) applied energy level USC = = eta*evg; -charging voltage effect over the molecule mu1 = = Ef (1.0 eta)*eVg ; -first contact energy level mu2 = = Ef + (eta*eVg); -second contact energy level ep = = ep0 + USC; -molecular energy level f1 = = 1.0/(1.0 + exp((ep-mu1)/kT)); -Fermi level at the first contact f2 = = 1.0/(1.0 + exp((ep mu2)/kT)); -Fermi level at the second contact N1 = = 2.0*f1; -number of charge carriers at first contact N2 = = 2.0*f2; -number of charge carriers at second contact IL = = (2.0*g1*q/hbar)*(N1 N); -right current IR = = (2.0*g2*q/hbar)*(N-N2); -left current IF (IL = IR) USE -Equilibrium conditions verification routine N = = (2.0*((g1*f1)+(g2*f2))/(g1 + g2)); I = = (2.0*q/hbar)*(g1*g2/(g1 + g2))*(f1 f2); ELSE N = = ((USC/U)+2.0*N0) N; -charging condition I = = IR-IL; END USE ; END ARCHITECTURE behav;

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52 Figure 9: Conductance and Current Vari ations from the VHDL-AMS Model Level-0 Regarding the adequate scales and the i ndependent variable se lected, these results show a proper concordance with the other mode ls. A discrete variation in conductance is reflected in a corresponding variation in the cu rrent. The main conclusion is that the molecular device is not showing the classical continous response Instead, the response is discrete following the quantized ap proach revealed in the theory. The VHDL-AMS simulations displayed a difference with respect to the other simulations, which is evident in simulati on time and the amount of computational resources required. This simulation was performed in 20 milliseconds using a laptop equipped with an Intel Pentium M processor at 1.5 GHz. Analysis from the VHDL-AMS entity-archi tecture pair was perf ormed. Different aspects are considered: A charging coefficient equal to zero means no current flow. A non-linear response was displayed when the charging coefficient was incr eased from 0 to 1, which corresponded to an increasing current. Current variations are not

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53 directly related to charging coefficien t variations. These non-linearities are effective in a remarkable manner when current values are small. No significant differences can be noted for currents higher than 100 mA, which is not the case of most of nanodevices. These results are pres ented in Figure 10. Figure 10: Current Variations with the Charging Coefficient from the VHDL-AMS model Level-0 The results for molecular nature and its potential energy level, (ep0 in the VHDL-AMS coding) are related to the c onsideration of the potential energy. They are presented in Figure 11. Signif icant differences in the current flow can be noted when the molecular energy level is less than the Fermi level. Starting at 20 uA, current differences of approximately two orders of magnitude can be discerned between simulations with ep0Ef. 0,0000001 0,000001 0,00001 0,0001 0,001 0,01 0,110,000010,00010,0010,01 0,1 1Time (s) Current (A) logarithmic eta=0.25 eta=0.5 eta=0.75 eta=1.0

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54 Figure 11: Current Variations wi th the Molecular Potential Energy Level from the VHDL-AMS Model Level-0 A similar situation can be noted with va riations of the Fermi energy level with respect to the molecular potential energy level. Results of these investigations are presented in Figure 12. While maintaining ep0 constant at -5.5 eV, reduction of the Fermi energy to a level more negative than ep0 yields a higher transistor current of approximately one or two orders of magnitude. As with previous observations, this fact was ve rified at low current levels. For currents higher than 10 mA, this situation was not verified. Current Variations with Molecular Potential Energy Level 1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 0,000000001 0,0000000 1 0,0000001 0,000001 0,00001 0,0001 0,00 1 0,0000 1 0,000 1 Time (s) Current (A) ep0=-10eV ep0=-8eV ep0=-6eV ep0=-4eV ep0=-2eV ep0=0eV ep0=1.0eV ep0=2.0eV ep0=4.0eV

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55 Figure 12: Current Variations wi th the Fermi Energy from the VHDL-AMS Model Level-0 As theoretical principles reveal, the effect of broadening corresponds to the coupling of the molecular energy levels with the source and drain regions. For the case of symmetrical transistors where the broadening effect can be considered the same at both interfaces, this broa dening produces higher charge carrier amounts flowing through the transistor. Increasing of the gamma coefficient was followed by proportional current increasings After approximately 10 mA, this situation was maintained only for gamma values less than 0.01. The results of these investigations are presented in Figure 13. Current Variations with Fermi Energy0,000001 0,00001 0,0001 0,001 0,00001 0,00010,001 0,01 Time (s) Current (A) Ef=-10eV Ef=-5eV Ef=0eV Ef=5eV

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56 Figure 13: Current Variati ons with Broadening Coeffici ent Gamma (Symmetric Case) from the VHDL-AMS Model Level-0 In the case of a non-symmetric transist or, which means that gamma 1 and gamma 2 are different, similar current variati ons were obtained when gamma 1 and gamma 2 values were interchanged. This fact reinforces the idea of the absolute broadening factor, which was incl uded in the Matlab code as: g=g1+g2. (63) The effect of broadening at each contact was linearly added to the broadening effect of the other. Temperature effects were also analyzed. The results of these investigations are presented in Figure 14. The logarithmic de pendence of the current with respect to 1E-10 0,000000001 0,00000001 0,0000001 0,000001 0,00001 0,0001 0,001 0,01 0,1 1 0,000010,00010,0010,010,11 Time (s)Current (A) gamma=10^(-4) gamma=10^(-3) gamma=10^(-2) gamma=10^(-1) gamma=0,5 gamma=1,0Current Variations with Broadening Co efficient Gamma (Sy mmetric Broadening)

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57 the temperature is displayed. An additional current increase was obtained in high temperature regimes and for current ranges no greater than 1 mA. A difference of hundreds of nanoamperes was displayed at high temperaures. According to the applied model very small differences can be obtained for higher current ranges. Figure 14: Current Variations with Temperature VHDL-AMS Model Level-0 The next question with respect to the model was the verification of noise being generated inside the device. Th e results of this investigation are presented in Figure 15. It was verified that with a minimum amount of evg, up to 7.5 e-21 eV, the noise current was not greater than 2.5 e-48 A. CurrentvariationswithTemperature0,0000001 0,000001 0,00001 0,000010,00010,001 Time (s)Current(A) T=200 T=300 T=400 T=500 T=600 T=1000 CurrentvariationswithTemperature0,0000001 0,000001 0,00001 0,000010,00010,001 Time (s)Current(A) T=200 T=300 T=400 T=500 T=600 T=1000

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58 Figure 15: Noise Generation In side the VHDL-AMS Model Level-0 Finally, the molecular transistor model was exported, with a SCORM utility, to be converted to a shareable lear ning object. Figure 16 presents the appearance of the model using a standard in ternet browser. Figure 16: SCORM Translated Mo del viewed from a Web Browser

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59 For an adequate model translation to SCORM format a main document was taken as a basis for the Hypertext Markup Language (HTML) and Extensible Markup Language (XML) files. The HTML file a ppears exactly like the original VHDL-AMS code, which was simulated with the hAMSter tool. The XML file contains a set of metadata definitions needed to translate the original format and to achieve a proper presentation of the information inside a L earning Management System (LMS). Appendix C presents the XML coding required for pr esentation of the VHDL-AMS model for the LMS. An open source LMS, called Moodle, was used in order to demonstrate the applicability of the nanodevice models at undergraduate and graduate engineering courses at Universidad Dist rital Francisco Jose de Caldas, Bogota, Colombia. From these results, more complicated stru ctures can be modeled and simulated. Simplifications can be modified in order to obtain a more accurate response. However, simulation time and computational resources may be higher, which will make their incorporation with ot her CAD tools and hardware de scription languages expensive. 4.3. Circuits Including the Proposed Model The next step in the validation process was to construct elect rical circuits where the proposed model was included. The purpose of this investigation was to demonstrate applicability of the model inside mo re complex designs, demonstrate model interoperability with other existing common devices and demonstrate model validity for other integration scales. Th e proposed circuits are simp le but allow verifying the applicability of the original nanodevice w ithin a more comple x architecture.

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604.3.1. Analog Circuits Test Bench Figure 17 provides a schematic view of an analog circuit including the already tested device. Figure 17: Analog Circuit Test Be nch Including the Original Model The VHDL-AMS code was organized as follows. Initially, the original molecular transistor entity was included. Following th e proposed modeling methodology, the initial molecular transistor device model needed to be related with other entities. Following the code presented in the following paragraphs it s hould be noted that the original entitiy now has an explicit line coding the interface ports. Prior formulation of the transistor had a purely functional architecture without major concerns for its connectivity. Now, the entity-architecture pairs need a dditional properties in order to connect with other entities. This is verified by the inclusion of PORT statements at the beginning of the entity definition. Then, the other entities related to micro or macroscale devices are defined. It is ensuring the reusability and compatibilit y of the proposed nanodevice models. Proper Vsin Molectrans R n1 n3 n2 V_ i_1 + + +

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61 PORT MAP statements need to be written in order to achieve an electrical connectivity among circuit elements. The test bench conclude s the statements of the complete circuit. Table 4 shows the complete coding. Table 4: VHDL-AMS Code Integrating the Models LIBRARY IEEE, DISCIPLINES; USE IEEE.MATH_REAL. ALL ; USE DISCIPLINES.ELECTROMAGNETIC_SYSTEM. ALL ; ENTITY molectrans IS PORT ( TERMINAL gate, drain, source: ELECTR ICAL); -Interface ports END ENTITY molectrans; ARCHITECTURE behav OF molectrans IS QUANTITY Vgate ACROSS gate TO electrical_ground; QUANTITY Vdraingate ACROSS Idrain THROUGH drain TO gate; QUANTITY eVg, mu1, mu2, USC, N, ep, f1, f2: REAL; QUANTITY N0, N1, N2, IL, IR, Vdrain, Vsource: REAL; CONSTANT eta: REAL := 0.5; -charging coefficient. it could be 0 molecular -potential energy level in equlibrium CONSTANT Ef: REAL := -5.0; - Fermi level CONSTANT hbar: REAL := 1.1356e-15; --Planck's constant CONSTANT g1: REAL := 0.1; -Broadening coefficient gamma1 CONSTANT g2: REAL := 0.1; -Broadening coefficient gamma2 CONSTANT U: REAL := 0.001; -charging constant CONSTANT kT: REAL := 0.083; -Boltzman's constant at room temperature CONSTANT q: REAL := 1.602e-19; -electron charge BEGIN Vsource = = 0.0; -source grounded Vdrain = = Vdraingate Vgate; N0 = = 2.0/(1.0 + exp((ep0 + Ef)/kT)); -electrons in equilibrium state eVg = = Vgate; -(electronvolts) applied energy level USC = = eta*evg; -charging voltage effect over the molecule mu1 = = Ef-(1.0 eta)*eVg; -first contact energy level mu2 = = Ef + (eta*eVg); -second contact energy level ep = = ep0 + USC; -molecular energy level f1 = = 1.0/(1.0 + exp((ep mu1)/kT)); -Fermi level at the first contact f2 = = 1.0/(1.0 + exp((ep mu2)/kT)); -Fermi level at the second contact N1 = = 2.0*f1; -number of charge carriers at first contact N2 = = 2.0*f2; -number of charge carriers at second contact IL = = (2.0*g1*q/hbar)*(N1-N); -right current IR = = (2.0*g2*q/hbar)*(N N2); -left current -Equilibrium conditions verification routine IF (IL = IR) USE N = = (2.0*((g1*f1) + (g2*f2))/(g1 + g2)); Idrain = = (2.0*q/hbar)*(g1*g2/(g1 + g2))*(f1 f2); ELSE N = = ((USC/U) + 2.0*N0) N; -charging condition Idrain = = IR-IL; END USE ; END ARCHITECTURE behav;

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62 Table 4 (Continued) LIBRARY IEEE; USE IEEE.MATH_REAL. ALL ; ENTITY vsin IS PORT ( QUANTITY v_in: REAL); END ENTITY vsin; ARCHITECTURE behav OF vsin IS BEGIN v_in = = sin(1.0e12*now); END ARCHITECTURE behav; ----------------------------------------------------------------------------------------------------LIBRARY DISCIPLINES; USE DISCIPLINES.ELECTROMAGNETIC_SYSTEM. ALL ; ENTITY resistor IS GENERIC (resistance: REAL); -resistance value given as a generic parameter PORT ( TERMINAL p,m: ELECTRICAL); --Interface ports END ENTITY resistor; ARCHITECTURE behav OF resistor IS QUANTITY r_e ACROSS r_i THROUGH p TO m; BEGIN r_i = = r_e/resistance; END ARCHITECTURE behav; -The test bench is the mechanism employed to simulate a VHDL-AMS design entity. -Test bench LIBRARY DISCIPLINES, IEEE; USE DISCIPLINES.ELECTROMAGNETIC_SYSTEM. ALL ; USE IEEE.MATH_REAL. ALL ; ENTITY Test_bench_Level0 IS END ENTITY Test_bench_Level0; ARCHITECTURE behav OF Test_bench_Level0 IS TERMINAL n1, n2, n3: ELECTRICAL; QUANTITY v_input: REAL; QUANTITY v_1 ACROSS n1 TO n3; QUANTITY i_1 THROUGH n2 TO n1; BEGIN VSource: ENTITY vsin(behav) PORT MAP (v_in => v_1); T1: ENTITY molectrans(behav) PORT MAP (gate => n1, drain => n2, source => n3 ); R1: ENTITY resistor(behav) GENERIC MAP (resistance => 500.0) PORT MAP (p => n2, m => electrical_ground); v_1 = = v_input; END ARCHITECTURE behav; Without regard to the circuit gain, wh ich depends on the size of the connected load, a proper current-voltage response at the load was obtained.

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63 Figure 18: Analog Circu it Test Bench Response 4.3.2. Digital Circuits Test Bench A very simple two input NAND gate circ uit using two molecular transistors has been formulated. Figure 19 shows the sche matic circuit correspondi ng to this gate. Applied voltage Current Voltage

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64 Figure 19: Basic NAND2 Gate Schematic using Two Molecular Transistors The NAND2 gate is following the true tabl e as was expected. As it can be seen on Figure 20, the two inputs (brown and blue traces) yield the output (green trace). Figure 20: Input – Output Be havior for the NAND2 Gate Time 0s 10ms 20ms 30ms 40ms 50ms In p ut 1 V ( M1: g) : In p ut 2 V ( M2: g) : 0V 2.0V 4.0V 6.0V Time 0s 10ms 20ms 30ms 40ms 50ms Out p ut V ( R1:1 ) : -4.0V 0V 4.0V 8.0V M1 R1 1k V1 5 VC 0 0 VC M2 V2 TD = 5m TF = 10n PW = 10m PER = 20m V1 = 0 TR = 10n V2 = 5 V3 TD = 1n TF = 10n PW = 10m V1 = 5 TR = 10n V2 = 0 0 0 V V V

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65 The VHDL-AMS coding for the two input NAND gate is presented in Table 5. The next step was to model and simulate a four transistor NAND gate. The schematic circuit is presented in Figure 21 and the VHDL-AMS coding for the circuit is presented in Table 6. Table 5: VHDL-AMS Code for a Simple Two-Input NAND Gate LIBRARY IEEE; USE ieee.all; USE work.all; ENTITY Test_bench_Level0 IS END Test_bench_Level0 ; ARCHITECTURE behav1 OF Test_bench_Level0 IS TERMINAL d,g,s,d1,g1,s1,vc: ELECTRICAL ; QUANTITY VXTOG ACROSS IXTOG THROUGH g TO electrical_ref; -Applied signal at M1 Gate QUANTITY VXTOG1 ACROSS IXTOG1 THROUGH g1 TO electrical_ref; -Applied signal at M2 Gate QUANTITY a,b: real; BEGIN sino: ENTITY v_sin PORT MAP (seno=>a,coseno=>b); -square signals mapping vgs: ENTITY v_constant GENERIC MAP (level=>5.0) PORT MAP (pos=>vc,neg=>electrical_Ref); nmos: ENTITY molectrans PORT MAP (drain=>d1,gate=>g1,source=>s 1); -Transistor M1 nmos1: ENTITY molectrans PORT MAP (drain=>s1,gate=>g,source=>electrical_ref); -Transistor M2 re: ENTITY resistor GENERIC MAP (resistance => 1000.0) PORT MAP (p=>vc, m=>d1); -Resistor R1 IF a>0.0 USE vxtog==5.0; ELSE vxtog==0.0; END USE ; IF b>0.0 USE vxtog1==0.0; ELSE vxtog1==5.0; END USE ; END ARCHITECTURE behav1;

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66 Figure 21: Two Input NAND Circuit with Four Molecular Transistors Table 6: VHDL-AMS Code for a Two-Input, Four-Tra nsistor NAND Gate LIBRARY IEEE; USE IEEE.MATH_REAL.ALL; LIBRARY IEEE; USE IEEE.ELECTRICA L_SYSTEMS.ALL; USE work.all; ENTITY Test_bench_Level0 IS END Test_bench_Level0 ; ARCHITECTURE behav1 OF Test_bench_Level0 IS TERMINAL g1,g2,g3,g4,vc,s1,s2: electrical; QUANTITY VXTOG1 ACROSS IXTOG1 THROUGH g1 TO electrical_ref; QUANTITY VXTOG2 ACROSS IXTOG2 THROUGH g2 TO electrical_ref; QUANTITY VXTOG3 ACROSS IXTOG3 THROUGH g3 TO electrical_ref; QUANTITY VXTOG4 ACROSS IXTOG4 THROUGH g4 TO electrical_ref; QUANTITY a,b: real; BEGIN sino: ENTITY v_sin PORT MAP (seno=>a,coseno=>b); vgs: ENTITY v_constant GENERIC MAP (level=>5.0) PORT MAP (pos=>vc, neg=>electrical_Ref); T1: ENTITY molectrans PORT MAP (drain=>vc,gate=>g1, source=>s1); T2: ENTITY molectrans PORT MAP (drain=>vc,gate=>g2, source=>s1); T3: ENTITY molectrans PORT MAP (drain=>s1,gate=>g3, source=>s2); T4: ENTITY molectrans PORT MAP (drain=>s2,gate=>g4, source=>electrical_ref); vxtog4==5.0-vxtog2; vxtog3==5.0-vxtog1; IF a>0.0 USE vxtog1==5.0; ELSE vxtog1==0.0; END USE ; IF b>0.0 USE vxtog2==0.0; ELSE vxtog2==5.0; END USE ; END ARCHITECTURE behav1; M1 0 VC M2 V TD = 5m TF = PW = 10m PER = 20m V1 = 0 TR = V2 = 5 V TD = 1n TF = 10n PW = 10m PER = 20m V1 = 5 TR = 10n V2 = 0 0 0 M4 M5 V

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67 A similar response to the two-transistor ci rcuit was obtained in this case. It shows that both formulations can be used, but the second one is electrically more appropriate than the first one.

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68 CHAPTER 5 CONCLUSIONS The methodology applied to model na noscale devices and systems simplifies calculations for integrated simulation envi ronments. In addition the methodology also incorporates consideration of quantum effects, which are always present in these kinds of systems. Results were comparable with ot her reports obtained when using more complex computational facilities and mo re elaborated mathematical models. Results were also in accordance with experimental results from other research groups. The applied methodology was implemented through various step s using different tools. However, the implementation can be reordered depending on th e initial conditions of the particular system definition. The application of quasi-continuum models as a middle point between quantum and continuum models was validated. The validation process can require different tools in order to achieve a better selection of system parameters and depends on the specific anal ysis performed. The modularity of the proposed methodology ensures an efficient validation process and the interaction from different stages of the process to various verification tools. “Lumped” models, where small sets of electric circuit elements represent the behavior of devices, were test ed. These models were shown to be of limited use when nano-devices coexist with microdevices, spec ifically when beam thickness is large

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69 compared to features thickness. Other propert ies can be analyzed in order to apply the models in other domains. Nano-device models can be represen ted using common hardware description languages such as VHDL-AMS. The use of VHDL-AMS provides affordable results, which can be applied to common design engine ering environments with typical operating conditions. The application of VHDL-AMS is compatable with current requirements imposed by industry and will evolve to alwa ys be compatable with industry demands. Nano-device models can be properly translated to st andard object oriented formats in order to be shareable as a web resource. In accordance with the feedback obtained from educational experiences, nanodevi ce models have been successfully reused inside various simulation environments. In addition, these models were used in more complex designs, which were more easily understood by undergraduate students in electrical engineering programs. Working conditions of most molecular na notransistors were properly translated into a VHDL-AMS modeling and simulation environment. The VHDL-AMS environment yielded affordable results in accordance with common operating conditions reported for current electronic nanodevices. The nano-device models developed were pr operly translated to standard object oriented formats in order to be shareable fr om standard Learning Management Systems. As a consequence, the models can be shared as a web resource and can be reused in environments such as collaborative re search, development groups and various educational situations.

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70 REFERENCES [1]. Shilkrot L.E., and Miller R.E., “A Coupled Atomistic-Continuum Model of Defects in Solids”, Journal of the Mechanics and Physics of Solids, Vol 50, pp 2085 2106, 2002 [2]. Jenkins J.W., Sundaram S. and Makhija ni V.B., “Coupling between Nanoscale and Microscale Modeling for Microfluidic devices”, CFD Research Corporation [3]. Yu Z., Dutton R. and Kiehl R., “Circuit/Device Modeling at the Quantum Level”, IEEE Transactions on Electron Devices, Vol. 47, No. 10, October 2000 [4]. Senturia S., “Microsystem Design”, Kl uwer Publishers. ISBN: 0-7923-7246-8, 2001 [5]. Ai S. and Pelesjo J., “Dynamics of a Canonical Electrostatic MEMS/NEMS System”, School of Mathemathics and CDSNS, Georgia Institute of Technology [6]. Lurie S., Belov P., Volkov-Bogorodsky D. and Tuchkova N., “Nanomechanical Modeling of the Nanostructures and Di spersed Composites”, Computational Materials Science, vol 28, pp. 529 539, 2003 [7]. Ferry D. and Goodnick S., “Transport in Nanostructures”, Cambridge Univ. Press, 1999 [8]. Pierret R., “Advanced Semiconductor Funda mentals”, Modular Series on Solid State Devices, Addison-Wesley, 1989 [9]. Damle P., “Nanoscale Device Modeling: from MOSFETS to Molecules”, Purdue University Ph.D. Thesis, at http://falcon.ecn.purdue.edu:8080/ publications/#theses, 2003 [10]. Liu W., “Computational Nanomechanics of Materials”, Handbook of Theoretical and Computational Nanotechnology, Amer ican Scientific Publishers, 2005 [11]. Liu W., “The Spectral Grid Method: A Novel Fast Schrdinger-Equation Solver for Semiconductor Nanodevice Simulation” IEEE Transactions on ComputerAided Design Of Integrated Circuits and Systems, Vol. 23, No. 8, August 2004

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71 [12]. Sano N., “Device Modeling and Simula tions Toward Sub-10nm Semiconductor Devices”, IEEE Transactions on Nanotec hnology, Vol. 1, No. 1, March 2002 [13]. Kosina H., “Comparison of Numerical Quantum Device Models” Internat. Conf. on Simulation of Semiconductor Proce sses and Devices, (SISPAD), pp. 171 174, 2003 [14]. Schweizer W., “Numerical Quantum Dyna mics”, Kluwer Academic Publishers, 2001 [15]. Mello P. and Kumar, N., “Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuatio ns”, Oxford University Press, 2004 [16]. Belsky V., Beall M., Fish J., Shephard M. S. and Maa S.G., “Computer-Aided Multiscale Modeling Tools For Composite Ma terials and Structures”, Computing Systems in Engineering, Vol. 6, No. 3, pp. 213 223, 1995 [17]. Voigt G., Schrag G. and Wachutka P ., “Microfluidic System Modeling using VHDL-AMS and Circuit Simulation”, Microe lectronics Journal, Vol 29, pp. 791797, 1998 [18]. Dewey A., Srinivasan V., Icoz E. “Vis ual Modeling and Design of Microelectromechanical System Transducers”, Microe lectronics Journal, Vol 32, pp. 373 381, 2001 [19]. Gregoire O., Souffland D., Gauthier, S. and Schiestel, R., “Towards Multiscale Modeling of Compressible Mixing Flows”, Acad emic Science, T. 325, Series II b, pp. 631-634, Paris, 1997 [20]. Endeman A., and Dunnigan M., “System Level Simulation of a Double Stator Wobble Electrostatic Micromotor”, Sens ors and Actuators, A 99, pp. 312 – 320, 2002 [21]. Voigt P., Schrag G. and Wachutka G., “Electrofluidic Full-System Modeling of a Flap-Valve Micropump Base d on Kirchhoffian Network Theory”, Sensors and Actuators, A 66, pp. 9 14, 1998 [22]. Garcia S., “Mixed-Mode System design: VHDL-AMS”, Microelectronic Engineering, vol 54, pp. I71 I80, 2000 [23]. Solomon P. and van Heeren H., “Worldwide Services and Infrastructure for Nano and Microproduction Nanotsunami”, at http://www.voyle.net/Guest%20Writers/ Patric%20Salomon/Patr ic%20Salomon%202004-0001.htm

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72 [24]. Wilson P., “Multiple Domain Behavioral Modeling using VHDL-AMS”, IEEE, ISCAS, 2004 [25]. Nikitin P., “Modeling Partial Differential Equations in VHDL-AMS”, IEEE, 2003 [26]. Schlegel M., “Analyzing and Simulation of MEMS in VHDL-AMS Based on Reduced-Order FE Models”, IEEE, 2005 [27]. Karayannis T., “Using XML for Representati on and Visualization of Elaborated VHDL-AMS Models”, IEEE, 2000 [28]. http://nanotitan.org [29]. Barniol N., “A Mass Sensor with Attogram Sensitivity using Resonating Cantilevers”, Poster, 10th MEL-ARI/NID Wo rkshop, Helsinki-Finland, July 1-3, 2002 [30]. Born M. and Huang H., “Dynamical theory of crystal lattices”, Oxford University Press, 1954 [31]. Gerousis J., “Nanoelectronic single-electr on transistor circuits and architectures” International Journal of Ci rcuit Theory and Applications, Vol 32, pp. 323 338, 2004 [32]. Albella J., Martinez J. and Agullo, F ., “Fundamentos de Microelectronica, Nanoelectronica y Fotonica”, Pearson-Prentice Hall. Madrid, 2005 [33]. Stan M., Franzon P., Goldstein S., Lach J. and Ziegler M. “Molecular Electronics: From Devices and Interconn ect to Circuits and Architecture”, IEEE Invited Paper 0018-9219, DOI 10.1109/JPROC, 2003 [34]. Repinsky S., “Organized Molecular Assemblies: Crea tion and Investigation of their Functional Properties”, e-Journal of Surface Science and Nanotechnology, Vol.1, pp. 7 19, 2003 [35]. Henderson S., Johnson E., Janulis J. and Tougaw P., “Incorporating Standard CMOS Design Process Methodologies in to the QCA Logic Design Process”, IEEE Transanctions on Nanotechnology, Vol. 3, No. 1, March, 2004 [36]. Snider G.and Williams S., “Nano/CMOS Architecture using a FieldProgrammable Nanowire Interconnect”, Nanotechnology, Vol. 18, pg. 11, 035204 DOI: 10.1088/0957-4484/18/3/035204, 2007

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73 [37]. Ma X., Strukov D., Lee J. and Likharev K. “Afterlife for Silicon: CMOL Circuit Architectures”, Proceedings of the 5TH IEEE Conference on Nanotechnology, 2005 [38]. Strukov D. and Likharev K., “CMOL FP GA: A Reconfigurable Architecture for Hybrid Circuits with Two-Terminal Nanodevices”, Nanotechnology, Vol. 16, pp. 888-900, DOI: 10.1088/0957-4484/16/6/045, 2005 [39]. Winfree E., “Algorithmic Self-A ssembly of DNA”, Proceedings of the International Conference on Microt echnology In Medici ne and Biology, DOI: 1-4244-0338-3/06, 2006 [40]. Pecheux F., Lallement C. and V achoux A., “VHDL-AMS and Verilog-AMS as Alternative Hardware Description Languages for Efficient Modeling of Multidiscipline Systems”, IEEE Transactions on Computer-Aided Design, DOI: 10.1109/TCAD.841071, 2004 [41]. Lyshevski S., “Nanotechnology and Super High-Density Three-Dimensional Nanoelectronics and NanoICs”, IEEE, DOI: 0-7803-7976-4/03, 2003 [42]. Lundstrom M.,“ Nanotransistors: A Bottom-Up View”, IEEE, DOI: 1-42440301-4/06, 2006 [43]. Ravariu C., “A NOI-N anotransistor”, IEEE, D OI: 0-7803-9214-0/05, 2005 [44] Pregaldiny F., Kammerer J. a nd Lallement C., “Compact Modeling and Applications of CNTFETs for Analog and Digital Circuit Design”, IEEE, DOI: 1-4244-0395-2/06, 2006 [45]. Drexler E., “Toward Inte grated Nanosystems: Fundamental Issues in Design and Modeling”, Journal of Computational and Th eoric Nanoscience, Vol. 3, pp. 1-10, 2006 [46]. Mylvaganam K. and Zhang L., “Bal listic Resistance Capacity of Carbon Nanotubes”, Nanotechnology, Vol. 18, pg. 4, 475701 DOI: 10.1088/09574484/18/47/475701, 2007 [47]. Schlegel M., Bennini F., Mehner J., Herrmann G., Mller D. and Dzel W., “Analyzing and Simulation of MEMS in VHDL-AMS Based on Reduced Order FE-Models”, IEEE Sensors, 2nd IEEE Intern. Conf. on Sens ors, Toronto, Canada, 2003 [48]. Schlegel M., Herrmann G. and Mller D., “Application of a Multi-Architecture Modeling Design Method in System Level MEMS Simulation”, DTIP 2003, Mandelieu-La Napoule, France, 2003

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74 [49]. Karray M., Desgreys P. and Ch arlot J., “VHDL-AMS Modeling of VCSEL Including Noise”, IEEE, DOI: 0-7803-8135-1, 2003 [50]. Ewing R. and Cabrera E., “Scaling Is sues Addressed for Mixed-Signal Design by use of Dimensional Analysis Methodology and VHDL-AMS”, IEEE, DOI: 0-7803-5491-5/99, 1999 [51]. Wilson P., Ross, J. Brown A. and Rushton A., “Multiple Domain Behavioral Modeling Using VHDL-AMS”, IEEE 2004, 0-7803-8251-X/04, Proceedings of the 2004 International Symposium on Circui ts and Systems ISCAS '04, Vol. 5, pp. 23-26 May 2004 [52]. Trofimov M. and Mosin S., “The R ealization of Algorithmic Description on VHDL-AMS”, Proceedings on Modern Problems of Radio Engineering, Telecommunications and Comput er Science, 2004, TCSET’ 2004 [53]. Nikitin P., Shi R. and Wan B., “ Modeling Partial Differential Equations in VHDL-AMS”, IEEE 2003, DOI: 0-7803-8182-3/03, Proceedings of IEEE International SOC Conference, pp. 345 348, 17-20 Sept. 2003 [54]. Rhew J., Ren Z. and Lundstrom M. “Benchmarking Macroscopic Transport Models for Nanotransistor TCAD”, Journal of Computational Electronics, Vol. 1, pp.385 388, KluwerAcademicPublishers, 2002 [55]. Rengel R., Pardo D. and Martin M., “Towards the Nano Scale: Influence of Scaling on the Electronic Transport and Small-Signal Behavior of MOSFETs”, Nanotechnology, Vol. 15, pp. 276 282, 2004. [56]. Kumar S., Kumar A. and Choudhury S., “Soft Computing Tools for the Simulation of Efficient Nanodevice Models”, IEEE, DOI: 0-7803-7749-4/03, 2003 [57]. Shear L., Singleton C ., Haertel G., Mitchell K. and Zaner S., “Measuring Learning: A Guidebook for Gathering and Interpreting Evidence”, Center for Technology in Learning, SR I International, 2007 [58]. NANOMAT Project No., ETIS-CT2003-508695, Deliverable 3.1, European State of Art Report, 2003

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75 APPENDICES

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76Appendix A: Matlab Code of Molecular Transistor Model % MATLAB CODE OF MOLECULAR TRANSISTOR MODEL clear all % Constants definition Hbar = 1.055e-34; q = 1.602e-19; I0 = q*q/hbar; % maximum conductance % Parameters definition U0 = 0.25; % charging energy in eV kT = 0.025; % energy in eV %at room temp T=300K mu = 0; ep = 0.2; % in eV N0 = 0; Alphag = 0; % molecular coupling Alphad = 0.5; % molecular coupling %Energy grid NE = 501; E = linspace(-1,1,NE); dE = E(2) E(1);

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77Appendix A: (Continued) g2 = 0.005*ones(1,NE); % gamma 2 g1 = g2; % gamma 1 g = g1 + g2; % absolute broadening factor %Bias IV = 101; VV = linspace(-0.8,0.8,IV); % applied voltage for iV = 1:IV Vg = 0; % gate voltage Vd = VV(iV); Vg = VV(iV); mu1 = mu; mu2 = mu1 Vd; UL = -(alphag*Vg) (alphad*Vd); U = 0; % self-consistent field dU = 1; while dU>1e-6 f1 = 1./(1 + exp((E mu1)./kT)); f2 = 1./(1 + exp((E mu2)./kT)); D = g./(2*pi))./(((E – ep – UL U).^2)+((g./2).^2)); D = D./(dE*sum(D)); N(iV) = dE*2*sum(D .*((f1.*g1./g) + (f2.*g2./g)));

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78Appendix A: (Continued) Unew = U0*(N(iV) N0); dU = abs(U Unew); U = U + 0.1*(Unew U); end I(iV) = dE*2*I0*(sum(D.*(f1 f2).*g1.*g2./g)); end hold on h = plot(VV,I); grid on

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79Appendix B: VHDL-AMS Code of Molecular Transistor Model The VHDL-AMS coding for the Molecular Tr ansistor Model is presented in this appendix. A complete entity-a rchitecture pair for the molecu lar transistor is presented. -VHDL-AMS MODEL OF A MOLECULAR TRANSISTOR -University of South Florida -College of Engineering -"Nano Scale Based Model De velopment for Mems To Nems Migration", Ph.D. Thesis -in Electrical Engineering -Copyright Andres Lombo-Carrasquilla -Model name: Molecular Transistor Level-0 -This is a discrete model of a molecular transistor -The molecular resistance is associ ated with the in terface between the -narrow wire and the wide contacts -Ballistic transport model: -No scattering is assumed -Contact 1 is grounded -Low bias is assumed -Minimum broadening of mol ecular energy levels is assumed -This code is optimized to be simulated with Hamster tool by Ansoft -Corporation -Constant Fermi level assumed

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80Appendix B: (Continued) -The output voltage obtained represents only the positive part LIBRARY IEEE; USE IEEE.MATH_REAL.ALL; -entity definition ENTITY MOLCtoy IS QUANTITY mu1: REAL; QUANTITY mu2: REAL; QUANTITY eVg: REAL; QUANTITY USC: REAL; QUANTITY N: REAL; QUANTITY ep: REAL; QUANTITY N0: REAL; QUANTITY f1: REAL; QUANTITY f2: REAL; QUANTITY N1: REAL; QUANTITY N2: REAL; QUANTITY IL: REAL; QUANTITY IR: REAL; QUANTITY I: REAL; QUANTITY G: REAL; CONSTANT eta: REAL := 0.5; -charging co efficient, which could be 0
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81Appendix B: (Continued) CONSTANT ep0: REAL := -5.5; - molecular -potential energy level in -equlibrium CONSTANT Ef: REAL := -5.0; -< electronvolts > Fermi level CONSTANT hbar: REAL := 1.1356e-15; --Planck constant CONSTANT g1: REAL := 0.1; -Br oadening coefficient gamma1 CONSTANT g2: REAL := 0.1; -Br oadening coefficient gamma2 CONSTANT U: REAL := 0.001; -charging constant CONSTANT kT: REAL := 0.025; -Boltzman constant at room -temperature CONSTANT q: REAL := 1.602e-19; -electron charge END ENTITY MOLCtoy; ARCHITECTURE Level-0 OF MOLCtoy IS BEGIN N0 == 2.0/(1.0 + exp((ep0 + Ef)/kT)); -electrons in equilibrium state eVg = = now; -(ele ctronvolts) applied energy level USC = = eta evg; -charging voltage effect over the -molecule mu1 = = Ef-(1.0 eta) eVg; -first contact energy level mu2 = = Ef + (eta eVg); -second contact energy level ep = = ep0 + USC; -molecular energy level

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82Appendix B: (Continued) f1 = = 1.0/(1.0 + exp((ep mu1)/kT)); -Fermi level at the first contact f2 = = 1.0/(1.0 + exp((ep mu2)/kT)); -Fermi level at the second contact N1 = = 2.0 f1; -number of charge carriers at first contact N2 = = 2.0 f2; -number of charge carriers at second -contact IL = = (2.0 g1 q/hbar) (N1 N); -right current IR = = (2.0 g2 q/hbar) (N N2); -left current G = = I'dot; -conductance equilibrium conditions -verification routine IF (IL = IR) USE N = = (2.0 ((g1 f1) + (g2 f2))/(g1 + g2)); I = = (2.0 q/hbar) (g1 g2/(g1 + g2)) (f1 f2); ELSE N = = ((USC/U) + 2.0 N0) N; -charging condition I = = IR IL; END USE; END ARCHITECTURE Level-0; --END OF VHDL-AMS MODEL OF A MOLECULAR TRANSISTOR

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83Appendix C: XML Code of Molecular Transistor Model + <langstring>Molecular Transistor Model</langstring> Predeterminado Entrada de catlogo predeterminada sp Predeterminado Predeterminado Palabra clave predeterminada

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84Appendix C: (Continued)
1.1 LOMv1.0 Final ADL SCORM 1.2 text/html molctoy.htm

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85Appendix C: (Continued)
LOMv1.0 Simulation LOMv1.0 no

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86Appendix C: (Continued) LOMv1.0 no
Model Description
LOMv1.0 Educational Objective

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87Appendix C: (Continued) Molecular transistor model
Simplified Non-equili brium Green function


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ABOUT THE AUTHOR Andres Lombo Carrasquilla received a Bachelor’s Degree in Electrical Engineering from Universidad Distrital Franci sco Jose de Caldas in 1994 and a M.S. in Electrical Engineering from Un iversidad de Los Andes in 1998. Andres started as a faculty member at the College of Engineering at Universida d Distrital in 1997, where he joined the Optoelectronics and Microelectroni cs Research Group at this university in 1994. Andres is currently the Prin cipal Investigator of the group. Andres was supported by Universidad Di strital during his Ph.D. program at the University of South Florida, in Tampa, Fl orida, where he was involved in research activities in MEMS and nanot echnology related areas. Andr es has made several paper presentations in Colombia, Latin America, and other international meetings related to the nanotechnology area.


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ABSTRACT: A novel integrated modeling methodology for NEMS is presented. Nano scale device models include typical effects found, at this scale, in various domains. The methodology facilitates the insertion of quantum corrections to nanoscale device models when they are simulated within multi-domain environments, as is performed in the MEMS industry. This methodology includes domain-oriented approximations from ab-initio modeling. In addition, the methodology includes the selection of quantum mechanical compact models that can be integrated with basic electronic circuits or non-electronic lumped element models. Nanoelectronic device modeling integration in mixed signal systems is reported. The modeling results are compatible with standard hardware description language entities and building blocks. This methodology is based on the IEEE VHDL-AMS, which is an industry standard modeling and simulation hardware description language.^ ^The methodology must be object oriented in order to be shared with current and future nanotechnology modeling resources, which are available worldwide. In order to integrate them inside a Learning Management System (LMS), models were formulated and adapted for educational purposes. The electronic nanodevice models were translated to a standardized format for learning objects by following the Shareable Content Object Reference Model (SCORM). The SCORM format not only allows models reusability inside the framework of the LMS, but their applicability to various educational levels as well. The model of a molecular transistor was properly defined, integrated and translated using SCORM rules and reused for educational purposes at various levels. A very popular LMS platform was used to support these tasks. The LMS platform compatibility skills were applied to test the applicability and reusability of the generated learning objects.^ ^^Model usability was successfully tested and measured within an undergraduate nanotechnology course in an electrical engineering program. The model was reused at the graduate level and adapted afterwards to a nanotechnology education program for school teachers. Following known Learning Management Systems, the developed methodology was successfully formulated and adapted for education.
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