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A self-consistent numerical model for bipolar transport in carbon nanotube field-effect transistors Clifford, Jason Paul 2003

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A SELF-CONSISTENT NUMERICAL M O D E L FOR BIPOLAR TRANSPORT IN C A R B O N N A N O T U B E F I E L D - E F F E C T TRANSISTORS by JASON P A U L C L I F F O R D B . A . S c . (Computer Engineering) University of Bri t ish Columbia, 2002 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Electrical and Computer Engineering We accept this thesis as conforming to thp-j=^auired standard THE UNIVERSITY OF BRITISH COLUMBIA December 2003 © Jason Paul Clifford, 2003 L i b r a r y A u t h o r i z a t i o n In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Jason Clifford December 15, 2003 Name of Author (please print) Date Title of Thesis: A Self-Consistent Numerical Model for Bipolar Conduction in Carbon Nanotube Field-Effect Transistors Degree: Master's of Applied Science Year: 2003 Abstract Carbon nanotube (CNT) field-effect transistors (CNTFETs) utilize a semiconducting CNT channel controlled by an isolated electrostatic gate. The essential physics of these devices is captured in a numerical model that allows calculation of energy band diagrams and current-voltage (I-V) relationships to describe their behavior. The numerical CNTFET device model is based on a solution to Poisson's equation that links the electrostatic potential everywhere in the device to charge induced in the CNT. This charge is a function of the local electrostatic potential and can be calculated from the carrier distribution and the local density of states for the nanotube. Under equilibrium conditions the carrier distri-bution within the CNT channel is known precisely. However, under non-equilibrium conditions (when a source-drain bias is present) carrier distributions in the channel are distorted from their equilibrium forms due existence of hot electrons and holes emitted from the source and drain, respectively. Quasi-equilibrium Fermi statistics are used to approximate non-equilibrium carrier distributions using an equilibrium distribution function shifted in energy. Solving Pois-son's equation self-consistently for charge and the electrostatic potential provides a profile of potential energy along the length of the CNT channel. Current flow in the device is a function of the ability of carriers to tunnel through or be thermionically emitted over the potential bar-riers in the channel. The contribution of both electrons and holes is considered when solving Poisson's equation and calculating the drain current. The numerical CNTFET model is a flexible framework for the examination of a wide variety of device geometries and materials parameters. It employs a finite element package to solve Poisson's equation in a two-dimensional projection of a cylindrical model space with azimuthal symmetry. The technique of conformal mapping is used to allow exact electrostatic solutions in unbounded structures without imposing artificial boundary conditions. The geometry and materials parameters of the CNTFET are systematically varied and the effects of these changes are observed in the performance predictions of the model. The model is also used to explore the behavior of more complex experimental architectures, such as partially gated CNTFETs, and provides new insight into their operation. ii Table of Contents Abstract 1 1 Table of Contents iii List of Figures v Acknowledgments vii Chapter 1. Introduction 1 1.1 A New Electronic Materials System 1 1.2 C N T F E T Modeling 2 1.3 Thesis Outline 4 Chapter 2. Background 7 2.1 Carbon Nanotubes 7 2.1.1 Structure and Electronic Properties of Graphene 8 2.1.2 Structure of C N T s 10 2.1.3 Electronic Properties of C N T s 13 2.2 Carbon Nanotube Field-Effect Transistors 15 2.2.1 S B - C N T F E T s 16 2.2.2 C M - C N T F E T s 18 2.3 Exist ing C N T F E T Models 19 2.3.1 Guo Model 20 2.3.2 Castro Model 22 2.3.3 John Model 24 Chapter 3. The Numerical C N T F E T Model 26 3.1 Structure and Materials 26 3.1.1 Geometry 26 3.1.2 Materials 28 3.1.3 Schottky Barriers 30 3.2 Equil ibr ium Solution 30 3.2.1 Poisson's Equation and Boundary Conditions 31 3.2.2 C N T Surface Charge 32 3.2.3 Solving the P D E 33 3.2.4 Band Diagrams 34 3.3 Self-Consistent Quasi-Equilibrium Solution 35 3.3.1 Carrier Tunneling 35 3.3.2 Quasi-Fermi Levels and Flux Balancing 37 3.3.3 Current Calculations 42 3.3.4 The Iterative Method 43 3.4 Modeling Open Boundaries 43 3.4.1 Conformal Mapping 45 i i i Table of Contents 3.4.2 Transformation of Laplace's Equation 41 3.4.3 The FEM-Optimized Conformal Transformation 4' Chapter 4. Results 4! 4.1 General Model Behavior 4! 4.1.1 Benchmark Device Configuration 4! 4.1.2 Equilibrium Conditions 51 4.1.3 Comparison With Analytic Equilibrium Solution 5'. 4.1.4 Non-Equilibrium Conditions 5' 4.1.5 General I- V Characteristics 5! 4.1.6 Comparison With Experimental I- V Characteristics 61 4.1.7 Contribution of Electrons and Holes to Breakdown Current 6i 4.2 Dependence on Materials Parameters 6' 4.2.1 Contact Work Functions 61 4.2.2 Dielectric Permittivity 6' 4.2.3 Dielectric Trapped Charge 7 1 4.3 Dependence on Device Geometry 71 4.3.1 Contact Radius 7, 4.3.2 Gate Dielectric Thickness 7-4.3.3 Gate Length and Gate-Contact Spacing 71 4.4 Channel Modulation CNTFET Devices 71 4.4.1 Architecture 7\ 4.4.2 Equilibrium Conditions 8 4.4.3 P-type Operation 8! 4.4.4 N-type Operation 8: Chapter 5. Conclusions 8! Bibliography 8i Appendix A . Compact Expressions Based on the Maxwell-Boltzmann Distribu tion Function 91 A . l Surface Charge Density 9: A. 2 Solving the Flux Balance Equations to Locate the QFLs 9' Appendix B . F E M L A B 9< B. l Model Specification and Configuration 9 B . l . l Environment 9 B.l.2 Geometry 9-B.1.3 Boundary Conditions 9. B.1.4 PDE 9 B.1.5 Mesh 10 B.1.6 Initial Values 10' B.1.7 Solver 10 B.2 3D Solutions 10 B.3 2D Solutions 10 iv List of Figures 2.1 The 2D graphene lattice 8 2.2 Graphene dispersion relation 10 2.3 Section and elevation views of zig-zag, armchair, and chiral CNTs 12 2.4 The chiral and translational vectors of a (4,2) CNT 13 2.5 Enhanced SEM image of a back-gated SB-CNTFET 17 2.6 Energy bands along the length of a SB-CNTFET 17 2.7 Enhanced SEM image of a top-gated CM-CNTFET 19 2.8 Energy bands along the length of a CM-CNTFET 19 2.9 Cross-sections of the cylindrical Guo CNTFET geometry 21 2.10Cross-sections of the closed, cylindrical CNTFET geometry of Castro and John 23 3.1 Isometric and cross-sectional views of the cylindrical CNTFET model geometry 27 3.2 Energy band diagram along the longitudinal axis of the CNTFET 35 3.3 Energy band diagram along the transverse axis of the CNTFET 36 3.4 Detailed electron distribution and Fermi-Dirac distribution shifted in energy. 38 3.5 Energy band diagram of the CNTFET under non-equilibrium conditions 39 3.6 Iterative process for obtaining self-consistent electrostatics solutions 44 3.7 The z-plane and conformally mapped w-plane 46 4.1 The closed cylindrical CNTFET geometry. 49 4.2 Longitudinal band diagram for a benchmark CNTFET 51 4.3 Conduction band profiles calculated using analytic and numerical models 53 4.4 Absolute difference between the numerical heterogeneous and homogeneous solutions. 53 4.5 Energy band diagrams showing the positions of the QFLs for increasing VDS 55 4.6 Electron component, hole component, and total drain current at VGS = 0.3 V 56 4.7 Dependence of QFL splits L\Epn and AEpp on bias 56 4.8 Total drain current for VGs at 0.0, 0.1, 0.3, and 0.5 V 61 4.9 Total drain current for VGs at 0.2, 0.3, 0.4, and 0.5 V at low VDS 61 4.10Log of total drain current for VGS at 0.0, 0.1, 0.3, and 0.5 V 61 4.11Misewich CNTFET total drain current and log of total drain current 61 4.12Component currents and the product of the component currents as a function of VQS- 64 4.13Infra-red emission intensity in the Misewich CNTFET as a function of VGS 64 4.14Energy band diagrams for contact-CNT work function differences 66 4.15Total drain current current for various contact-CNT work function differences 67 4.16Band diagram near the source for various values of eTjOX and increasing VGS 68 4.17Drain current for various values of er<ox and at VGS = 0.0 V 69 4.18Drain current for various values of eT)OX and at VGS — 0.3 V 69 4.19Drain current for various values of eTiOX and at VGS — 0.1 V 69 4.20Drain current for various values of tTfiX and at VQS = 0.5 V 69 4.21Band diagram for e r iOX = 3.9 at VGs = 0.5 V and VDS = 0.6 V 71 4.22Drain current components for various values of eT<ox 71 4.23Band diagram at for er>ox = 25 at VGs = 0.5 V and VDS = 0.6 V 71 4.24Quasi-Fermi level splits 5Epn and 5Epp for various values of er>ox 71 v List of Figures 4.25Band diagram near the source for positive and negative dielectric trapped charge 72 4.26Equipotential contours near a source contact with a radius of 10 x Rcnt 74 4.27Equipotential contours near a source contact with a radius of R c n t 74 4.28Band diagram near the source for increasing contact radii 75 4.29Drain current components for C N T F E T s with contact radii of 10 x Rcnt and R c n t - . . . 76 4.30Total drain current for C N T F E T s with contact radii of 10 x R c n 4 and R c n t 76 4.31Band diagram near the source for increasing gate radii 77 4.32Band diagram near the source for increasing gate-source spacing 78 4.33Equipotential contours near the source for a gate-source gap of 20 nm 79 4.34Equipotential contours near the source for a gate-source gap of 250 nm 79 4.35Band diagrams near the source for various gate and contact lengths 80 4.36Band diagram for the C M - C N T F E T with negative dielectric charge at VGS = 0.0 V . . . 82 4.37Band diagram for the C M - C N T F E T with negative dielectric charge at VGS = 1-3 V . . . 82 4.38Band diagram for the C M - C N T F E T with negative dielectric charge at VGS = 1-3 V . . . 84 4.39Band diagram for the C M - C N T F E T with negative dielectric charge at VGS = 1.6 V . . . 84 vi Acknowledgments I would like to thank my supervisor, Dr. David Pulfrey, for his guidance during the course of my research and the writing of this thesis. I am especially grateful to my colleagues in the UBC Nanoelectronics Group, David John, Leonardo Castro, and Paul Pereira for the opportunity to have many thoughtful conversations on on carbon nanotubes. I also want to thank my girlfriend, Wendy, for her undying support and encouragement throughout my graduate studies at UBC. vii Chapter 1 Introduction 1.1 A New Electronic Materials System Decades of scaling and the ongoing demand for smaller feature sizes in integrated circuits is leading toward the physical limits of bulk semiconductor materials. A wide variety of new semi-conductor material systems are currently under investigation in an attempt to find a candidate to replace traditional bulk materials such as silicon and the Group-III-V semiconductors. One of the most promising new material arrangements is carbon nanotubes (CNTs). Discovered by the Japanese electron microscopist Sumio Iijima in 1991 [1], these cylindrical carbon molecules have unique electrical properties which are related to their one-dimensional (ID) structure. The diameter of a semiconducting CNT is approximately two orders of magnitude smaller than the minimum feature size of current bulk semiconductor technology (2003), making these devices very attractive for use in highly integrated semiconductor devices. The link between the physical structure and electronic properties of CNTs is a consequence of the small diameter of the tubes (order of nanometers) which quantizes the allowed electronic states in the circumferential direction. The diameter of the tube, and the orientation of carbon bonds in the tube wall both affect the energy band structure and differentiate semiconducting tubes with a finite bandgap from metallic tubes with zero bandgap. The combination of semi-conducting and metallic tubes could allow CNTs to form a complete material system for the fabrication of electronics circuits: semiconducting tubes fashioned into transistors, which are, in turn, connected using metallic tubes. 1 Chapter 1. Introduction Besides their small size, other advantages of C N T s which make them attractive for use in integrated circuits are their current- carrying ability and chemical compatibility with other ma-terials such as oxides. Transistors fabricated from semiconducting C N T s have shown transcon-ductances (per unit width) an order of magnitude greater than state-of-the-art M O S F E T s [2], while metallic C N T s used as an electrical transmission medium have demonstrated sustained current densities of up to 10 1 0 A / c m 2 [3]. These current-conducting abilities can be attributed to very low rates of carrier scattering in the tube [4] and could allow single tubes or small groups of C N T s to replace much larger structures in integrated circuits. Complete bonding of carbon atoms in the surface of C N T s prevents the dangling bonds and surface states common to bulk semiconductor materials. This bonding arrangement eliminates the need for passivation of the tubes and allows compatibility with a variety of oxides and other dielectric materials [2, 4]. Field-effect transistors (FETs) based on a semiconducting C N T channel have been demon-strated experimentally by a growing number of researchers [5, 6, 7]. C N T F E T s can be classified into two types, based on the way drain current is controlled. The Schottky barrier C N T F E T ( S B - C N T F E T ) modulates the thickness of potential barriers at the source and drain ends of a C N T channel in order to control the rate of carrier tunneling through the barriers. The channel modulated C N T F E T ( C M - C N T F E T ) adjusts the height of a thick potential barrier adjacent to the gate in order to create a channel for carrier movement. Both types of C N T F E T s can be fabricated to have either p-type or n-type characteristics, indicating that C N T s could be used to directly replace conventional F E T s in both digital complementary logic and analog designs. Complementary logic inverters based on C N T F E T s have already been demonstrated [2, 8], indicating the feasibility of integrating these devices. 1.2 C N T F E T Modeling Despite a growing number of experimental C N T F E T devices, there are few published C N T F E T device models. The models that do exist are targeted toward very specific C N T F E T geometries and thus lack generality. The first C N T F E T model, developed by Guo et al., uses the gradual channel approximation to determine the energies of the bands in a C N T channel as a function 2 Chapter 1. Introduction of the gate potential [9]. This model is limited to C M - C N T F E T s as it is unable to take into account the effects of Schottky barriers at the source-CNT and dra in -CNT interfaces. Two more recent models account for the presence of these Schottky barriers in order to examine SB-C N T F E T s [10, 11]. The former is a compact model which estimates potential barrier shapes near the source and drain and self-consistently calculates the resulting drain current for both electrons and holes. The latter model is capable of calculating the exact shape of the potential barriers near the source and drain but is limited to equilibrium bias conditions, thus limiting it's usefulness for calculating drain current. A more thorough discussion of these models is presented in Section 2.3. Each of the existing C N T F E T models is capable of explaining the operation of a particular type of C N T device using various assumptions and simplifications. However, these models are limited to representing a small range of possible C N T F E T configurations and are not capable of including more complex architectural features found in experimental devices. There is a need for a general, flexible, and accurate model which examines the equilibrium and dynamic behavior of CNT-based transistors. Such a model should be able to examine the performance of different types of C N T F E T operation and relate this to the device architecture. The numerical C N T F E T model developed in this work addresses these demands by providing a framework based on a finite element method ( F E M ) electrostatics solver and a self-consistent calculation for carrier concentration in the C N T . The numerical electrostatics solver allows a variety of materials and device geometries to be examined by reducing the constraints on device configurations imposed by analytic electrostatics solutions. This flexibility allows the model to measure the sensitivity of C N T F E T s to component properties and allows experimental device architectures to be examined, providing an opportunity to validate the behavior of the model through direct comparisons to experimental results. The model can simulate both S B - C N T F E T and C M - C N T F E T device architectures through appropriate combinations of materials and component geometry. The problem of self-consistently solving for electrostatic potentials and carrier distributions in the C N T F E T is partitioned into two modules. This segmented design should allow the numerical C N T F E T model to provide the basis for future models, as more 3 Chapter 1. Introduction sophisticated algorithms for calculating carrier distributions can be fitted into the framework as the understanding and architecture of C N T F E T s progresses. Besides the generality of the numerical C N T F E T model, its novel features include the use of conformal mapping to achieve accurate solutions of open boundary electrostatics problems, and the application of quasi-equilibrium statistics to approximate charge distributions under non-equilibrium conditions. Conformal mapping allows the model space of an open boundary, 2D electrostatics problem to be truncated near bodies of interest without affecting the accuracy of the solution. Combined with the azimuthal symmetry found in coaxial C N T F E T s , this tech-nique allows electrostatics solutions to be calculated in small model spaces and eliminates the use of artificial boundary conditions, thus improving the accuracy and efficiency of the solution. Quasi-equilibrium carrier statistics have been used previously to model highly non-equilibrium carrier distributions in H B T s [12, 13, 14]. These conditions are similar to those predicted in the channel of C N T F E T s . The use of quasi-equilibrium statistics allows carrier distributions to be approximated by analytical expressions and eliminates the need to simultaneously solve both Schrodinger and Poisson's equations. 1.3 Thesis Outline Chapter 2 introduces the physical and electronic properties of C N T s , discusses the two types of C N T F E T s , and examines existing C N T F E T models. A brief introduction to C N T s shows how their structure and electronic behavior can be derived from graphene, the 2D sheets of carbon atoms which form bulk graphite. The C N T density of states is derived from a graphene bonding model, providing the link between the physical and electrical properties of the C N T molecule. A n overview of the operating principles of S B - C N T F E T s and C M - C N T F E T s shows how current control in CNT-based transistors is linked to the structure and materials properties of the device. Finally, an examination of published C N T F E T models reveals the principles used to model C N T F E T s and introduces three recently published models. A discussion of the strengths and weaknesses of these models shows there is a need for a more robust C N T F E T model capable of simulating the behavior of a wider range of device architectures. 4 Chapter 1. Introduction Chapter 3 presents the CNTFET device model developed in this research. The numerical CNTFET model is based around a nonlinear Poisson equation which links electrostatic potential and charge on the surface of the CNT to applied potential differences at the source, drain, and gate electrodes. The model uses a cylindrical structure, as opposed to the planar structures common to the experimental CNTFETs introduced in the previous chapter. The cylindrical geometry of the CNTFET model provides better channel control due to increased capacitive coupling between the gate and CNT, and combined with azimuthal symmetry, allows the three-dimensional CNTFET structure to be described using a two-dimensional projection, reducing the resources required to find an electrostatics solution. Poisson's equation is solved using a FEM numerical partial differential equation (PDE) solver and yields the electrostatic potential throughout the CNTFET structure. The electrostatic potential on the surface of the CNT is used to construct an energy band profile which allows carrier movement through the CNT to be calculated using a novel, quasi-equilibrium, flux-balancing formulation. The calculation of carrier movement leads to a recalculation of the carrier distributions, which are, in turn, used in the solution of Poisson's equation, allowing an iterative, self-consistent solution of potential, charge, and carrier transport in the CNTFET. Chapter 4 discusses the predictions of the CNTFET model and compares the results to those of prototype experimental devices and an analytical model. The first section discusses the general behavior of the CNTFET model, as applied to SB-CNTFETs, and introduces bipolar conduction and drain-induced barrier thinning, two unusual characteristics of CNTFET oper-ation. A direct comparison between the results of the numerical CNTFET model and those of an analytic model show the accuracy of the numerical electrostatics solution. The second and third sections provide a detailed examination of the effects of materials parameters and component geometry on device behavior. Understanding the impact of dielectric permittivity, contact work functions, gate and contact geometry, and other key parameters of the CNTFET is the first step toward optimizing device performance. A discussion of CM-CNTFET devices shows hows the CM-CNTFET architecture is a variation of the SB-CNTFET architecture with a large gate-contact spacing and an additional agent to control the potential energy of the 5 Chapter 1. Introduction CNT in the gate-source and gate-drain gaps. The energy band diagrams of these devices show that Schottky barriers no longer control carrier movement, rather the potential energy of the channel becomes the regulating feature. Finally, predictions of CM-CNTFET behavior are compared with the characteristics of experimental CM-CNTFET devices, showing the validity of the numerical model as applied to this type of CNTFET device. The final chapter summarizes the findings of this project and presents ideas for future work. 6 Chapter 2 Background 2.1 Carbon Nanotubes Carbon nanotubes are hollow, cylindrical molecules with a surface composed of one or more separate layers of carbon bonded in a hexagonal lattice. The structure of these layers is nearly identical to graphene, but in a C N T the layers are rolled up to form coaxial, cylindrical shells capped by hemispherical carbon structures resembling fullerenes. C N T s have diameters of the order of l n m and lengths that can exceed 10 um. [15]. This results in a structure with a very high aspect ratio and I D behavior. C N T s are classified as either single-wall ( S W C N T ) or multi-wall ( M W C N T ) , depending on the number of layers of material forming the walls of the cylinder. In this work only single-wall tubes are considered, but the structure and properties of the layers are common to both types of tubes. A brief examination of graphene is used as a starting point for the development of a model of the inter-related physical and electronic properties of C N T s . The physical structure of a C N T is described in terms of a sheet of graphene rolled up along a chiral vector, which indicates the orientation of the carbon lattice relative to the axis of the tube. The orientation of the carbon lattice is critical in determining the electronic properties of the C N T . This is one of the most significant properties of C N T s and is unique to solid state physics [16, page v]. The electronic properties of C N T s are based on the electronic properties of graphene with additional constraints resulting from the quantization of allowed electronic states in the direction of the tube circumference. The electronic properties of graphene are derived from the nearest-3 7 Chapter 2. Background a) 0~O o ->>3 O-"g>^ p - e .©•<! ^ &••<$ ^ - d o - # CM3 b-Q) £>C| o d p - 4 p a o<* o o «>€ O O O O 0 a t>'6 00 ;©-o &-c) o e b o i><3 o d !>o <>d 'O-s o<S t>c 0-0 e- c Figure 2.1: The 2D graphene lattice showing: (a) the location of carbon atoms and carbon-carbon bonds, (b) a schematic representation of a small section of the lattice showing the lattice vectors d\ and a^. Subfigure a) Plotted using WebLab Viewer [17] from coordinates generated by TubeGen [18]. neighbor tight-binding model and yield a dispersion relation that can be extended geometrically to C N T s and used to calculate the C N T density of states (DOS). 2.1.1 Structure and Electronic Properties of Graphene Graphene is a two-dimensional (2D) hexagonal lattice of covalently bonded carbon atoms. Each atom is bonded to three adjacent atoms in the same plane. Figure 2.1(a) shows the arrangement of carbon atoms in a graphene sheet and Figure 2.1(b) shows the corresponding schematic representation of the graphene lattice including the lattice vectors, d\ and 0 2 , that define the lattice geometry. The lattice vectors are defined in terms of the lattice constant, a, which is a geometric function of the carbon-carbon bond distance, a c _ c (0.142 nm) [16, page 25], |a! | = |ci21 = a = \ / 3 a c - c - (2-1) Carbon-carbon bonding in the graphene lattice occurs v ia the formation of hybridized sp2 orbitals between neighboring atoms. In each atom, the 2s, 2px, and 2py orbitals combine to form three directional covalent o bonds in the zy-plane, leaving 2pz electrons to form valence 7r orbitals perpendicular to the plane. A first-order approximation of the electronic properties of this lattice is based on each atom contributing one valence n orbital occupied by one valence 8 Chapter 2. Background electron, and neglects the contribution of the o orbitals to the valence bands. These assumptions of limited electron interaction constitute a tight-binding model of the system. Assuming that interactions between w orbitals are limited to adjacent atoms further simplifies the model, yielding the nearest-neighbor tight-binding model for graphene. This model of electron orbital interaction is used to calculate the dispersion (E(k)) relation for graphene [16, 19], ^ky) = (2-2) 1 + sw(kx, ky) e2p - tw(kx,ky) 1 - sw(kx,ky) where t is the transfer integral, s is the overlap integral, e^v is the 2p orbital energy within each atom, and w(kx, kv)={l + Acos (^Aj c o s + W • represents the remaining components of the eigenvalues of the secular equation, t and s char-acterize the interaction of the electrons of neighboring atoms in the graphene lattice and are calculated from the Bloch functions used to represent the behavior of the electrons. Assigning values of t = —3.033 eV, s = —0.129, and e^p = OeV [16, page 27], completes the specification of the dispersion relation and allows the energies of the n bonding orbital and the TT* anti-bonding orbital to be calculated relative to the 2p orbital energy. The dispersion relation for a single hexagonal cell in the reciprocal graphene lattice is shown in Figure 2.2. The reciprocal lattice lies in the xy-plane and energy is plotted in the z-direction. The 7r bonding orbital is the lower surface and the n* anti-bonding orbital is the upper surface. These surfaces meet at the corners of the hexagonal reciprocal lattice cell, indicating that the bandgap of graphene is zero. When the temperature of the system is at absolute zero, the lower orbital is filled by the two valence electrons from the two carbon atoms in the graphene unit cell and the higher energy 7r* orbital is empty. Note that the valence band of graphene consists of all of the bonding orbitals, while the conduction band consists of all of the anti-bonding orbitals. 9 Chapter 2. Background Energy Figure 2.2: Dispersion [E(k)) relation for graphene showing the energies of the 7T bonding orbital and the 7r* anti-bonding orbital within the hexagonal reciprocal lattice cell. The reciprocal lattice lies on the xy-plane and energy is plotted in the z-direction. The dispersion relation given by Equations (2.2) and (2.3) can be further simplified using the Slater-Koster scheme to set the overlap integral to zero [20]. This makes the n bonding orbital and the n* anti-bonding orbital symmetric about €2P- Using €2P as the reference energy, the dispersion relation reduces to [16, 19] E(kx, ky) = = * j l + 4coS (^^j c o s (hi^ + 4cos2 j ' . (2.4) 2.1.2 Structure of C N T s The high degree of bonding symmetry in graphene allows a wide variety of lattice orientations when a graphene sheet is rolled to form a CNT. The only constraint is that atomic positions on one edge of the sheet must align with the atomic positions on the other edge of the sheet so that these positions are coincident in the closed structure. Note that the term "rolling" is used to describe how the structure of a CNT can be obtained from a graphene sheet and does not 10 Chapter 2. Background reflect how C N T s are actually formed. Figure 2.3 shows three general orientations of the carbon lattice in a S W C N T : zig-zag, armchair, and chiral. In a zig-zag tube, one set of bonds is parallel to the longitudinal axis of the tube. In an armchair tube, one set of bonds is parallel to the transverse axis of the tube. The "zig-zag" or "armchair" patterns can be seen by following a ring of bonds around the circumference of the tube. A chiral tube has bonds oriented at an angle between the armchair and zig-zag cases. Note that there is no distortion of the hexagonal carbon lattice in any of these orientations, other than a moderate bending to account for the curvature of the cylinder. The carbon-carbon bond length in the C N T is also increased slightly from that of graphene ( a c _ c = 0.144 nm) [16, page 38]. The alignment of bonds in a sheet of graphene rolled into a C N T is determined by the chiral vector, where d[ and a~2 are the graphene lattice vectors and n and m are the integer indices of the tube. The resulting tube is identified as a (n,m) C N T and has a radius of The coefficients n and m are limited to the range 0 < \m\ < n since any chiral vector outside this range is equivalent to a vector within the range due to the symmetry of the hexagonal lattice. A n armchair C N T has equal chiral vector coefficients (n = m), whereas a zig-zag C N T has only an n coefficient (m = 0). A l l other chiral vectors correspond to chiral tubes. Figure 2.4a) shows the chiral vector for a (4,2) C N T on the graphene lattice. Also shown are the lattice vectors, a"\ and a~2, and the directions of the chiral vectors for zig-zag (n,0) and armchair (n,n) L — ntii + ma2, (2.5) C N T s . The translational vector, T , is perpendicular to the chiral vector and defines the direction of the longitudinal axis of the C N T , (2.7) 11 Chapter 2. Background chiral (12,5) semiconducting Figure 2.3: Section and elevation views of the three varieties of CNTs showing the orientation of the graphene lattice. Each of these tubes has roughly the same diameter. Plotted using WebLab Viewer [17] from coordinates generated by TubeGen [18]. where d/j is the greatest common divisor of 2m + n and 2n + m. The chiral and translational vectors define the CNT unit cell. This cell spans the entire cir-cumference of the CNT, a portion of the length, and is the smallest repeating structure in the CNT geometry. The area of the CNT unit cell can be used to calculate the number of graphene unit cells in the CNT unit cell, N = LxT (2.8) m x a2\ For CNTs with very long lengths, the tube can be assumed to be infinitely long and the unit cell provides sufficient geometric detail to develop a model of the electronic properties of the tube. 12 Chapter 2. Background ! a2s L FT1 a 2 Figure 2.4: (a) The chiral and translational vectors of a (4,2) C N T plotted on the graphene lattice, (b) The reciprocal lattice for the same C N T showing the orientation of the allowed states (short line segments) relative to the chiral and translational vectors. 2.1.3 Electronic Properties of CNTs Assuming that the nearest-neighbor tight-binding model for graphene holds when the 2D sheet is rolled into a tube, the electronic properties of graphene can be used to define those of the C N T [16, 19]. The fundamental difference between C N T s and graphene is that there is a constraint on allowed electronic states in the circumferential direction of the C N T resulting from the periodic boundary condition imposed by the closed cylindrical form. If the C N T is assumed to be infinitely long, the only constraint on allowed states in the longitudinal direction is an equivalence of states in each C N T unit cell. Electron states in a C N T are represented by a 2D wave vector, K = iKL + jKT, (2-9) where Kr, and K? are reciprocal space vectors in the directions of the chiral and translation vectors in the C N T , respectively. The magnitudes of vectors Kr, and KT are defined in terms of the geometry of the C N T unit cell. The quantization of allowed states in the circumferential 13 Chapter 2. Background direction requires that ^ (2-10) r l and i is an integer in 1 < i < N. The equivalence of states along the length C N T requires that 2TT T (2.11) and j is a real number in —1/2 < j < 1/2. The magnitude of KT, is the separation in k-space between allowed electronic states in the circumferential direction. Each of the ./V graphene unit cells in the C N T unit cell contributes 2 allowed states but only one distinct value of IKL- There is no restriction on integer values of i> N, but these states are equivalent to states for i < N. The magnitude of KT is the length of the I D first Bri l louin zone in the C N T . Since j can take on real values, there is no restriction on the value of KT within the Bri l louin zone and energy bands are continuous in this direction. Figure 2.4 (b) shows the 28 allowed values of K for a (4,2) C N T along the length of L . Note that the location of the right-most allowed value of K coincides with the center of a reciprocal lattice cell. This state is equivalent to the state at KL = 0 since they are integer multiples of each other in k-space. For any value of K in the reciprocal lattice, there are two corresponding energy values defined by the dispersion relation for graphene: J5(tf(0) = ± i { l + 4 « » ( ^ . (2.12) Thus, for each i there is a K(i) which contributes a ir and TT* energy band to the I D dispersion relation for the C N T . This can be visualized by applying the graphene E(k) shown in Figure 2.2 to each of the reciprocal lattice cells in Figure 2.4 and taking cross-sections of the E(k) along the allowed state lines. A l l of the cross-sections combine to form the I D E(k) of the first Bri l louin zone for the C N T . 14 Chapter 2. Background The location of each allowed state line in the graphene reciprocal lattice cell determines the energy of the corresponding TX and IT* bands and the energy gap between them. The two closest 7T and 7T* bands determine the bandgap of the C N T . If an allowed state line coincides with the corner of a reciprocal lattice cell, where there is zero gap between the bands, the C N T wil l be metallic, otherwise it wi l l be semiconducting. Once the dispersion relation has been established it is relatively straight-forward to calculate the density of states (DOS) for the C N T [19]: N n(E) oc £ i This derivation constitutes the nearest-neighbor tight-binding D O S ( T B D O S ) for C N T s . It is also possible to derive the D O S using a simplified expression for the graphene E(k) that allows a completely analytic expression of the D O S for any C N T geometry [21]: n[E)= fa~ f , | £ | (2.14) where acc is the carbon-carbon bond length, Vppv is the tight-binding energy, Sm is the lowest energy of the energy band with index m, and r is the radius of the C N T . This universal D O S was developed by J . W . Mintmire and C . W . White, and is referred to as the Mintmire and White D O S ( M W D O S ) . Although less accurate than the T B D O S model, especially for higher energy bands, the M W D O S allows for very compact expressions for carrier concentrations when combined with the Maxwell-Boltzmann distribution function. These expressions are developed in Appendix A . In this work, the T B D O S is used exclusively to calculate carrier densities. The T B D O S allows the use of bias voltages that populate higher energy bands thus increasing the flexibility of the C N T F E T model. 2.2 Carbon Nanotube Field-Effect Transistors A t the qualitative level, the basic structure and operation of C N T F E T s are similar to tradi-tional bulk F E T s , as both types of devices use a gate-source potential difference to control carrier movement between the source and drain through a semiconducting channel. Examples 15 dE{K{i)) dK (2.13) Chapter 2. Background of experimental C N T F E T s show how structural and materials differences distinguish Schottky barrier C N T F E T s ( S B - C N T F E T s ) from channel modulation C N T F E T s ( C M - C N T F E T s ) and allow different features of the device to regulate the movement of carriers between the source and drain [22]. 2.2.1 S B - C N T F E T s The simplest C N T F E T architecture consists of a single C N T on top of a conducting plane, separated by a thin dielectric. Metal contacts deposited on top of each end of the tube form the source and drain of the device and the conducting plane forms the gate. Figure 2.5 shows an enhanced S E M image of an experimental C N T F E T with this type of architecture [23]. The figure is a composite of a scanning electron microscope (SEM) image of the surface of the device and a sectional drawing showing the dielectric and gate layers. The figure is approximately to scale and indicates the relative sizes of the device components. The channel length of the device in Figure 2.5 is approximately 300 nm, which is several times longer than the channel lengths of current bulk F E T devices. Unlike bulk F E T devices, however, transport in the C N T is ballistic and the length of the channel does not limit the drain current. The device in Figure 2.5 is a back-gated S B - C N T F E T . This type of C N T F E T incorporates a gate that spans the full length of the channel and provides strong capacitive coupling to all regions of the C N T . Strong coupling between the gate and the regions of the tube close to the source and drain contacts allow gate modulation of the thickness of potential barriers that control carrier movement into and out of the channel. This interfacial control is the characteristic feature of the S B - C N T F E T . The term back-gated refers to the location of a gate below the C N T channel. Typical back-gates are not localized to a particular C N T , rather they extend over large areas of the substrate that supports the device [5, 6, 23]. Use of an expansive back-gate simplifies construction of C N T F E T s by eliminating the need for tubes to be located precisely on the substrate, but also limits the integration of C N T F E T devices. Figure 2.6 shows the location of the energy bands along the length of the S B - C N T F E T for 16 Chapter 2. Background s i l i c o n s u b s t r a t e (gate) 0 20 40 60 80 100 120 Distance from source (nm) Figure 2.5: Enhanced S E M image of a back-gated S B - C N T F E T . The source/drain separation and dielectric thickness are ap-proximately 300 nm and 120 nm, respectively. Source: Appenzeller et al., Short-channel like effects in Schottky barrier C N T F E T s . In IEDM Tech. Digest, pages 285-288, 2002. Figure 2.6: Energy bands along the length of a S B - C N T F E T for VGS = 0.0 V and VGS = 0.3 V . Note the reduction of the potential energy of the conduction and valence bands along the length of the C N T for a positive Vcs- Th in potential barriers in the conduc-tion band at the source and drain allow elec-trons to tunnel between the contacts and the C N T . applied gate-source potentials of VGs = 0.0 V and VQS — 0-3 V . The work functions of the C N T and the metal electrodes are assumed to be identical such that the bands in the C N T are flat in the absence of any bias between the electrodes. Application of a positive bias between the gate and the source causes an accumulation of charge in the C N T that reduces the potential energy of the conduction and valence bands in the C N T channel. The fixed potential of the bands at the source and drain metallurgical junctions leads to band bending and the formation of spikes in the conduction band near the source and drain. These spikes are sufficiently thin to allow electrons to tunnel between the contacts and the tube. Increasing the gate-source potential reduces the thickness of the barriers and increases the number of electrons able to tunnel between the contacts and the C N T . If a drain-source potential is present there wil l be a net current between the drain and source which can be regulated by the potential applied to the gate. 17 Chapter 2. Background 2.2.2 CM-CNTFETs C M - C N T F E T s use a gate that partially spans the channel, resulting in reduced capacitive coupling between the gate and the regions of the C N T near the source and drain contacts. This configuration reduces the ability of the gate to control carrier movement by modulating the thickness of potential barriers at the source and drain contacts. Instead, C M - C N T F E T s modulate the potential energy of the gated region of the C N T and it's availability as a channel for carriers. C M - C N T F E T s are also known as bulk-switched C N T F E T s [22, 24]. In addition to the common F E T components, C M - C N T F E T s require an additional agent to control the height and thickness of the potential barriers that exist in the ungated regions of the C N T channel. This component can take the form of a gate dielectric with trapped charge [2], additional electrodes [22], or some other type of charged material in close proximity to the C N T . The net effect of the agent is to induce large quantities of charge in the C N T resulting in a shift in the C N T energy bands that reduce the thickness of barriers to carrier movement in the ungated regions. Figure 2.7 shows a sectional view of a C M - C N T F E T with a single gate and a dielectric with trapped charge. Figure 2.8 shows the energy bands along the length of C M - C N T F E T with trapped positive charge in the gate dielectric. The bands in the ungated regions of the C N T are extensively shifted as a result of very large concentrations of positive charge induced in the tube. Potential barriers are still present at the source and drain, but the barrier thickness is dramatically reduced and held constant, making the metal -CNT junctions essentially ohmic. In the mid-channel (gated) region of the C N T , the location of the bands remains a function of the applied gate potential due to capacitive coupling between the gate and the C N T . Increasing the gate-source potential difference causes negative charge to accumulate in the gated region of the tube, creating a channel for electrons between the source and drain. This device shows n-type behavior. A n p-type device could be constructed using trapped negative charge in the gate dielectric to reduce the potential barriers to holes. C M - C N T F E T s potentially offer several advantages over S B - C N T F E T s . Firstly, the thickness 18 Chapter 2. Background Gate oxide (20 nm)\JJ Source ( T i p f-C N T / P** S i Drain •~~<7I) S i 0 3 1 um Figure 2.7: Enhanced SEM image of a top-gated CM-CNTFET. Note that the gate does not span the entire length of the channel. Adapted from: Wind et al., Lateral Scaling in Carbon-Nanotube Field-Effect Transistors. Phys. Rev. Lett, pages 058301-1-058301-4, 2003. 0.4 0.2 0 > S. 0.2 3N D) | 0.4 LU 0.6 0 8 1 Vgs « 0.3/ Vgs = 0V EFdrainj I J 200 200 400 600 800 1000 1200 Distance from source (nm) Figure 2.8: Energy bands along the length of a CM-CNTFET with trapped positive charge in the dielectric for VGS = 0.0 V and VGS = 0.3 V. Note the reduction of the height of the potential energy barrier in the conduction band in the region of the CNT adjacent to the gate for a positive VGS- The barriers at the source and drain are not affected by the gate. of the potential barriers at the source and drain metallurgical junctions is dramatically re-duced. This reduces impediments to carrier movement that limit transistor conductance and current capacity [25]. Secondly, the sensitive metal-nanotube interface is no longer the current-controlling feature of the device [22]. This interface is highly sensitive to process variations, and could make consistency in SB-CNTFET devices difficult to achieve. 2.3 Exis t ing C N T F E T Models A CNTFET model allows predictions of internal and externally observable device behavior such as charge distributions and drain current based on applied potential biases and the properties of the device materials. In order to accomplish this, the model must solve Poisson's equation self-consistently for the electrostatic potential and charge everywhere within the device. The charge in a CNTFET is often limited to induced charge on the surface of the semiconducting CNT but can also include fixed charge in materials such as the gate dielectric. Induced charge in the CNT is a function of the local electrostatic potential and the distribution of carriers in 19 Chapter 2. Background the tube. Poisson's equation is solved in a model space that represents the physical structure of the device and is often a simplification or abstraction of an experimental device geometry. Bias potentials applied to the terminals of the device are typically represented as boundary conditions at the edges of the model space. The complexity of such a model arises from the fact that the electrostatic potential and charge distributions in the C N T F E T are interdependent. Under equilibrium conditions, the distribu-tion of carriers in a semiconductor can be defined analytically as a function of electrostatic potential using the density of states of the material and a statistical distribution function such as the Maxwell Boltzmann or Fermi-Dirac distribution. However, under non-equilibrium condi-tions there is no analytic relationship based on local electrostatic potential and the distribution of carriers must be found using more rigorous methods. A self-consistent solution of a C N T F E T needs to satisfy both Poisson's equation and the charge-potential relationship for carriers in the C N T . A n examination of three existing C N T F E T models show how simplified architectures and l im-ited bias conditions can be used to limit the complexity of solving the heterogeneous Poisson equation. Models developed by Guo et al. [9], Castro et al. [10], and John et al. [26] are pre-sented in the following sections with a focus on the basic operating principles and assumptions used to construct each model. The strengths, shortcomings, and generality of the models are also discussed. 2.3.1 Guo Model One of the first detailed C N T F E T models was published by Jing Guo, Prof. Mark Lundstrom, and Prof. Supriyo Datta [9]. The Guo model is based on a coaxial architecture with a long, cylindrical gate surrounding a semiconducting C N T . The C N T is intrinsic in the gated region and heavily doped at either end outside the gate, forming a C M - C N T F E T . Figure 2.9 shows cross-sections of the Guo device geometry. The Guo architecture allows several important simplifications to be made: 20 Chapter 2. Background Source Gate Dielectric n + CNT CNT n+ CNT £ Drain Figure 2.9: Longitudinal and transverse cross-sections of the cylindrical Guo CNTFET geom-etry showing the gate, source, and drain contacts, gate dielectric, heavily doped CNT regions, and the intrinsic CNT channel. 1. the channel is sufficiently long for transverse electric fields to dominate and the effects of longitudinal electric fields can be neglected; 2. the cylindrical device architecture provides azimuthal symmetry in the transverse electric fields; 3. the highly doped end regions of the CNT form ohmic contacts with the the source and drain electrodes. The long channel allows use of the gradual channel approximation to relate the charge density in the CNT to the potential applied to the gate. The applicability of the gradual channel approximation to FET devices with ballistic channels has been reported previously [27]. Com-bined with the azimuthal symmetry of the device, this approximation limits the solution of Poisson's equation to a single dimension in a direction normal to the surface of the tube. The concentration of electrons in the channel (the contribution of holes is ignored) is based on the injection of electrons from thermalized distributions in the source and drain electrodes. In the absence of Schottky barriers at the source-tube and drain-tube interfaces, the movement of electrons is limited only by the potential energy of the conduction band in the channel. 21 Chapter 2. Background The total electron density in the C N T is n = T,ni = Y , r ~P-[f(E)+f(E + qVDS)]dE (2.15) where EQ is the mid-gap energy in the C N T , A , is the bottom of the ith conduction band relative to E0, D(E) is the Mintmire and White density of states [21], and the source Fermi level is used as the reference energy. The drain current is simply the difference in the flux originating from the source and the flux originating from the drain. The shortcomings of this model are the limited variety of C N T F E T architectures which can be considered and, more importantly, the inability to take into account the effects of drain-source potential differences large enough to cause short-channel effects. The Guo model is only applicable to channel-modulated devices where the effects of the Schottky barriers at the source and drain can be overlooked. 2.3.2 Castro Mode l Leonardo Castro, David John, and Prof. David Pulfrey extended the Guo model by taking into account the metal-semiconductor interfaces at the source and drain contacts of a cylindrical C N T F E T [10]. This model allows S B - C N T F E T s to be examined using the long-channel princi-ples introduced by Guo et al. The Castro model uses a coaxial cylindrically gated geometry but terminates the intrinsic C N T channel with planar metal source and drain contacts, as opposed to heavily doped C N T sections. This structure creates a closed cylindrical box bounded on all sides by metal contacts. Figure 2.10 shows longitudinal and transverse cross-sections of the model geometry. This model uses a hybrid solution of Poisson's equation to calculate the electrostatic poten-tial on the surface of the C N T . Guo's gradual channel approximation is used to calculate the mid-channel potential, and for the regions of the C N T close to the source and drain contacts (within two contact radii of the metallurgical junction), the potential is approximated by an ex-ponential expression. This expression fits the potential profile to a complete analytical solution of Poisson's equation. The details of this analytical solution are introduced in the following 22 Chapter 2. Background Figure 2.10: Longitudinal and transverse cross-sections of the closed, cylindrical CNTFET geometry common to the Castro and John models. The gate forms the curved surface of the outer cylinder, and the source and drain contacts form the two ends. Note that the source and drain contacts are not separated from the gate at the corners of the cylindrical box. Adapted from: John et al., Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors. IEEE Trans. Nanotechnol, 2(3):175-180, 2003. section. While not optimal, this method of combining solutions allows the potential on the surface of the nanotube to be approximated under non-equilibrium conditions. The potential on the surface of the tube is used to construct an energy band diagram which defines the shape and height of potential barriers in the conduction band of the CNT channel. Carrier movement into and out of the channel is limited by the ability of electrons to tunnel through or be thermionically emitted over these barriers. The electron concentration in the channel is found by including the probabilities of electron tunneling and reflection in the integral introduced in Equation 2.15: n = g{E) {/+(£) (J- - l) T* + f»(E + qVDS) (J- - ij 7* j dE, (2.16) where EQ is the bottom of the conduction band, g(E) is the NNTB density of states, jfvjJT"(E) is the right-moving distribution of electrons in the source, and fp (E) is the left-moving distribu-tion of electrons in the drain. T$ and Tjj are the tunneling probabilities for the source and drain, respectively, and T* is the overall transmission probability (T* = TsTrj/(Ts + Xb — TSTD))-Similar to the Guo model, the current through the CNT channel is the difference in the electron flux entering the channel from the source and that entering the channel from the drain. 23 Chapter 2. Background When originally published, the Castro model did not take into account the possibility of hole contribution to the drain current. This feature was added later [26], allowing the observation of exponential increases in drain current with increasing source-drain bias after a period of drain current saturation. The Castro model, while similar in methodology, cannot be compared directly to the Guo model due to the fact that these models represent C M - C N T F E T s and S B - C N T F E T s , respectively. The shortcomings of this model include those noted for the Guo model, wi th the exception that the model is limited to the analysis of S B - C N T F E T s . This model also relies on a closed, and un-realizable cylindrical architecture which does not allow for electrical isolation of the source and drain contacts from the gate at the corners of the cylindrical box. There is also the possibility of inaccuracies in electrostatic potential near the contacts resulting from approximation of the solution to Poisson's equation. Since electron tunneling, and current in the device, is highly sensitive to the shape of the potential barriers in these regions, small errors in potential can result in much larger errors in drain current. 2.3.3 John Mode l David John, Leonardo Castro, Jason Clifford, and Prof. David Pulfrey used a complete, ana-lytical solution of Poisson's equation to calculate the location of the energy bands of cylindrical S B - C N T F E T s under equilibrium conditions [26]. The model is based on the same closed cylin-drical geometry as the Castro model, shown in Figure 2.10. The electrostatic solution within the C N T F E T is achieved through the superposition of solutions to Laplace's equation and Poisson's equation. In the case of Poisson's equation, the Green's function formalism is employed. Under equilibrium conditions, the concentration of electrons and holes in the C N T can be described as a function of the local electrostatic potential using an equilibrium statistical carrier distribution, such as the Fermi-Dirac distribution, and the local density of states. Integrating over the energy of the conduction band gives the net carrier concentration: roo n(z)= 9(E){f(E + qV(z))-f(E-qV(z))}dE, (2.17) JEc 24 Chapter 2. Background where EQ is the bottom of the conduction band, g(E) is the N N T B density of states, f(E) is the Fermi-Dirac distribution, and V(z) is electrostatic potential on the surface of the nanotube. Note that equation 2.17 includes both electrons and holes where the previous models considered only the contribution of electrons to the electrostatic solution. The equilibrium results of the John model are compared directly to those of the numerical C N T F E T model in Section 4.1.3. This model can be used to calculate the drain current under sub-threshold conditions by as-suming that there is no charge on the C N T . This assumption allows Laplace's equation to be solved in place of Poisson's equation when calculating the electrostatic solution of the C N T -F E T . The assumption is justified on the grounds that the amount of charge induced on the C N T for small gate-source biases is negligible. The current through the channel is calculated in the same manner as the Castro model. The John model also takes into account work function differences between the C N T and the source, drain, and gate contacts and allows the dielectric permittivity, channel length, and gate radius to be varied. The shortcomings of the John model are the limited variety of closed cylindrical device geome-tries which can be modeled and the inherent problems related to this unrealizable architecture, as noted previously. Additionally, the model uses a dielectric permittivity within the nanotube equal to that of the surrounding dielectric material. This assumption is an artifact of the ana-lytic method used to solve Poisson's equation and results in an overestimation of the thickness of the potential barriers at the source and drain. However, the greatest limitation is that the model is only applicable to equilibrium and sub-threshold non-equilibrium bias conditions. 25 Chapter 3 The Numerical C N T F E T Model 3.1 Structure and Materials The numerical C N T F E T model is built around a geometric structure that represents the com-ponents of the C N T F E T device. The dimensions of the components are parameterized, allowing a variety of C N T F E T device geometries to be examined. Appropriate boundary conditions and P D E s are assigned to the geometric objects and the surrounding space in order to provide them with the characteristics of the materials they represent. The boundary conditions and P D E s are also parameterized to allow the effects of applied biases and materials selection on C N T F E T performance to be observed. 3.1.1 Geometry The C N T F E T model includes the five basic components common to most F E T devices, namely the channel, source, drain, gate, and gate dielectric. The C N T F E T model geometry is com-pletely cylindrical and components are represented by cylinders of various radii and length, located on a common longitudinal axis. Isometric and cross-sectional views of the C N T F E T model geometry are shown in Figure 3.1. The source and drain electrodes are solid cylinders attached to either end of a cylinder representing the C N T channel. The gate consists of a cylindrical band around the C N T channel, separated by a layer of dielectric material. The dielectric is omitted in Figure 3.1a) for clarity, but it would occupy the space between the gate and the C N T surface and extend to the source and drain contacts. Note these views are not drawn to scale and the dimensions in the longitudinal direction have been compressed to allow the features of the device to be discerned. 26 Chapter 3. The Numerical CNTFET Model Figure 3.1: Isometric and cross-sectional views of the cylindrical CNTFET model geometry showing the basic components and dimensions of the device. The cylindrical geometry, although more difficult to fabricate than the planar structures used in current experimental CNTFETs, increases electrostatic coupling between the channel and the gate electrode and is likely necessary for the ultimate performance limit of CNTFETs to be reached [28]. The use of coaxial cylindrical components yields a CNTFET structure with azimuthal symmetry that can be exploited to represent the 3D device structure using only the radial and longitudinal components of a cylindrical coordinate system. This simplification reduces the degrees of freedom of the model and the complexity of solving Poisson's equation. The cylindrical geometry has been used in three previous CNTFET models where Poisson's equation was solved analytically [9, 10, 26]. 27 Chapter 3. The Numerical CNTFET Model Figure 3.1b) shows a cross sectional view of the C N T F E T model geometry and its dimensions. These dimensions can be specified independently, with the exception of the C N T radius which is linked to the electronic properties of the nanotube. The radius and length of the gate elec-trode can be modified in order to measure the effects of geometry on gate-to-channel coupling. Similarly, the radius of the source and drain contacts can be modified to determine the role of contact geometry on the behavior of the Schottky barriers at the contact-nanotube interface. 3.1.2 Materials There are three distinct types of materials used in the C N T F E T model. The source, drain, and gate contacts are metallic conductors, the gate dielectric is an insulator, and the C N T is a semiconductor. The relevant properties of contacts and dielectric are their work functions and permittivity, respectively, while the important C N T properties are the density of states, work function and permittivity of the tube. The interface between the C N T and the metal source and drain contacts forms a Schottky barrier. The height and shape of this barrier is a function of the contact work function and the bandgap and work function of the C N T . Metal Contacts The metal contacts in the C N T F E T are assumed to be ideal conductors in order to isolate the behavior of the device from secondary effects resulting from resistance and electric fields within the contacts. This allows bias potentials applied to the C N T F E T to be represented as Dirichlet (strong) boundary conditions on the surface of the contacts: V(surface of contact) = Vapviied uas ~ 4>contact, (3-1) where V is the vacuum potential, VappUed bias is the bias potential applied to the contact, and Q<Pcontact is the work function of the contact. Contact work functions are used to establish the location of the vacuum potential relative to the applied potential at the contacts. Unique work functions, q<f>s, q4>D, and qcftd are specified for the source, drain, and gate, respectively. The vacuum potential forms the solution to Poisson's equation and takes into account the effects of both work functions and applied potentials. Note 28 Chapter 3. The Numerical CNTFET Model that the vacuum potential is also referred to as the electrostatic potential when discussing electrostatic solutions of C N T F E T devices. Work function differences between the contacts and the C N T are very important as they have a direct impact on the shape of the energy bands near the contact-CNT interfaces. Another important assumption is that the contacts behave as I D metals due to quantization of allowed energies in the transverse direction. It is assumed that the diameter of the contacts is sufficiently small to constrain electron movement in all directions other than the longitudinal direction, effectively creating a I D structure similar to the C N T . The I D approximation allows the use of I D equations for tunneling and current at the contact-CNT junctions and has been used in previous C N T F E T models [10, 26]. Gate Dielectric The dielectric is assumed to be an ideal insulator, such that charge is not allowed to leak between the gate and any other component. The relative permittivity of the dielectric (er>ox) can be adjusted in order to examine the use of so-called high-K dielectric materials in the C N T F E T . The electron affinity (qXdielectric) of the dielectric does not need to be specified in order to calculate the vacuum potential since the local potential in the dielectric does not affect the solution. C N T Many of the electronic properties of the C N T used in the C N T F E T model are derived from the D O S described in Section 2.1.3. The D O S is used to calculate the surface charge density on the C N T , the location of the band edges, and the contribution of energy bands to conduction in the C N T channel. Although there are multiple bands in semiconducting C N T s , the lowest energy bands play the greatest role in determining C N T F E T behavior as they are the most likely to be occupied and the most likely to be used as channels for carrier movement. Accurate predictions of the location and density of these bands is necessary to ensure the accuracy of the C N T F E T model. 29 Chapter 3. The Numerical CNTFET Model The numerical C N T F E T model uses a (16,0) C N T with a diameter of 1.25 nm and bandgap of 0.639 eV. This is representative of the size and bandgap of semiconducting C N T s used in experimental C N T F E T devices [6, 29, 30, 31, 32, 33]. The lowest energy bands of the (16,0) C N T are doubly degenerate. Besides increased contribution to the D O S , this degeneracy provides two channels for conduction of electrons and two channels for conduction of holes in the conduction and valence bands, respectively. Carrier scattering within the C N T is not included in the model and transport within the tube is assumed to be ballistic. Addit ional properties of the C N T used in the model are the work function (4>CNT) and relative permittivity ( e r , c7vr ) of the tube. 3.1.3 Schottky Barriers The interface between a C N T and a metal contact forms a Schottky barrier. Unlike many of the interfaces between bulk semiconductors and metal contacts, the Schottky barriers associated with C N T s do not show Fermi-level pinning [34] and the height of the barriers remain a direct function of the work functions of the two materials. The height of the Schottky barrier (<?</>B) at a CNT-meta l interface can be calculated from the bandgap of the C N T (Eg) and any difference between the work functions of the metal (<J</>M) and the C N T (q</>cNT)- In the case of an intrinsic C N T half the bandgap energy is added to the difference in work functions of the two materials, # s = q<f>M - q<f>cNT + - y - (3-2) 3.2 Equi l ib r ium Solution Under equilibrium conditions (no applied source-drain potential difference, V^s — 0) the elec-trostatic solution of the C N T F E T is relatively straight-forward to calculate since the distri-bution of carriers in the tube is known. Both electrons and holes occupy states based on a thermalized distribution function. This situation allows the simplest possible solution for the electrostatic potential in the device and is an excellent starting point for developing a more complete model of C N T F E T behavior. 30 Chapter 3. The Numerical CNTFET Model The fundamental equations governing the electrostatic behavior of the C N T F E T are Poisson's equation, the I D D O S for C N T s , and a thermal distribution function such as the Maxwell-Boltzmann or Fermi-Dirac distribution. Together they form a nonlinear P D E which relates charge on the C N T (ps) and the electrostatic (vacuum) potential (V) everywhere in the model space. F E M L A B , a commercial finite element method ( F E M ) solver [35], is used to solve the P D E . The electrostatic potential on the surface of the C N T , along with the work functions of the C N T , and source, drain, and gate contacts, are used to calculate the energy band diagram of the C N T F E T . 3.2.1 Poisson's Equation and Boundary Conditions The electrostatic potential at any point in the C N T F E T satisfies Poisson's equation and the boundary conditions at the source, drain, and gate electrodes. The general form of Poisson's equation and the boundary conditions for the C N T F E T , using the source contact as a reference, are: - V - e V V = PV(V,EF) (3.3) V(surface of source) = —(ps (3-4) V(surface of drain) — VJJS — <PD (3-5) V(surface of gate) = VQS — <t>G, (3-6) where e is the dielectric permittivity, V is the vacuum potential, pv is the net volume charge density in the C N T , EF is the Fermi level in the C N T , VDS is the drain-source potential, and V G S is the gate-source potential, qcfts, Q&Di and qcfic are the contact work functions for the source, drain, and gate, respectively. It should be noted that while equation (3.3) is representative of the relationship between po-tential and charge in the C N T , the actual implementation of the potential-charge relationship is slightly different. Equation (3.3) is based on volume charge density (pv), but the charge in a C N T is distributed only in a thin band near the tube's surface and can be approximated better by a surface charge density (ps). In the context of a F E M solution, a Neumann boundary 31 Chapter 3. The Numerical CNTFET Model condition on the surface of the tube, - e V V = P s , (3.7) is used to relate the rate of change of the electrostatic potential (V) across the boundary to the surface charge density (ps) on the boundary [36, page 219], where e is the dielectric permittivity. 3.2.2 C N T Surface Charge Net charge in the C N T is the sum of electron and hole concentrations and is represented by a surface charge density function that depends on the local electrostatic potential and the location of the Fermi energy. The concentration of individual carriers is calculated from the density of states per unit area per unit energy (g(E))(see appendix A) and a statistical distribution function (f(E, Ep, VCNT{Z))) that is shifted from the Fermi energy by the local potential on the C N T (VCNT(Z))- Carrier concentration at a point (z) on the C N T is the product of these functions integrated over the the energy range of the conduction or valence band: rtop of band ns(z)= f(E,EF,qVCNT(z)) g{E) dE, (3.8) JEc where Ep is the Fermi energy in the C N T and VCNT(Z) = V(Z)+<PCNT. (3-9) The two density of states for C N T s introduced in Section 2, the Mintmire and White D O S ( M W D O S ) [21] and the nearest-neighbor tight-binding D O S ( T B D O S ) [16, 19], were examined for use with the C N T F E T model. When combined with the Maxwell-Boltzmann (MB) distri-bution function, the M W D O S forms a very compact representation of the net surface charge density, / \ / \ / \ nc /-IVCNT{Z)- EF tqVcNT{z)+EF Ps{z) = ps{z) - ns{z) « Ci(exp( — ) - exp( — )), (3.10) where kg is Boltzmann's constant, T is the ambient temperature, and C\ is a constant derived from the density of states. This equation is derived in Appendix A . 32 Chapter 3. The Numerical CNTFET Model While the M B - M W D O S method for calculating charge as a function of potential provides a very compact expression, its usefulness is limited by the accuracy of the M W D O S and to a greater extent the range of energy values over which the M B approximation of the carrier distribution is valid. Using an M B distribution precludes the use of a large range of useful C N T F E T bias voltages and thus severely limits the usefulness of the model. A better solution is to use the Fermi-Dirac (FD) distribution and the T B D O S to provide the greatest accuracy and flexibility in the model. The T B D O S is calculated numerically for a range of energy values, so it cannot be combined analytically with the F D distribution. Instead, the expression for the net surface charge density is an integral over the energy range of the conduction and valence bands. Due to the symmetry of the D O S for electrons and holes, this can be combined into a single integral over one band: Hop of band ps(z) = ps(z) - ns(z) = / (f(E,EF,qVCNT(z)) - f(E, -EF, -qVCNT(z))) g(E) dE. JEc (3.11) Despite the integral form of this FD-TBDOS-based expression for surface charge density, it does not significantly increase the computational resources required to calculate the electrostatic solution in the C N T F E T . The FD-TBDOS-based expression for surface charge density is used for all of the model results presented in Chapter 4. 3.2.3 Solving the P D E The nonlinear P D E introduced in Section 3.2.1 can be solved analytically for a limited number of closed geometries [26], however, more complex C N T F E T geometries require the use of numerical solutions. The general purpose F E M package F E M L A B easily incorporates the geometry and P D E s used to describe the C N T F E T and is used for all device configurations in the numerical C N T F E T model [35]. A detailed description of the implementation of the C N T F E T model in the F E M L A B environment, and how solutions were obtained, is presented in Appendix B . 33 Chapter 3. The Numerical CNTFET Model 3.2.4 Band Diagrams Solving Poisson's equation provides the electrostatic potential at every point inside the C N T -F E T model. This potential is used to locate the energy bands of the C N T F E T . Since the C N T F E T is azimuthally symmetric, the electrostatic potential on the surface of the C N T is also azimuthally symmetric and varies only along the longitudinal axis (z). This allows a I D electrostatic potential along the surface of the C N T to be extracted from the 2D or 3D representation of the C N T F E T . The conduction band (Ec) and valence band (Ev) of the C N T are located relative to the the vacuum potential by the C N T work function and bandgap (Eg). In an intrinsic tube, E Ec(z) = -qVcNT(z) + qcpcNT + y E Ev(z) = -qVCNT{z) + q4>CNT - y -Figure 3.2 shows the energy band diagram along the longitudinal (z) axis of a C N T F E T with a 100 nm long nanotube channel. The source and drain Fermi levels are shown in the regions - 2 0 < z < 0 and 100 < z < 120, respectively. The region 0 < z < 100 shows conduction and valence bands, and the intrinsic Fermi level of the C N T . There are no work function differences between the materials of the C N T F E T in this figure, but an applied source-gate bias of 0.5 V is shifting the bands of the C N T in the mid-channel region. Figure 3.3 shows the energy band diagram along the transverse axis of a C N T F E T with a 5nm nanotube gate oxide on either side of the C N T . The gate Fermi level is shown in the regions —10 < z < — 5 and 5.6 < z < 10.6, and the gate dielectric conduction band is shown in regions —5 < z < 0 and 0.6 < z < 5.6. The region 0 < z < 0.6 shows conduction and valence bands, and the Fermi level of the C N T . There are no work function differences in the materials for this figure, but an applied gate bias of 0.5 V is causing the band bending in the gate dielectric and a shift in the bands of the C N T . Note that there is no band bending within the C N T in the transverse direction since all charge is located on the surface of the tube. 34 Chapter 3. The Numerical CNTFET Model ir -source fCNT qV(z) drain -vac ^Fdr, 20 40 60 80 100 120 Distance from source (nm) Figure 3.2: Energy band diagram along the longitudinal axis of the C N T F E T for VGS — 0.5 V and V D S = 0.0 V . 3.3 Self-Consistent Quasi-Equil ibrium Solution Solutions of the C N T F E T under equilibrium conditions are straight-forward to calculate but cannot account for changes in electron and hole distributions resulting from an applied source-drain potential difference. In order to model the device under these conditions, the charge distribution in the C N T must account for both hot and thermalized carriers. In this work it is assumed that these distributions can be represented by quasi-equilibrium Fermi statistics. The hole and electron quasi-Fermi levels are allowed to split at the source-CNT and dra in-CNT metallurgical junctions, allowing carrier fluxes to be calculated at each interface. Balancing the net flux of each carrier at the the source and drain allows a self-consistent solution of charge and potential in the C N T F E T under non-equilibrium conditions. 3.3.1 Carrier Tunneling The movement of electrons and holes in S B - C N T F E T s is controlled by potential barriers in the conduction and valence bands at the source-CNT and dra in -CNT metallurgical junctions. The profile of these barriers is determined by the electrostatic solution of the C N T F E T . Carriers are able to surpass these obstacles, in the classical sense, by thermionic emission over the barrier or, 35 Chapter 3. The Numerical CNTFET Model 5 4 Vac 3 > <u q<t>, q<t>, 2 source CNT c LU 0 ^ F g a t e ^ F g a t e 10 5 0 5 10 Distance from metallurgical junction (nm) Figure 3.3: Energy band diagram along the transverse axis of the C N T F E T for V G S — 0.5 V and VDS = 0.0 V . quantum mechanically, via direct tunneling through the barrier. Using the ID representation of the metal contacts and the C N T , the probability of tunneling through the potential barrier at a particular energy is calculated using the W K B approximation [37, page 281], where h is Planck's constant, m is the effective mass of an electron in the current energy band, and U(z) is the potential energy of the barrier as a function of position. The limits of integration are the edges of the barrier at energy E. The W K B method provides an approximate solution to the Schrodinger equation in instances where the potential energy of the barrier is changing slowly relative to the wavelength of the particle, and has been used in previous C N T F E T device simulations [10, 26]. This method gives a tunneling probability which approaches zero as the barrier thickness increases and approaches one as the barrier thickness reaches zero. Quantum mechanical reflection above the barrier is not taken into consideration. (3.12) 36 Chapter 3. The Numerical CNTFET Model 3.3.2 Quasi-Fermi Levels and Flux Balancing Quasi-Fermi levels (QFLs) allow non-equilibrium carrier distributions to be approximated by an equilibrium statistical distribution function such as the Fermi-Dirac distribution. Application of the Q F L method to C N T F E T s was originally proposed by Prof. David Pulfrey and was developed in collaboration with David John. The use of Q F L s provide a compact representation of carrier distributions that are easily incorporated into flux and current equations and prevent the need for more rigorous calculations of the carrier distributions. The Q F L method is based on the concept that quasi-equilibrium conditions hold in the pres-ence of small potential energy gradients and that non-equilibrium carrier distributions can be approximated by equilibrium distributions shifted in k-space (energy-space). The magnitude of the shift is represented by the split of the Q F L from the equilibrium Fermi level at a potential barrier. The dashed curve in Figure 3.4 shows a detailed electron distribution in the conduction band of a C N T F E T calculated directly from electron fluxes at the source and drain barriers us-ing the method of Castro et al. [10]. The solid curve shows the shifted Fermi-Dirac distribution used to approximate the detailed electron distribution. Both distributions are for equivalent device geometries at the same bias conditions (VGS = 0.3 V , VDS = 0.3 V ) . The distribution of carriers in energy is noticeably different for these two methods, however, both methods predict the same total number of carriers in the channel for the same bias conditions. Since the solution of Poisson's equation is not sensitive to the energy of the carriers in the C N T , only the total number of carriers, similar electrostatics solutions are calculated for both distributions. Despite being conceived for use in quasi-equilibrium conditions, Q F L s have been found to be effective for describing carrier distributions under highly non-equilibrium conditions such as those encountered in extremely short base-width H B T s [12, 13, 14]. Abrupt junction, Npn, H B T devices are characterized by a potential spike in the conduction band at the emitter-base junction, a feature resulting from discontinuities in the electron affinity and bandgap at the metallurgical junction. Electrons are injected from the emitter into the base via tunneling and thermionic emission at this barrier. Due to the short base width (relative to the electron mean-37 Chapter 3. The Numerical CNTFET Model Figure 3.4: Electron distributions functions representing the probability of state occupancy as a function of energy above the conduction band edge. free-path length) there is very little opportunity for scattering, resulting in excess numbers of hot electrons, which profoundly distort the carrier distribution in the base from its equilibrium form. Despite the very non-equilibrium nature of this situation, it is possible to get good agreement between solutions based on Q F L distributions of carriers and those based on a complete solution of the Boltzmann Transport Equation [38]. The conditions at the metal-semiconductor junctions at the source and drain contacts of C N T -F E T s are similar to the emitter-base hetero-j unction of H B T s and thus lend themselves to analysis using Q F L s . Injection of electrons from the source contact into the C N T occurs via tunneling and thermionic emission at the potential barrier in the conduction band. The as-sumption of ballistic transport in the C N T precludes scattering of these carriers, resulting in carrier distributions in the C N T which are distorted from their equilibrium form [10], as shown in Figure 3.4. A similar phenomenon occurs with holes in the valence band, resulting in separate electron and hole Q F L s used to represent the distribution of each carrier in the C N T . Q F L s are located relative to known Fermi levels in the source and drain electrodes using the 38 Chapter 3. The Numerical CNTFET Model 0.6 i i i i i i i i -20 0 20 40 60 80 100 120 Distance from source (nm) Figure 3.5: Energy band diagram of the C N T F E T under non-equilibrium conditions, showing: the electron and hole quasi-Fermi levels, EFn (dashed line) and EFp (stippled line), respectively; the splitting of the two Q F L s ; and the various electron and hole fluxes, Fi to F 4 , and F5 to F$, respectively. Source: J. Clifford et al., Bipolar conduction and drain-induced barrier thinning in carbon nanotube F E T s . IEEE Trans. Nanotechnoi, 2(3):181-185, 2003. technique of flux balancing [39]. This method requires that the net flux through a feature such as a barrier must be equal to the net movement of carriers away from the feature in the absence of recombination or generation of carriers. When applied to the C N T F E T , and assuming there is no carrier recombination in the C N T , this requires that the net flux into the channel must equal the net flux out of the channel and the net flux through the barrier at the source must equal the net through the barrier at the drain. This relationship can be used to locate each of the Q F L s in the C N T . There are four electron fluxes to be considered in the C N T F E T : Fi, the flux from the source into the C N T ; F2, the flux from the C N T into the source; F3, the flux from the C N T into the drain; F4, the flux from the drain into the C N T . There are four corresponding fluxes for holes, F5 through F&, respectively. Figure 3.5 shows the electron and hole fluxes at each of the barriers in the C N T F E T . 39 Chapter 3. The Numerical CNTFET Model The magnitude of each flux is a function of the ability of carriers to traverse the potential barriers via tunneling or thermionic emission, and the carrier distribution. The Landauer formula is commonly used to calculate net current through a I D elastic scattering barrier [40], W = - T E / T ^ ~ f(E ~ qV)) dE, (3.13) i J ^bottom % where Etop % and Ebottom i a r e the upper and lower energy limits of the \ t h sub-band, V is the difference in electrostatic potential across the barrier, T(E) is the tunneling probability as a function of energy, and q is the magnitude of the electronic charge. The total current is the sum of the currents through each sub-band. However, due to the inter-band spacing and the bias voltages considered in the C N T F E T model, only the doubly degenerate, lowest energy band becomes sufficiently populated to contribute to the drain current. The contribution of higher energy bands is not considered and the summation in equation 3.13 can be replaced by a constant, 2, representing the degeneracy of the lowest energy band. A further simplification is to extend the upper limit of integration of the integral to infinity since the distribution function approaches zero as E —> oo. The Landauer equation can be adapted to calculate directional fluxes by considering the carriers originating from each side of the barrier separately: 2 f°° FposMve-k = - r / T(E)f(E)dE (3.14) ™ JEC Fnegative-k = ~T / T(E)f(E-qV)dE, (3.15) ™ JEC where Fvositive-k and Fnegative-k represent fluxes due to carriers occupying states in positive k-space and negative k-space, respectively. This can also be thought of as right-moving and left-moving fluxes in a I D system. The four electron fluxes can now be defined in terms of the 40 Chapter 3. The Numerical CNTFET Model distributions in the various regions of the C N T F E T , 2 f°° Fi = — / TSn(E)f(E)dE (3.16) ™ JEc 2 f°° F2 = — / TSn(E)f(E - EFn)dE (3.17) JEC 2 f°° F 3 = — / TDn(E)f(E - EFn)dE (3.18) nri JEC 2 f°° F4 = — TDn{E)f{E + qVDS)dE, (3.19) ™ JEC where EFn is the electron Q F L and Tsn{E) and Ton{E) are the electron tunneling probabilities at the source and drain, respectively. The tunneling probability at each barrier is calculated by applying the W K B method to the conduction band profile near the source-CNT and dra in-CNT junctions. EQ is the energy of the conduction band edge in the mid-tube region of the C N T where the band is essentially flat. This is the lower energy limit for electron movement through the C N T . The source Fermi level is used as the reference energy level for the system. The method of Q F L s allows state occupancy in all regions of the C N T F E T to be represented by Fermi-Dirac statistical distributions and the occupancy of states in positive and negative k-space to be represented by herni-Fermi-Diracian distributions. A n y difference between two distributions is a result of shifting one distribution relative to the other in k-space and is de-scribed by a difference in the Fermi-levels or Q F L s between the two distributions. For two carrier distributions approaching the same barrier from opposite directions, any net flux is a result of differences in the Fermi-levels or Q F L s on either side of the barrier since the transmis-sion probability calculated using the W K B method is the same when approaching from either direction. Applying these principles to the C N T F E T , the difference between fluxes F i and F2 is a result of the split in the electron Q F L (EFn) from the source Fermi level at the source-CNT junction. Similarly, the difference between fluxes F3 and F4 is a result of the split in EFn from the drain Fermi level at the dra in -CNT junction. The drain Fermi level is located relative to the source Fermi level by the applied drain-source potential difference (Vps) and since VDS is known and the net flux through the source and drain barriers must be equal, it is possible to solve for EFn. 41 Chapter 3. The Numerical CNTFET Model Separate flux balance equations for electrons and holes describe the relationships between fluxes at the barriers, Right-moving electron fluxes are treated as positive values and left-moving fluxes as negative. For holes, the opposite sign convention is used to reflect the movement of an oppositely charged carrier. Equations (3.20) and (3.21) are solved separately to yield EFn and EFp, respectively. Newton's method is used to find the electron and hole Q F L s which satisfy these equations. For low bias conditions it is also possible to use Maxwell-Boltzmann statistics to describe the carrier distributions. In this case there is an analytic solution for EFn and EFp, which is derived in Appendix A . 3.3.3 Current Calculations The drain current for each carrier is calculated from the tunneling probability and the distri-bution function using the Landauer formula. Each carrier current can be calculated once the locations of the Q F L s within the C N T are known, and can be found directly from the expression for net flux through either the source or drain barrier. The total drain current is the sum of these two carrier currents. F\ — F2 - F% — F4 (3.20) 1*6 — F5 — F& — Fj. (3.21) iDn = -q(F1-F2) q— TSn(E)(f(E) - f(E - EFn))dE 2 f°° (3.22) IDU — -JEC I D P = -q{Fe-F5) 2 fEv !Dp = -q-z \ (3.23) ID = IDU +1 Dp (3.24) 42 Chapter 3. The Numerical CNTFET Model 3.3.4 The Iterative Method The starting point for the quasi-equilibrium solution of of the C N T F E T is the calculation of the electrostatic solution, as described in Section 3.2, using the applied source-drain bias and best-guess Q F L values. Tunneling probabilities at the source and drain barriers are calculated from the electrostatic potential on the surface of the C N T and the flux balancing equations are solved to yield the new locations of Epn and EpP- These Q F L s are used in a successive electrostatic solution. After each iteration, the Q F L s calculated from the electrostatic solution are compared to those used to obtain the solution. If the values agree with each other, within an acceptable margin of error (0.00025 eV for the results presented in Chapter 4), a self-consistent solution has been found. Once self-consistency is achieved, the potential on the tube and the drain current are recorded as the solution of the C N T F E T under the given bias conditions. Figure 3.6 summarizes this process in a flow chart and shows the values being passed between different stages of the iterative process. 3.4 Model ing Open Boundaries The cylindrical C N T F E T geometry used by John et al. [26], and Clifford et al. [41], consists of a closed model space with strong boundary conditions on all sides. This closed form model is readily solved by both analytic and numerical means but does not lend itself to realizable device geometries. The requirement that the source and drain electrodes are separated from the gate by an infinitesimal spacing does not allow for an insulator between these two conductors with independent potentials. However, placing an insulating space at this location leaves a poorly defined boundary region known as an open boundary. There are a variety of techniques for dealing with open boundaries in F E M models. One of the most commonly used methods is to truncate the model space some distance beyond the bodies of interest using an artificial boundary [42]. This boundary can be of either Dirichlet or Neumann type, as both yield similar solutions near the bodies of interest. However, the extension of the boundaries to a sufficient distance so as not to cause inaccuracy in the solution can significantly increase the size of the model space and the resources required to find a solution 43 Chapter 3. The Numerical CNTFET Model r new E f n and E p p Initial guess for QFLs E F n and Epp Solve Poisson's equation in FEMLAB Iterative PDE solver E F n and Epp potential profile Use Flux Balance method to calculate QFLs Newton's method new Ep„ and E F p Iteration through back substitution No Are the new and existing QFLs equal within limits of accuracy? Yes Consistent solution Figure 3.6: Stages and data flow of the iterative process used to obtain self-consistent electro-statics solutions for the C N T F E T under non-equilibrium conditions. numerically. In order to avoid increasing the size of the model space when imposing an artificial boundary, it is necessary to have a boundary condition that renders the boundary essentially invisible and allows the solution to be calculated as if the boundary were not present. Conformal mapping can be used to impose these boundary conditions by calculating the solution in both the region of interest and in a second finite model space which represents all the space beyond the artificial boundary [42, 43]. 44 Chapter 3. The Numerical CNTFET Model 3.4.1 Conformal Mapping Conformal mapping involves the use of an analytic, one-to-one function to transform a complex value in one plane to a unique complex value in a second plane. This technique is often employed as a means to redefine a problem in a domain in which the solution is easier to obtain. In the context of F E M , conformal mapping can be used to transform the infinite space sur-rounding a model into a finite region in which the model equations can be solved using a finite number of elements. The first step is to establish a boundary which encloses the bodies of interest of the model. The area within the boundary is the near-space and the remaining area outside the boundary is the far-space. A l l points in the far-space are mapped into a second finite region in the conformed plane. The boundaries of the near-space and far-space regions are coupled through the use of equivalent nodes at the boundaries of each region, allowing a consistent solution to be calculated in both regions simultaneously. Equivalent nodes require that both the value and the derivative of the electrostatic potential to be the same in both nodes. For example, the space outside of a unit circle in the complex plane can be mapped into a second unit circle in another complex plane using the function w = f(z) = - , (3.25) z where z = x + iy and w = u + iv. The singularity at z = 0 is not encountered since only the region 1 < \z\ is mapped into the second plane. Figure 3.7 shows two unit circles, the first in the z-plane and the second in the w-plane. The z-plane represents unmapped space and the w-plane represents the corresponding transformed or mapped space. Point P in the z-plane of Figure 3.7 lies in far-space and is transformed into a corresponding point Q in the w-plane. Typically, the dimensions of the model and the boundary separating the near and far-space regions are chosen so that all of the bodies of interest, which include all sources, are contained within near-space, while the free space surrounding these bodies is mapped into the transformed far-space. This method has the advantage that the bodies of interest can be defined in terms 45 Chapter 3. The Numerical CNTFET Model z-plane w-plane > u Figure 3.7: The z-plane and conformally mapped w-plane. The area within the unit circle in the z-plane is the near space. The area outside the unit circle in the z-pane is the far space and is mapped into the unit circle in the w-plane. of conventional z-plane coordinates and the solution around the bodies of interest remains in terms of z-plane coordinates. If the far-space solution is needed, it can be mapped back into the z-plane using the inverse of the mapping function 1 * = / - » w (3.26) 3.4.2 Transformation of Laplace's Equation Equations corresponding to regions in the far-space of the z-plane are mapped into the w-plane in the same way that points in the far-space of the z-plane are mapped into the transformed space. For the C N T F E T model, Poisson's equation is used to determine the electrostatic potential as a function of position in all regions of the model space. In the region surrounding the device there are no sources of charge and Poisson's equation reduces to Laplace's equation. In a Cartesian system, such as that shown in Figure 3.7, the original and transformed versions of Laplace's equation are identical, due to the invariance of Laplace's equation. In the numerical C N T F E T model a 2-D projection of cylindrical space is used, resulting in an additional term in Laplace's equation, V 2D cyl. dr2 z dz dz2 (3.27) 46 / Chapter 3. The Numerical CNTFET Model Transformation of Laplace's equation between cylindrical coordinate planes was investigated by David John [44], yielding the transformed Laplace equation, VL c y l . = (u2 + v 2 ) ^ + ^ - 2u^ + j ^ = 0. (3.28) 3.4.3 The F E M - O p t i m i z e d Conformal Transformation Placing all bodies of interest in the near-space unit circle prevents the transformation of poten-tially complex geometries in the far-space region. However, this constrains the dimensions of model objects to fitting within a unit circle. These limitation can be overcome by scaling the model so as to normalize its dimensions, but this requires additional work when the model is defined and when the results are analyzed, and limits direct observation of model behavior. The use of unit circles to define the boundaries of the near and far field regions results from a requirement of the F E M solver that boundaries defined as equivalent must have the same length. In this case the near and far-field boundaries must have the same circumference and hence the same radius. Using the 1/z transform to map the space outside a circle with radius r results in a transformed space circle with radius 1/r, and obviously a different circumference. To avoid these limitations and complexities, a new mapping function based on the 1/z transform was developed. The FEM-opt imized transformation, w = & , (3.29) z allows a circle with an arbitrary radius (RB) to be drawn around the bodies of interest and a second circle of the same size is used to represent the far-space solution in the w-plane. RB is chosen so that the near-space circle is slightly larger than the minimum required to enclose the C N T F E T device being modeled. Simulations showed that the use of larger values of RB did not affect the electrostatic solution within the limits of accuracy of the numerical F E M tool. This transformation yields the same transformed Laplace's equation as calculated for the 1/z transform, 47 Chapter 4 Results 4.1 General M o d e l Behavior Before examining the relationship between C N T F E T architecture and performance, it is im-portant to understand the basic behavioral characteristics predicted by the numerical model. A S B - C N T F E T with a simple, closed cylindrical geometry is used as a reference architecture for an examination of the response of the model to equilibrium and non-equilibrium biasing. Band diagrams, equipotential plots, and current-voltage (I- V) plots are used to illustrate these predictions. I- V plots are particularly useful, as they allow comparison of the results of the model to the measured performance of experimental devices. This section also introduces bipo-lar conduction and drain-induced barrier thinning, two unusual characteristics predicted by the model that explain recent experimental C N T F E T observations. 4.1.1 Benchmark Device Configuration A benchmark C N T F E T device configuration is used as a control for simulations in which de-vice parameters are being adjusted, and serves as a default device for all other simulations. The geometry of this configuration is also identical to that used in a recent analytic C N T F E T model [26], allowing direct comparison of the results of both models. The benchmark configu-ration consists of a closed cylindrical geometry and materials parameters which simplify device analysis. The benchmark device geometry is a simplified, closed form of the general cylindrical geometry introduced in Section 3.1.1 and was originally selected to allow an analytic solution for potential 48 Chapter 4. Results Figure 4.1: The closed cylindrical CNTFET geometry. Adapted from: John et al., Electro-statics of coaxial Schottky-barrier nanotube field-effect transistors. IEEE Trans. Nanotechnol, 2(3):175-180, 2003. in the CNTFET [26]. Figure 4.1 shows sectional and end views of this configuration, which is also referred to as the analytic CNTFET geometry. In this configuration, the gate surrounds the entire length of the CNT and is separated from it by a cylindrical layer of dielectric material. Planar source and drain contacts form the ends of the cylindrical structure and are butted against the edge of the gate contact. This geometry forms a closed model space with known electrostatic potentials applied as strong boundary conditions on all perimeter surfaces. The radial dimensions of this device are based on the radius (Rc nt) of a (16,0) CNT and the longitudinal dimensions are measured in nanometers. The gate radius is 10 x R c n ( (6.26 nm) leaving a dielectric thickness of 9 x RCnt (5.64 nm). A 100 nm long CNT is used. The length of the CNT was chosen to provide a reasonably small model size while maintaining separation between the electrostatic fields localized at the source and drain. This separation eliminates direct interaction and any overlap of the band bending occurring near the source and drain terminals. Materials used for the benchmark device configuration are selected so as to simplify device behavior or to reflect the materials used in experimental CNTFET devices. The metals used for the source, drain, and gate contacts are assumed to have an identical workfunction to the 49 Chapter 4. Results C N T so as to eliminate band bending in the device in the absence of applied bias potentials. The dielectric permittivity of the C N T is set to that of free space [45], and the relative permittivity of the gate dielectric is set to 3.9, corresponding to Si02, a common gate dielectric material used in C N T F E T s [6, 5, 23]. 4.1.2 Equilibrium Conditions The electrostatic solutions of the C N T F E T for increasing gate-source biases are used to develop an understanding of carrier movement control in S B - C N T F E T s . While there is no drain current under equilibrium conditions, the features of the energy bands that regulate carrier movement in the device are present. Band Bending Figure 4.2 shows the energy bands of the benchmark configuration C N T F E T for applied gate biases of VGS = 0.0 V , 0.1 V , 0.3 V , and 0.5 V . Note the presence of Schottky barriers in both the conduction and valence bands at the source and drain metallurgical junctions. These barriers fix the potential energy of the bands at these junctions relative to the energy of the Fermi level in the source and drain. In the mid-channel region of the C N T , displacement of the conduction and valence bands occurs as a result of capacitive coupling with the gate. In order to satisfy the strong boundary conditions at the ends of the tube and the capacitive coupling in the central region of the channel, the potential energy of the band must vary along the length of the tube, resulting in band bending near the source and drain contacts. For small applied gate biases (VGS < 0.3 V ) there is a corresponding small shift in the energy bands in the mid-channel region of the C N T . The amount of charge induced on the C N T by this shift in the bands is negligible and the solution to Poisson's equation inside the C N T is not appreciably different from the solution to Laplace's equation. Under these conditions there are no significant transverse electric fields in the mid-channel region of the C N T since the local potential on the surface of the tube matches the potential of the gate. For larger applied gate biases, shifts in the bands of the mid-channel region of the C N T result in a significant accumulation of charge on the C N T surface. The presence of this charge reduces the magnitude 50 Chapter 4. Results X - E c (VGS=0.0V) 1 K 4 T E c ( v G S = o . i v M Schottky Barrier -\V r R E c (v G S =o.3V)// -A ^ - E C (VGS=0.5V) E F d r a i r -Schottky Barrier k A • W ^E V (V G S =O.3VJ/J -1.8 ' ' ' ' ' ' 1 1 20 0 20 40 60 80 100 120 Distance from source (nm) Figure 4.2: Longitudinal band diagram of a C N T F E T with benchmark materials and geometry parameters for applied gate biases of VQS — 0.0 V , 0.1 V , 0.3 V , and 0.5 V . of the shift in the local potential on the tube. Thus for larger gate biases, the local potential on the C N T no longer matches the gate potential and a transverse electric field exists between the C N T and the gate in the mid-channel region of the C N T . Tunneling Barrier Formation Reducing the potential energy of the conduction and valence bands in the mid-channel region of the C N T reduces the thickness of the Schottky barriers at the source and drain metallurgical junctions creating tunneling barriers. The thickness of these barriers ranges from Onm at the top of the barrier to the order of the diameter of the contacts at the bottom of the barrier (6 nm) and is small enough to allow quantum tunneling of carriers. The height of the barriers remains unchanged, as it is a function of the contact and C N T work functions and the bandgap of the C N T . The thickness of the tunneling barriers is a function of the displacement of the bands in the C N T channel which is dependent upon the applied gate potential. This relationship allows the gate potential to control the ability of carriers to move into and out of the channel by 51 Chapter 4. Results controlling the thickness of the tunneling barriers and the probability of quantum tunneling at a given energy. Equilibrium Carrier Movement Note that under equilibrium conditions there is thermal movement of electrons and holes, but no net carrier movement due to the lack of a potential difference between the source and drain. In the absence of an applied gate potential (VGS = 0.0 V ) , only those electrons or holes with sufficient energy to be thermionically emitted over the potential barriers at the metallurgical junctions are able to move into the channel from either the source or drain. This wil l be a very small proportion of the carriers since the height of the barrier is roughly 0.319 eV for the (16,0) C N T considered here. For positive gate potentials, the probability of tunneling through the potential barriers in the conduction band dramatically increases, as does the electron flux into and out of the channel. Conversely, hole movement into the channel is reduced for positive gate potentials due to an increase in the height of the potential barrier the valence band seen by holes in the source and drain. 4.1.3 Comparison With Analytic Equilibrium Solution A comparison of the equilibrium electrostatic solutions predicted by the numerical model to those of an analytic model provides an opportunity to verify the accuracy of the numerical solution to Poisson's equation. In order to directly compare solutions of the numerical and analytic models, the inability of existing analytic models to specify different permittivities for the C N T and surrounding gate dielectric must be taken into account. The analytic C N T F E T model of John et al. [26], and other analytic models [46], require a single dielectric permittivity to be used for the entire model space in order to avoid accounting for an abrupt change in permittivity at the CNT-dielectric interface. The relative permittivity of undoped semiconducting C N T s is 1 [45], while the relative permit-tivity of dielectric materials used in C N T F E T s is 3.9 to 19.5 [6, 2]. The numerical C N T F E T model is able to apply different permittivities to each region of the device allowing a direct comparison to analytic results and an opportunity to examine the impact of approximating the 52 Chapter 4. Results Distance from source contact (nm) Figure 4.3: Conduction band profiles near the source contact calculated using analytic and numerical models for VGS = 0.0 V to VGS = 0.5 V . The inset shows a close-up view of the bands for VGS = 0.3 V . x 103 VGS=0.5V 7/ VGS=0.3V 17 — - < \ \ \ O / G S ' ° 1 V V °0 5 10 15 20 Distance from source contact (nm) Figure 4.4: Absolute difference in energy be-tween conduction bands of the heterogeneous and homogeneous solutions of the numerical model for erfiX = 3.9 and VQS = 0.0 V to VGS — 0.5 V in increments of 0.1 V . permittivity of the C N T with that of the surrounding dielectric material. Figure 4.3 shows the conduction bands near the source for V G S = 0.0 V to VQS = 0.5 V in incre-ments of 0 . 1 V . Two results from the numeric model, one with heterogeneous and the other with homogeneous permittivities within the C N T F E T , are presented along with the the results from the analytic model (homogeneous permittivity). In each case a gate dielectric with a relative permittivity of 3.9 was used. The homogeneous numeric solution uses a relative permittivity of 3.9 within the C N T , while the heterogeneous solution uses a relative permittivity of 1. There is excellent agreement between the solutions of the analytic model and the numeric model with homogeneous permittivity. The difference between the heterogeneous and homogeneous solu-tions using the numeric model is small, relative to the magnitude of the band bending, and is limited to the regions where band bending is taking place. Figure 4.4 show the absolute difference in conduction band energy between the heterogeneous and homogeneous solutions using the numeric model. Similar to the previous figure, results for VQS — 0.0 V to VGS — 0.5 V in increments of 0.1 V are presented. The largest discrepancies between the bands correspond to the higher VGS values. 53 Chapter 4. Results 4.1.4 N o n - E q u i l i b r i u m Cond i t i ons Introduction of a source-drain potential difference creates an opportunity for net carrier move-ment to contribute to a drain current. Control of carrier movement is described in terms of the potential barriers in the conduction and valence bands introduced in the previous section. Effects of Increasing Source-Drain Potential Differences Figure 4.5 shows a series of six band diagrams for a C N T F E T with a fixed gate potential (VGS = 0.3 V ) and progressively increasing source-drain potential differences. These band diagrams illustrate the response of the device to non-equilibrium bias conditions and allow a qualitative description of carrier movement and the effect of changing carrier concentrations on the energy bands of the C N T . Figure 4.6 provides the corresponding quantitative representation of carrier movement in the C N T F E T in the form of an I- V relationship. This figure shows electron and hole components as of the drain current as well as the total drain current, respectively. Figure 4.5a) (Vps = 0.0 V ) shows the C N T F E T under equilibrium conditions. The applied gate bias (VGS — 0.3 V ) has reduced the mid-channel potential and the thickness of the Schottky barriers at the source and drain contacts leaving thin spikes in the conduction band at the contacts. The gate-source and gate-drain potential differences are equal and the source and drain barriers have identical height and thickness (assuming identical geometry and materials in both contacts.) In Figure 4.5b) (VDS = 0.1 V ) a potential difference is introduced between the source and drain contacts of the C N T F E T . This results in a reduction of the height of the conduction band spike at the drain, relative to the mid-channel potential. Due to its Schottky nature, the height of this barrier is fixed relative to the drain Fermi level, which is now 0.1 eV below E F S - The shape of the potential barrier at the source and the mid-channel potential energy in the conduction and valence bands remain relatively unchanged from the equilibrium case. The changes in the valence band are not significant since its profile is dominated by the mid-channel potential energy. 54 Chapter 4. Results -O.B • -1 • -20 0 20 40 60 80 100 120 Distance from source (nm) (a) VDS = 0.0 V -0.8 • -1 • -20 0 20 40 60 80 100 120 Distance from source (nm) (c) VDS = 0.3 V -0 .8 --1 -I i i i i ] i I -20 0 20 40 60 80 100 120 Distance from source (nm) (e) VDS = 0.6 V -0.8 --1 --20 0 20 40 60 80 100 120 Distance from source (nm) (b) VDS = 0.1 V -0.8 -1 -20 0 20 40 60 80 100 120 Distance from source (nm) (d) VDS = 0.5 V 0.2 0 V _ 5-0.2 0) Si-0.4 <D 5-0.6 -0.8 -1 ^ --20 0 20 40 60 80 100 120 Distance from source (nm) (f) VDS = 0.8 V Figure 4.5: Energy band diagrams showing the positions of the Q F L s , Epn (dashed line) and Epv (stippled line), for VGS = 0.3 V and increasing values of Vps-55 Chapter 4. Results vGS=o.o v - ' ' VGs=0.1_v' v v vGS=p.3v - j : vGS=oTo V vGS=o.i v , - - - '-_-_-}} VG"s=0.3V vGS=oVv }.4 0.5 0.6 0.7 0.8 S<V> Figure 4.6: Electron component, hole compo- Figure 4.7: Dependence of Q F L splits AEpn nent, and total drain current at VQS = 0.3 V . (dashed line) and AEpP (stippled line) on bias for VGS values of 0.0, 0.1, 0.3, and 0.5 V . The introduction of a potential difference between the source and drain creates a net flow of electrons toward the drain, which is moderated by the barriers in the conduction band. The location of Epn satisfies the flux balance relationship which ensures the net flows of electrons into the channel at the source and out of the channel at the drain are equal. Note that Epn splits from the Fermi levels at both the source and drain. The discontinuity is an indication of net movement of electrons in the direction of decreasing potential. Hole movement is limited by the mid-channel potential energy of the valence band. The location of Epp also satisfies the flux balance relationship and its split at both the source and drain show hole movement due to thermionic emission over the potential barrier. However, the height and, thickness of this barrier prevent significant quantities of net hole movement. In Figure 4.6 the electron component appears to be equal to the total drain current at VQS — 0.1 V , indicating that electrons are the only significant carrier. A t these bias conditions the C N T F E T is operating in the triode regime. Increasing drain potentials result in increased drain current for the same gate potential. In Figure 4.5c) (VDS = 0.3 V ) the drain-source potential difference is increased to same value as the gate-source potential difference resulting in the removal of the spike in the conduction band at the drain. Right moving electrons in the channel are now free to pass into the drain and the 56 Chapter 4. Results drain current increases correspondingly. In order to satisfy the flux-balance relationship, E F N is shifted downward to provide a larger split at the source and an increased net electron current through the potential barrier at the source. Lowering the potential energy of the drain decreases the height of the potential barrier to holes in the drain, but this barrier is sti l l equal to half the C N T bandgap and too large to allow significant numbers of holes to be thermionically emitted into the channel. The contribution of holes to the drain current remains insignificant, as shown in Figure 4.6. This bias condition marks the end of the triode regime and the beginning of saturation. Further lowering of the conduction band near the drain does not increase electron current, as it is now limited by the mid-channel potential of the C N T . In Figure 4.5d) (Vps = 0.5 V ) the source-drain potential is increased further so that the bands at the drain are bent down creating a potential spike in the valence band. The height of the barrier seen by holes in the drain is fixed (Schottky barrier), but the thickness of this barrier is reduced considerably allowing holes to tunnel into the channel. This is reflected in the increased split between E F P and the drain Fermi level. The hole component of the drain current can now be seen in Figure 4.6. Increasing the hole current coincides with increased hole concentration in the channel. In order to maintain the net charge in the mid-channel region of the C N T which satisfies Poisson's equation, there must be a corresponding increase in electron concentration. This is accomplished by shifting E F N toward E F S and the conduction band; a reversal of the trend of E F N movement away from E F S for increasing VDS-However, near the source contact E F P is closer to the valence band than in the mid-channel region due to localized band bending and there is a dramatic increase in hole concentration in this area. In order to satisfy Poisson's relationship, the local potential on the C N T is increased and the conduction and valence bands are shifted downward to increase the number of electrons and reduce the number of holes in this region. The bands are not shifted any further than in the mid-channel region, but are shifted to the mid-channel value over a shorter distance, effectively increasing the steepness of the band bending near the source and reducing the thickness of 57 Chapter 4. Results the potential spike in the conduction band at the source. There is a corresponding increase in electron current due to increased tunneling at the source, as seen in Figure 4.6. Simultaneously there is a similar situation occurring at the drain where Epn IS v e r v close to the conduction band causing an increase in electron concentration in this region and a corresponding thinning of the potential spike in the valence band. Increasing VDS causes increasing currents and carrier concentrations, which, in turn, further reduce barriers to carrier movement. This phenomenon has been labeled Drain-Induced-Barrier-Thinning (DIBT) and is responsible for the exponential increase in both electron and hole components of drain current for increasing VDS [41]. The onset of D I B T marks the end of the saturation regime and the beginning of the breakdown regime. In Figure 4.5e) (VDS = 0.6 V ) the drain-source potential difference is increased to two times the gate-source potential, causing the energy bands of the C N T to have identical barriers to electrons and holes. Under these bias conditions the hole current becomes equal to the electron current, as can be seen in Figure 4.6. As VDS continues to increase, the hole current exceeds the electron current, and the presence of holes in the channel begins to dominate the behavior of the C N T F E T . In Figure 4.5f) (VDS = 0.8V)the source-drain potential causes extensive D I B T in the barrier to electrons at the source and in the barrier to holes at the drain. Epn and EpP move past the conduction and valence bands, respectively, in the mid-channel region of the C N T indicating degenerate carrier concentrations which correspond to large electron and hole currents. The magnitude of these currents are a consequence of the very thin barriers at the source and drain ends of the channel. The C N T F E T is deep in the breakdown region of operation. QFL Behavior Under Increasing Source-Drain Potentials Figure 4.7 summarizes the behavior of the Q F L splits AEpn and AEFP under increasing source-drain bias. The Q F L splits are the location of Epn relative to EpSource a n d EFP relative to 58 Chapter 4. Results EFdraim as shown in Figure 3.5. Note that Z?p nand EFp are negative values. AEFn = E F n AEFp = EFp + qVDS Consider again the bias range VQS — 0.3 V and Vi )s=0 .0V to 0.8 V : For the triode regime (VDS=0.0 V to 0.3 V ) AEFp quickly saturates at « 0.02 e V while AEFn decreases linearly with increasing VDS- The increasing split of EFn from the source Fermi level shows electron current increasing with VDS hi this regime. The split of EFp is nearly constant since there is no increase in hole current with increasing VDS once the initial source-drain bias is introduced. This is due to the constant shape of the barriers to hole movement in the valence band in this bias range. For the saturation regime ( V D S = 0 . 3 V to 0.5 V ) AEFp increases linearly with increasing VDS while AEFn is relatively constant. Increasing AEFp shows that hole current is now increasing with VDS due to the diminishing barrier to holes at the drain. The slight reversal in the downward trend of AEFn is a response to the increasing concentration of holes in the channel and the result of maintaining the charge balance on the surface of the C N T . For the breakdown regime (VDS > 0.5 V ) there is a sharp rise in AEFn and a decrease in AEFp for increasing VDS- This is not an indication of decreasing current, but rather an indication of increasing electron and hole concentrations in the channel as the two Q F L s move closer to the conduction and valence bands, respectively. There is a dramatic increase in current due to the reduction of the source and drain Schottky barrier thicknesses. When VDS exceeds 0.5 V the trend of the Q F L s moving closer to the conduction and valence bands reverses and increasing VDS again leads to increasing Q F L splits. 4.1.5 General I- V Characteristics C N T F E T behavior under an increasing drain-source potential difference can be extended to a more general I- V characteristic by including the effects of the gate-source potential differences. Figures 4.8 and 4.10 show linear and semi-logarithmic plots of drain current as a function of 59 Chapter 4. Results source-drain bias (VDS) for VGS at 0.0, 0.1, 0.3, and 0.5 V. The I- V characteristics predicted by the CNTFET model are similar to those of bulk MOSFETs as they include triode and saturation modes of operation. However, the CNTFET includes an additional mode, breakdown, where current increases exponentially after VDS exceeds VGS-Restricting VDS to values less than VGS prevents breakdown behavior, giving the CNTFET an /- V characteristics equivalent to that of an n-type FET. Figure 4.9 shows an expanded view of the drain current relationship for VDS = 0.0 V to 0.2 V and VGs at 0.2, 0.3, 0.4, and 0.5 V. The semi-log plot of Figure 4.10 illustrates the relationship between ID and VGS in the CNT-FET. Considering only the non-zero gate biases, there is an exponential relationship between V G S and the drain current for both the triode and saturation operating regimes. This plot also shows that the length of the saturation region is a function of VGS- For larger gate biases the onset of breakdown occurs at larger values of V G S than for smaller gate biases. This can be explained by the fact that the breakdown phenomenon occurs as a result of an increasing hole current in the CNTFET. For larger values of VQS there are larger potential energy barriers in the valence band and the hole current is unable to become significant until VDS exceeds VGS-In the breakdown regime, the ability of the gate bias to control the current in the CNTFET is diminished with increasing VDS and the current becomes a function of only the drain-source potential. Referring again to Figure 4.10, the I- V curves for each of the four values of V G S approach a common asymptote as VDS approaches 0.8 V. Under these conditions the CNTFET ceases to function as a transistor and shows two-terminal behavior. 4.1.6 Comparison With Experimental I- V Characteristics The model I- V characteristics can be compared directly with those of experimental devices. Figure 4.11 shows a linear plot of drain current vs. VDS with an inset sub-figure showing a semi-log plot of drain current vs. VDS f° r an experimental SB-CNTFET device developed by Misewich et al. [33]. This device has the same basic architectural features as the CNTFETs considered by the numerical CNTFET model and serves as a good reference, both qualitatively 60 Chapter 4. Results Figure 4.10: Log of total drain current for VGS Figure 4.11: Misewich CNTFET total drain at 0.0, 0.1, 0.3, and 0.5 V. current and log of total drain current (in-set). Source: Misewich et al., Electrically induced optical emission from a carbon nan-otube FET. Science, 300:783-786, 2003. Gl Chapter 4. Results and quantitatively, for the performance predictions of the model. The Misewich device consists of an intrinsic, semiconducting S W C N T with metal contacts and a planar, degenerately doped, silicon back-gate. The C N T has a measured diameter of 1.4 nm and estimated bandgap of 0.6 eV that corresponds well to the 1.25 nm diameter and 0.639 eV bandgap of the (16,0) C N T used in the numerical model. The fundamental difference between the Misewich C N T F E T and the model C N T F E T is the shape and location of the gate electrode. Misewich et al. use a planar gate, separated from the C N T by 150 nm of SiC>2, where the analytic geometry used in the model consists of a cylindrical gate with a 6.26 nm of S i 0 2 separating the gate from the C N T . The numerical model shows very similar I- V characteristics to those of the Misewich device. The triode, saturation, and breakdown regimes of operation are visible for both devices, es-pecially in the semi-log plots. Both devices show a similar pattern of exponential increases in saturation current and later onset of breakdown for increasing VGS- Note also the similarity in the magnitudes of the currents: both devices show saturation currents in the range of 1 0 _ 1 to 10~ 5 ^ A . The Misewich device requires VGS values of 4 V to 8 V to achieve the same drain currents as those predicted by the model for VQS values of 0.1 V to 0.5 V . This difference in sensitivity to the gate-source bias is a result of reduced capacitive coupling between the gate and the C N T resulting from the planar gate structure and the thick gate dielectric. There is also an order of magnitude difference between the drain-source biases used in the Misewich device and those used in the model C N T F E T . Increased values of VDS in the experi-mental device could be necessary to overcome high contact-CNT junction resistances which are not included in the model. This explanation is speculative, as Misewich et al. do not comment on the resistances at the contacts. 62 Chapter 4. Results 4.1.7 Contribution of Electrons and Holes to Breakdown Current The breakdown operating regime is characterized by an exponentially increasing hole current resulting from diminishing potential energy barriers in the valence band. A novel feature of the numerical C N T F E T model is that it predicts the contribution of electrons as well as holes to the total breakdown current. The ratio of these two component currents remains a function of VGS and VDS-Figure 4.6 shows the exponentially increasing electron and hole currents in a C N T F E T for VGS = 0.3 V . As noted when discussing the response of the C N T F E T to increasing VDS, the component currents in the device are equal when VDS = 0-6 V . Equal electron and hole currents for VDS — 2VG5 is a common feature of S B - C N T F E T s with contact work functions equal to the C N T work function, and is observed generally in the numerical C N T F E T model. Large electron and hole currents under breakdown conditions are accompanied by high carrier concentrations in the C N T F E T channel. The simultaneous presence of electrons and holes in large quantities introduces the opportunity for recombination. Although not included in the numerical C N T F E T model, maximum recombination is likely to take place when the product of the electron and hole currents is a maximum. This For a fixed value of VDS this wi l l occur when VQS = VGS/2, as shown in Figure 4.12. This is the exact bias condition at which maximum radiative recombination is observed from a S B - C N T F E T [33]. The Misewich C N T F E T device introduced in the previous section exhibits optical emission at infra-red wavelengths under appropriate bias conditions. Figure 4.13 shows measured emission intensity function of gate voltage for VDS — 8 V and VDS = 4 V . The curves on this figure give the corresponding computed emission assuming that recombination is limited by the lesser of the hole or electron currents. Figure 4.12 shows electron and hole currents and their product as a function of VQS f ° r VDS = 0.8 V in the numerical C N T F E T model. In both figures VQS sweeps from 0 V to VDS and the values of VDS place both devices deep in the breakdown region. For VDS = 0.8 V the total drain current in the C N T F E T model is no longer controlled by VQS, 63 Chapter 4. Results 10, Figure 4.12: Electron component (dashed line), hole component (stippled line), total drain current, and the product of the com-ponent currents as a function of VGS f° r VD5=0 .8 V . Note that the product of the two components is not to scale. c 1 4 « 8 10 "~ Gale Voltogo (V) Figure 4.13: Infra-red emission intensity in the Misewich CNTFET as a function of VGs-Measured values shown as dots and calculated values shown as solid lines for V D S = 8 V and V D S = 4 V . Source: Misewich et al , Electri-cally induced optical emission from a carbon nanotube FET. Science, 300:783-786, 2003. as seen in the convergence of the I- V curves in figure 4.10 and by the essentially flat total current curve in Figure 4.12. However, Figure 4.12 also shows a linear relationship between the magnitude of the electron and hole components of the total drain current and VGS in the breakdown regime. 4.2 Dependence on Materials Parameters The electrical and electronic properties of the materials used to fabricate the CNTFET govern its behavior and performance. While the electronic properties of the CNT are determined by its geometry, the performance of the CNTFET also depends upon the materials used to form the contacts and gate dielectric. The parameters of these materials affect the energy bands of the CNT through interactions at metallurgical junctions and electrostatic coupling between proximal structures. The sensitivity of the model with respect to contact work functions, permittivity of the gate dielectric, and trapped charge in the gate dielectric are examined using energy band diagrams and the I- V characteristics of the device. 64 Chapter 4. Results 4.2.1 Contact Work Functions As discussed in Section 3.1.3, the height of the Schottky barriers at the source and drain ends of the C N T are directly related to the difference in the work functions of the source and drain contacts and the C N T . This unique situation allows tremendous opportunities for engineering the energy bands of C N T F E T s [2, 8, 31, 47, 48, 49] through manipulation of the contact work function. Materials used for C N T F E T source and drain contacts (and their native work functions) include gold ( g0 m = 4 . 3 e V ) [6], and platinum (g</>m=5.7eV) [25]. These contacts are typically exposed to ambient oxygen during device,fabrication, resulting in a reduction of the contact work function [47, 8, 49]. The effects of oxygenation are reversible by annealing the C N T F E T in an oxygen-free environment at moderately high temperatures [47, 8]. The work functions of intrinsic, small-diameter S W C N T s are about 4.5 eV to 4.8 eV [5, 50]. Since the exact work functions of the contacts and C N T s are difficult to ascertain with precision, it is more appropriate to discuss work functions in terms of relative differences between that of the C N T and a particular contact. Figure 4.14 shows the effects of contact work function differences of ±0 .2 eV on the energy bands of the C N T near the source contact. There are no applied potentials at any of the source, drain, or gate contacts ( V G S = Vbs — 0). For the case of no work function differences, q<ps = q<Pcnt, there is no band bending and both the conduction and valence bands are equidistant from the Fermi level along the length of the C N T . When qcf>s < q4>cnt the bands near the contact are bent down in order to maintain continuity of the vacuum potential when the contact and C N T Fermi levels are aligned. This results in a reduction of the conduction band potential energy and the creation of a potential barrier in the valence band which favours electron movement and presents an obstacle to hole movement. The complementary effect is seen when q<j>s > q4>cnt, resulting in the creation of a potential barrier in the conduction band and an increase in the potential energy of the valence band near the metallurgical junction. Identical effects are observed in the energy bands near the drain when work function differences are present. The effects of changes in the energy band profile at the source and drain contacts on the I-65 Chapter 4. Results -10 0 10 20 -10 0 10 20 -10 0 10 20 Distance from source (nm) Figure 4.14: Energy band diagrams showing the conduction and valence bands near the source contact for: a) q<ps < qficnt, b) qcps = # c n t , c) q<f>s > q4>cnt-V characteristics of the C N T F E T are shown in figure 4.15. Reduced work functions at the source and drain are considered independently, showing how work function engineering is used to modify the D C characteristics of the device. For q(j>s < q<t>cnt a n increase in current is observed for all VDS as a result of the decreased height of the potential barrier to electrons in the source and an increased flow of majority carriers. For qcpr) < q(j>cnu saturation is encountered at a lower values of VDS, resulting in an extended saturation region. Recall that saturation occurs when VDS bends the bands at the drain far enough to eliminate the spike in the conduction band. B y reducing the height of the potential barrier to electrons at the drain, smaller values of VDS are able to remove spike in the conduction band. In general, it is desirable to use both source and drain contacts with reduced work functions in order to maximize the performance of an n-type C N T F E T (device with predominant elec-tron conduction). Reducing the native potential barriers in the conduction band allows greater numbers of electrons to travel through the channel and improves the bias range over which 66 Chapter 4. Results - - - (1) * s = 4 - 5 e V * D = 4 - 5 e V ' /' / . (2) O s=4.5 eV * D =4.3 eV i l l III - - - (3) <£>s=4.3 eV *D=4^5 eV : • / . (4) O s=4.3 eV * D =4.3 eV 77 " 4 i I 1 / / / / / 11 i / / / - / - / 2 _ _ — I \ I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.15: Total drain current current for various contact-CNT work function differences at VQS = 0.5 V. Source: Clifford et al., Bipolar conduction and drain-induced barrier thinning in carbon nanotube FETs. IEEE Trans. Nanotechnol, 2(3):181-185, 2003. controllable conduction occurs. For a p-type CNTFET (device with predominant hole conduc-tion) reduction of the potential barriers in the valence band is necessary in order to improve the conduction of holes. This is accomplished by increasing the work functions of the source and drain contacts. 4.2.2 Dielectric Permittivity Capacitive coupling between the gate and the CNT is a function of both the gate geometry and the relative permittivity, e r>ox, of the dielectric separating the two structures. Silicon dioxide (SiC>2) is a very common dielectric material and is used extensively in back-gate CNTFET devices [5, 6, 29, 8, 49] as it is easily grown on the Si substrates that form the gate and support the device structures. Si02 can also be deposited over CNT devices using CVD to form top-gate structures [49]. Si02 has a relative permittivity of 3.9. High-K dielectrics have been developed for use in conventional bulk FET devices to improve capacitive coupling 67 Chapter 4. Results Distance from source (nm) Figure 4.16: Band diagram near the source for VGS ranging from 0.1 V to 0.7 V in 0.2 V incre-ments for: a) er<ox = 3.9, b) eTi0x = 19.5, c) eTi0X = 25. The uppermost conduction and valence bands in each figure correspond to VGS = 0.1 V with each lower band representing an increase in VGs by 0.2 V . between the gate and channel without reducing the dielectric film thickness. Ultra-thin films are unattractive due to their susceptibility to defects and direct (gate-channel) tunneling currents. Two promising high-*; materials are hafnium oxide (HfO^) and zirconium oxide (ZrO^) which have permittivities of 19.5 and 25, respectively. These permittivities are roughly 5 and 6.5 times that of Si02, respectively, which allow significant improvements in effective oxide thickness. Effective oxide thickness is the thickness of SiC>2 that has the same capacitive coupling as non-Si02 dielectric material of a given thickness. Experimental C N T F E T devices using an 8 n m Zr02 dielectric have been fabricated [2], thus demonstrating the compatibility of these materials with C N T s . Figure 4.16 a) shows the energy bands, near the source, of a C N T F E T (closed cylindrical geometry) with eTfiX = 3.9 under equilibrium conditions for VGS ranging from 0.1 V to 0.7 V in 0.2 V increments. Figures 4.16 b) and 4.16 c) show a similar devices with erfix = 19.5 and £r,ox — 25 under the same bias conditions. 68 Chapter 4. Results 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.17: Drain current at V G S = 0.0 V for £r,ox = 3.9, £ r , o x = 19.5, and (-T,ox = 25. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.19: Drain current at V G S = 0.1 V for £ r , o x = 3.9, £ r , o x = 19.5, and tr,ox = 25. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.18: Drain current at V G S = 0.3 V for ^ r , o x = 3.9, 6 r , o x ~ 19.5, and C r , o x ~ 25. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.20: Drain current at V G S = 0.5 V for £r,ox = 3.9, €r,ox = 19.5, and €r,ox = 25. While there is little difference in the mid-channel potentials of the C N T s for devices with different dielectric permittivities when VQS is small, the effects of improved capacitive coupling between the gate and the C N T become more apparent for higher gate biases. For VQS > 0.1 V the potential on the C N T is closer to the gate potential in the high-/c devices, resulting in a greater band bending and with increasing VGS- The I-V characteristics for these devices are shown in Figures 4.17 to 4.20. For V G S = 0.0 V and 0.1 V , drain currents for the three devices are essentially the same until breakdown occurs. In the breakdown regime the devices with high-K gate dielectrics show the 69 Chapter 4. Results same exponential increase in current for increasing VDS, but the currents are lower than those in the device with e r > o x = 3.9. For V G S = 0.3 V and 0.5 V , the devices with high-K dielectrics experience breakdown for higher values of VDS- For VQS = 0.5 V the drain current in the high-K devices is higher for low values of VDS and shows a more pronounced saturation region than the device with er,ox — 3.9. Both of these features improve device behavior by increasing the duration of the saturation regime. The improvements in the I- V characteristics of the devices with high-K dielectrics are due to improved gate-CNT coupling which shifts the conduction and valence bands to lower potentials for a given VGS- This increases the height of the potential barrier seen by holes attempting to enter the channel from the drain, limits hole current, and the effects of hole current on the energy bands, namely D I B T . Figures 4.21 and 4.23 show the energy bands at VGS = 0.5 V and VDS = 0.6 V for eTiOX = 3.9 and er>ox = 25, respectively. A t these bias conditions breakdown is starting to appear in the high-K device and is fully underway in the 1OW-K device, where significant barrier thinning is apparent in both the conduction and valence bands. Figure 4.22 shows the electron and hole drain currents at V G S = 0.5 V for each of the permittivity values considered. Note that holes do not contribute to the drain current in the high-K devices until VDS > 0.7 V and electron currents are much larger than hole currents at all biases considered. Figure 4.24 shows the quasi-Fermi level splits for electrons and holes at VQS = 0.5 V for e r ] 0 X = 3.9, e r ) 0 X = 19.5, and eTfiX = 25. For the high-K devices, the electron Q F L does not show the tendency to move toward the source Fermi levels for VDS > 0.5 V to the same extent as is observed in the 1OW-K device. This is a result of improved control of hole currents for increasing VDS- Reduced injection of holes in the high-K device reduces the need to increase electron concentrations in the mid-channel region of the tube in order to satisfy Poisson's equation. 4.2.3 Dielectric Trapped Charge The SiG-2 gate dielectrics used in C N T F E T devices are assumed to be charge free. Other materials may contain trapped ion impurities that result in a net charge within the dielectric. 70 Chapter 4. Results 20 40 60 80 Distance from source (nm) Figure 4.21: Band diagram for e r o x VQS = 0.5 V and V D S = 0.6 V . = 3.9 at Figure 4.22: Electron component (dashed lines) and hole component (stippled lines) of drain current at VGS = 0.5 V for er>ox — 3.9, 20 40 60 80 Distance from source (nm) 120 ^r,ox — 19.5, and 6r,ox 0.25 = 25. Figure 4.23: Band diagram at for er>l V G S = 0.5 V and V D S = 0.6 V . 25 at Figure 4.24: Quasi-Fermi level splits 6EFTL (dashed lines) and 6EFP (stippled lines) at VGS = 0.5 V for erfix = 3.9, e r , o x = 19.5, and ^•r.ox — 25-71 Chapter 4. Results 0.8 10 0 10 20 Distance from source (nm) 0 10 20 Distance from source (nm) Figure 4.25: Band diagram near the source for: a) Qox -0.005 to -0.020 nm 3 in -0.005 nm 3 increments, b) Qox 0.005 to 0.020 n m - 3 in 0.005 n m - 3 increments. The atomic layer deposition process used to grow high-K dielectric materials like ZrC>2 involves the use of precursor materials which are left behind in small quantities as impurities in the Zr02 film [2]. For ZrC>2 deposition, ZrCLj is used, leaving behind chlorine ions (negative charge). These impurity concentrations can be as high as 1 atom0/, [2] or 8.55 x 1 0 2 2 c m - 3 , equivalent to degenerate doping in a bulk semiconductor material. Trapped negative charges in the dielectric adjacent to a CNT induce complementary positive charges on the surface of the tube. This situation is equivalent to doping the tube with positive charges (hole doping). Figure 4.25 a) shows band diagrams for a closed cylindrical geometry CNTFET with trapped negative charge in the gate dielectric. The positive charge induced on the CNT requires a reduction in the local potential along the CNT surface and results in a shift of the conduction and valence bands toward higher energies in the central region of the CNT channel. Note that significant band bending occurs at the tube-contact junctions where the potential of the bands is fixed. Figure 4.25 b) shows the equivalent situation for trapped positive charge in the gate dielectric ( V G S — 0 V in all cases). 72 Chapter 4. Results Band bending due to dielectric charge results in two very significant features: formation of Schottky barriers at the source-drain metallurgical junctions and a change in the potential of the conduction and valence bands in the central region of the channel. For negative trapped charge in the dielectric, this allow holes to tunnel through the valence band barriers, effectively creating a p-type depletion device in which current flows through the channel unless blocked by positive VGS bias values'. 4.3 Dependence on Device Geometry The geometry of the components in a C N T F E T are as important as the properties of the materials in determining the behavior of the device. The radius of the source and drain contacts, the thickness of the gate dielectric, the gate length, and any gate-contact spaces all have a direct influence on the energy band profile of the C N T F E T and its performance. 4.3.1 Contact Radius The thickness of the tunneling barriers at the source-CNT and d ra in -CNT metallurgical junc-tions is a function of the gradient of the local potential near the junctions. Concentrating the electrostatic field lines in these areas by reducing the surface area of the contacts results in more rapid changes in-local potential and thinner potential energy barriers in the C N T [32]. Figure 4.26 shows the equipotential contours near the source of a C N T F E T with a contact radius of 10 x Rcnt, where R c „ t is the radius of the C N T . This is the closed cylindrical geometry of the benchmark C N T F E T . Figure 4.27 shows the equipotential contours near the source of a C N T F E T with a contact radius of Rcnt- Note the difference in the spacing of the equipotential contours in the C N T near the source-CNT metallurgical junction in these figures. Figure 4.28 shows the energy bands of the C N T F E T near the source for contact radii of: a) 10 x R e t , b) 5 x RCNT, and c) Rcnt- Small changes in the thickness of the barrier can be observed between a) and b) for all values of VQS, but the band profiles remain very similar. In Figure 4.28 c) the radius of the contact is reduced to Rcnt, forming so-called '"needle-contacts'" [32]. Changes in the energy bands are more pronounced and the barriers are noticeably thinner for 73 Chapter 4. Results Gate Source l i ' I , 1 : CNT i i i 1 1 1 52 -50 -48 -46 -44 -42 -40 Distance from center of gate (nm) £ 6 O b A W -32 -50 -48 -46 -44 -42 -40 Distance from center of gate (nm) Figure 4.26: Equipotential contours near the Figure 4.27: Equipotential contours near the source of a C N T F E T with a contact radius of source of a C N T F E T with a contact radius of 10 x r\cnt- Rent-all values of VGS-Figures 4.29 and 4.30 show the I- Vcharacteristics for C N T F E T s with contact radii of 10 x R c n t and Rent- There is roughly a 4x increase in current in the needle contact geometry vs. the planar contact geometry in the benchmark C N T F E T . This can be attributed to an increase in tunneling current though the potential barriers in the conduction band. 4.3.2 Gate Dielectric Thickness Capacitive coupling between the gate and the C N T is dependent upon the permittivity and thickness of the gate dielectric. Early C N T F E T devices with back-gates used dielectric thick-nesses of 100 nm to 300 nm [6, 5], while more recent top-gated devices have been developed with dielectric thicknesses of 5 to 15 nm [51, 49]. Decreasing the thickness of the dielectric improves capacitive coupling between the gate and the C N T and increases the ability of the gate to control the local potential on the surface of the tube. Figure 4.31 illustrates the effect of decreasing gate dielectric thickness on the energy bands of C N T F E T s near the source. In the nomenclature of the cylindrical C N T F E T geometry, the dielectric thickness is specified indirectly by the radius of the gate. The radius of the gate minus the radius of the C N T yields the thickness of the dielectric, thus the benchmark geometry with 74 Chapter 4. Results j gi , , 1 -o.gi • 1 1 -0 .9 '— : — 1 1 1 -10 0 10 20 -10 0 10 20 -10 0 10 20 Distance from source (nm) Figure 4.28: Band diagram near the source for contact radii of: a) 10 x R C nt, b) 5 x Rcn<, and c) Rcnt with VGS — 0.1 V , 0.3 V , 0.5 V , and 0.7 V . The uppermost conduction and valence bands in each figure correspond to V G S = 0 . 1 V with each lower band representing an increase in VGS by 0.2 V . a gate radius of 10 x R c n t has a dielectric thickness of 9 x R c n t or roughly 5.6 nm. Reducing the gate dielectric thickness in the closed cylindrical geometry of the benchmark CNTFET also reduces the radius of the source and drain contacts. This change in the contact geometry is responsible for the thinning of the Schottky barriers in the conduction band in Figure 4.31. 4.3.3 Gate Length and Gate-Contact Spacing The length of the gate is independent of the length of the CNT channel in most experimental CNTFET geometries. In a back gated device the gate extends far beyond the source and drain ends of the tube and spans the entire area of the wafer supporting the CNTFET [6]. Top gated CNTFET devices have been fabricated with gates covering the entire length of the channel [49], or only a portion of the length of the channel [2]. Partially gated CNTFET devices allow for spacing between the gate and contacts, reducing capacitive coupling and the possibility of 75 Chapter 4. Results Figure 4.29: Electron component (dashed Figure 4.30: Total drain current at VGS — lines) and hole component (stippled lines) of 0.5 V for C N T F E T s with contact radii of 10 x drain current at VGs = 0.5 V for C N T F E T s RCNT and R c n t . with contact radii of 10 x R c n t and RCnt-leakage currents between these structures [4]. However, this spacing reduces the effectiveness of the gate to control the thickness of the potential barriers near the source and drain. Figure 4.32 shows the energy bands of C N T F E T devices with gate-source and gate-drain spaces ranging from 0.0 nm to 20.0 nm. There is a linear increase in the thickness of the Schottky bar-riers with increasing separation between the gate and the source and grain contacts. Increasing barrier thickness results from reduced electrostatic coupling between the gate and the C N T in the gate-contact gap regions. This configuration allows the potential on the surface of the tube to transition from the value in the gated region to the value at the contact over a longer distance. Introduction of a gate-contact space opens the model space surrounding the C N T channel, and requires contact and gate dimensions to be completely defined. The radius of the source and drain contacts used in these simulations is 50 nm, and the thickness of the gate contact is « 4 4 nm. The thickness of the gate dielectric is unchanged from the benchmark device geome-try. These dimensions were selected to be comparable to those used in a recent experimental C N T F E T with gate-contact spacing [2]. 76 Chapter 4. Results > CD CU c LU 0.9 20 -10 c) 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 H -0.8 -0.9 20 -1 • I I • • 0 10 Distance from source (nm 0 10 20 Figure 4 .31: Band diagram near the source for gate radii of: a) 10 x R c n j , b) 5 x R c „ t , and c) 2 x R c n 4 with VGS — 0 . 1 V , 0.3 V , 0.5 V , and 0 .7 V . The uppermost conduction and valence bands in each figure correspond to V G S = 0 . 1 V with each lower band representing an increase in V G S by 0.2 V . When the gate-source and gate-drain gaps increase from the order of tens of nanometers to hundreds of nanometers, the potential in these regions becomes weakly coupled to fringing fields between the gate and contacts, forming a side-gated F E T structure [2]. Figures 4 .33 and 4 .33 show the Equipotential contours near the source of C N T F E T s with gate to source gaps of 20 nm and 250 nm, respectively. Note that the device with the long gap exposes the channel to the fringing fields between the upper surfaces of the gate and source contacts. These fringing fields are a function of the surface area of the gate, source, and drain contacts. The proportional size of the gate vs. that of the contacts determines the concentration of field lines incident on the contact, and in turn the rate of change of electrostatic potential in the vicinity of the contact and the thickness of the potential barrier. Figure 4 .35 shows the energy bands near the source for C N T F E T s with a gate-source spacing of 250 nm and various gate and contact lengths. For smaller contact lengths, and for larger gate lengths, the potential barrier at the contact becomes thinner due to an increased rate of 77 Chapter 4. Results Figure 4.32: Band diagram near the source for gate-source spacing of: a) 0.0 nm, b) 10.0 nm, and c) 20 nm with VQS = 0.1 V , 0.3 V , 0.5 V , and 0.7 V . The uppermost conduction and valence bands in each figure correspond to V G S = 0 . 1 V with each lower band representing an increase in VGS by 0.2 V . change of electrostatic potential near the contact. The opposite effect is seen for larger contact lengths and smaller gate lengths: the potential barrier at the contact becomes thicker due to a decreased rate of change of electrostatic potential near the contact. Moving the gate away from the source and drain contacts reduces the ability of the gate to modulate the thickness of the potential barriers at the contacts. If the gate-contact spacing is increased beyond the order of a few tens of nanometers, the thickness of the potential barriers increases to the point that there is no tunneling current except at the very top of the barriers. Thermionic emission of carriers becomes the dominant means for carrier movement into and out of the channel, and drain current will be reduced dramatically. 4.4 Channel Modulation C N T F E T Devices All of the numerical CNTFET model results discussed thus far have been for Schottky barrier devices where the drain current is regulated by the thickness of potential barriers at the CNT-78 Chapter 4. Results 300 •&250 H O 200 •5 to 150 c cu § 1 0 0 •*= 50 <u u S 0 \ Source Gate 500 -500 -400 -300 -200 Distance from center of aate (nm) 300 "E •E-250 h-O 200 •s to 150 c CO 100 *= 50 <u u £ 0 Source \ \ \ \ \\\\\\ 4£ — Gate 500 -500 -400 -300 -200 Distance from center of aate (nm) Figure 4.33: Equipotential contours near the Figure 4.34: Equipotential contours near the source of for a gate-source gap of 20 nm. source of a C N T F E T with a gate-source gap of 250 nm. contact interfaces. This final section focuses on channel modulated C N T F E T devices that control drain current by modulating the potential energy of the conduction and valence bands in the mid-channel region of the C N T . A n examination of the energy bands of C M - C N T F E T s shows how the features of this device evolve from those of the Schottky barrier devices considered previously and compares the predictions of the model to the results of an experimental model. 4.4.1 Architecture The architecture of the C M - C N T F E T considered here is based on the experimental device ar-chitecture of Javey et al. [2]. The Javey device consists of a planar, top-gated C N T with a high-K ZrC"2 gate dielectric that contains up to 1 atom of negative trapped charge. Trapped negative ions are a by-product of the atomic layer deposition process used to fabricate the dielectric. The layout of the device is not compact and admittedly not optimized. A 2000 nm gate length, 500 nm gate-source and gate-drain spacing create an unnecessarily long C N T chan-nel (3000nm). However, assuming ballistic transport in the C N T , this channel length should not directly degrade device performance. The features that make this a C M - C N T F E T are the large (500 nm) gate-contact spacing and a very large concentration of trapped charge in the gate dielectric. 79 Chapter 4. Results L . =500 nm L , =250 nm gate contact L . =500 nm L . =500 nm gate contact L . =1000 nm L . =250 nm gate contact 50 100 150 200 Distance from source (nm) 300 Figure 4.35: Band diagram near the source for C N T F E T s with gate-source spaces of 250 nm and various gate and contact lengths. VGS = 0.5 V and VDS = 0.0 V . The cylindrical representation of the Javey architecture translates transverse dimensions into radial equivalents and compresses the length of the device in order to limit the model space required. Planar component thicknesses are directly converted into radial dimensions. For example, the 50 nm thick planar contacts become 50 nm radius cylindrical contacts and the 8 nm thick planar gate dielectric becomes a cylindrical gate dielectric with a radial thickness of 8 nm. The length of the C N T , gate, contacts, and gate-contact gaps are reduced by a factor of two. These changes have no significant impact on device behavior, as the electric fields inside these regions are nearly constant. The critical regions of the device are the material interfaces and the edges of the contacts and gate, as this is where fields are changing and band bending occurs. As long as the lengths of the device components are kept long enough to prevent the effects of fringing fields from becoming significant, length reduction does not affect the device electrostatics or the location of the energy bands. Reducing the length of the C N T also helps to eliminate some of the problems associated with representing very-high aspect ratio structures in F E M simulations. 80 Chapter 4. Results A significant difference between the planar experimental device and the model is that the gate and gate dielectric surround the C N T in the cylindrical configuration and the SiC"2 surface of the substrate is not present. In the cylindrical device there is increased electrostatic coupling between the C N T and the trapped charge in the dielectric since the charge in the dielectric wi l l induce complementary charges in the C N T rather than inducing charges in both the C N T and the S i 0 2 surface. Additionally, in order to avoid inaccuracies in the local potential on the surface of the C N T , there is a constraint on the maximum charge concentration in the dielectric imposed by the C N T surface charge calculation. The density of states used for the C N T only considers allowed ground states rather than all allowed energy states in the tube. This approximation does not affect the low energy calculations encountered under typical C N T F E T bias, but the accuracy of the charge-potential relationship breaks down when the potential energy on the surface of the tube exceeds more than a few eV. A trapped charge concentration of 0.1 ions /nm 3 in the dielectric of the model satisfies both of these requirements. Despite the limit on the trapped charge in the dielectric, modeling the C M - C N T F E T archi-tecture sti l l pushes the limits of the numerical C N T F E T model. Very steep band bending at the edges of the ungated regions likely exceeds the capabilities of the shifted D O S approxi-mation which is used to calculate the carrier concentration-potential relationship. This limits the accuracy of a quantitative analysis of carrier tunneling through these barriers, however the equilibrium results are still useful for predicting the qualitative behavior of the device. 4.4.2 Equilibrium Conditions Figure 4.36 shows the band diagram of the C M - C N T F E T device with negative dielectric trapped charge and no applied potential differences between the contacts. Induced positive charge in the C N T requires an increase in the potential on the surface of the tube and a corresponding shift in the conduction and valence bands. The shift in the bands in the region of the C N T adjacent to the gate is moderated by capacitive coupling to the gate and the potential on the C N T in this region is similar to that calculated in Section 4.2.3. However, band shifting in the gate-source and gate-drain gap regions is not significantly affected by the gate and contact 81 Chapter 4. Results 3 2.5 2 1.5| 1 0.5 0 200 400 600 800 Distance from source (nm) 200 200 400 600 800 1000 1200 Distance from source (nm) Figure 4.36: Band structure for the C M - Figure 4.37: Band structure for the C M -C N T F E T with 0.1 n m - 3 negative trapped C N T F E T with 0.1 n m - 3 negative trapped charge in the dielectric at VGS = 0.0 V VDS = charge in the dielectric at V G S = 1.3 V VDS = 0.0 V . 0.0 V . potentials, and large changes in the potential energy of the bands result. This creates large potential barriers in the conduction band in the gap regions and very thin tunneling barriers in the valence band at the source-CNT and dra in-CNT metallurgical junctions. The widths of the potential barriers in the valence band at the contacts are not determined by the gate potential, but rather by the concentration of charge induced on the C N T by the trapped dielectric charge. These barriers are not able to control the tunneling of carriers, rather they present a small and constant restriction to their movement. The source-CNT and dra in -CNT metallurgical junctions become essentially ohmic contacts. This feature allows significant quantities of carriers to reach the C N T channel, where as in a similar architecture without trapped dielectric charge, the thickness of the barriers at the contacts precludes most carrier movement. Note that thermionic emission of electrons over the potential barrier in the conduction band is insignificant due to the height of the barrier. 4.4.3 P-type Operation For a negative source-drain potential difference and V G S = 0.0 V the potential energy of the valence band in the gated (mid-channel) region of the C M - C N T F E T is high enough to present no significant barrier to hole movement in the channel. Holes readily tunnel through the Schottky 82 Chapter 4. Results barriers at the source and drain and are free to cross both the gap and gated regions of the C N T channel. A positive potential of 1.3 V , applied to the gate, lowers the potential energy of the valence band in the gated region of the channel and create a barrier to hole movement, as shown in Figure 4.37. This is p-type depletion behavior. The thickness of the potential barriers at the contact-CNT interfaces of the C M - C N T F E T device is much smaller than the barriers seen in many S B - C N T F E T devices. This feature could allow greater carrier movement and higher drain currents in C M - C N T F E T s for similar bias conditions. This introduces the prospect that the ultimate performance of C N T F E T devices could be found in C M - C N T F E T architectures. 4.4.4 N-type Operation The C M - C N T F E T with trapped negative dielectric charge is optimized for hole conduction and p-type behavior, however, n-type behavior has also been observed in this C N T F E T architecture [2]. Ambipolar operation is a feature previously associated only with S B - C N T F E T devices [48]. A novel explanation for ambipolar conduction in C M - C N T F E T s is provided here, based on theoretical evidence for inter-band tunneling in C N T junctions [46]. Figure 4.39 shows a band diagram of the C M - C N T F E T device with for an applied gate-source potential difference of 1.6 V . Profound band shifting in the gate-source and gate-drain gap regions of the C N T , which opens the channel to hole movement in the valence band, also creates an opportunity for electron movement via inter-band tunneling. Very sharp band bending at the edges of the gap regions by energies larger than the bandgap of the tube create narrow potential barriers between the valence and conduction bands of the C N T . This introduces the possibility for electron movement from occupied states in the in the valence band of the gap regions to unoccupied states in the gated region of the C N T . The direct bandgap of C N T s and the symmetry of the conduction and valence bands aids in the efficiency of this tunneling [46]. As the gate potential increases, so does the window of overlap between the valence and conduction bands at the edge of the gated region. Increasing VGS should result in increasing electron conduction and drain currents in the presence of a positive source-drain potential 83 Chapter 4. Results 3 2.5 2 _ 1.5 > 1 0.5 0 -0.5 n 3 2.5 2 > 1 5 1 0.5 0 -0.5 1 \ [ \ f \ f \ 200 400 600 800 1000 1200 Distance from source (nm) 200 400 600 800 1000 1200 Distance from source (nm) Figure 4.38: Band structure for the C M - Figure 4.39: Band diagram for the C M -C N T F E T with 0.1 n m - 3 negative trapped C N T F E T with 0.1 n m - 3 negative trapped charge in the dielectric at VGS — 1-3 V VDS = charge in the dielectric at VGS = 1-6 V VDS — 0.0V. 0.0 V . difference, as seen in the Javey I- V characteristics. Javey suggests that the change from p-type to n-type behavior could be the result of an anneal-ing process applied to the C M - C N T F E T s before measuring their n-type I- V characteristics [2]. Annealing has become a standard practice for improving electron conduction in S B - C N T F E T s , as it reduces the work functions of the metallic source and drain contacts [47]. However, reduc-tion of the work functions of the contacts in a C M - C N T F E T device architecture is not likely to improve the ability of electrons to cross the contact-CNT interface, as the shape of the potential barriers at these is primarily determined by band bending induced by the trapped dielectric charge. The effect of the annealing is more likely to be observed through a change in the work function of the metallic gate. Reducing the work function of the gate allows smaller values of VGS to reduce the potential energy of the channel sufficiently to allow electron conduction. The experimental Javey device shows a p-type Vp of approximately 1.2 V and an n-type Vr of approximately 1.3 V after annealing. This shift in Vr could be explained by a change in gate work function, assuming the charge trapped in the dielectric is not affected by the annealing process. 84 Chapter 5 Conclusions This theoretical study has shown the usefulness of the numerical CNTFET model for predicting the behavior and performance of CNTFET devices. The conclusions drawn from the results of the model are organized into groups corresponding to the general behavior of the model, the dependence on materials parameters, the dependence on device geometry, and CM-CNTFET devices. Under equilibrium conditions, the model shows that capacitive coupling between the gate and the CNT channel allows the gate to modulate the shape of potential barriers at the source-CNT and drain-CNT metallurgical junctions. The equilibrium results of the numerical CNTFET model show excellent agreement with those of an analytic model for the same device geome-try and materials. Analysis of Schottky-barrier CNTFETs under quasi-equilibrium conditions shows that: 1. bipolar conduction occurs in SB-CNTFET devices despite the presence of Schottky bar-riers; 2. bipolar conduction leads to a new phenomenon, drain-induced barrier thinning (DIBT), in which tunneling of electrons at the source is enhanced through modification of the source-CNT potential barrier by large quantities of holes injected into the channel at the drain; 3. DIBT leads to exponential increases in the drain current for increasing VQS', 4. the simultaneous presence of large concentrations of electrons and holes in the nanotube 85 Chapter 5. Conclusions when VGS = Vbs /2 indicates that significant recombination of carriers may occur. Radia-tive recombination in C N T F E T s has been observed experimentally under the same bias conditions. The behavior and performance of S B - C N T F E T s is dependent on the properties of the materials used to construct the device components: 1. the work functions of the source and drain contacts have a direct effect on the ability of electrons and holes to contribute to the drain current in a S B - C N T F E T . These work functions can be modified in order to tailor the I- V characteristics of the device; 2. gate dielectrics with higher relative permittivities improve capacitive coupling between the gate and the C N T channel and provide increased control of carrier movement in the channel. High-K gate dielectrics are especially effective for delaying the onset of D I B T with increasing VDS\ 3. the presence of trapped charge in the gate dielectric adjacent to a C N T effectively dopes the tube with complementary charge. The resulting shift in the energy bands of the C N T can result in a C N T F E T device with depletion behavior. The behavior and performance of S B - C N T F E T s is also dependent on the geometry of the components: 1. decreasing the radius of the source and drain contacts results in steeper band bending near the contacts and thinner potential barriers. Thinner barriers increase the probabil-ity of carrier tunneling and lead to an increase in drain current under non-equilibrium conditions; 2. capacitive coupling between the gate and the C N T channel is directly dependent on the thickness of the gate dielectric. Reduction of the dielectric thickness results in improved capacitive coupling and control of the electrostatic potential on the C N T F E T by the gate; 3. the introduction of gate-source and gate-drain spaces reduces capacitive coupling between the gate and the regions of the C N T close to the contact resulting in increased potential barrier thicknesses; 86 Chapter 5. Conclusions 4. gate-contact spaces of the order of hundreds of nanometers result in capacitive coupling of the C N T in these regions to weak fringing fields between the gate and the contacts. Control of the potential barriers in the C N T , by the gate, is reduced and the shape of these barriers becomes dependent on the fringing fields. Architectural differences separate C M - C N T F E T s from S B - C N T F E T s by changing the way carrier movement and drain current are controlled in the device: 1. the combination of large gate-contact spaces and trapped charge in the gate dielectric reduces the thickness of the potential barriers at the contact-CNT interfaces sufficiently to create essentially ohmic contacts; 2. the elimination of control of the thickness of these barriers by the gate allows another feature of the device to control current movement in the C N T , namely the electrostatic potential of the gated, mid-channel region of the tube; 3. the very small thickness of the tunneling barriers at the contact-CNT interfaces should allow greater drain currents in C M - C N T F E T devices as compared to S B - C N T F E T s . This suggests that the ultimate performance of C N T F E T devices could be found in C M -C N T F E T architectures; 4. ambipolar operation in C M - C N T F E T devices can be explained based on theoretical evi-dence for inter-band tunneling in C N T junctions. Future development of the numerical C N T F E T model could include enhancements for exam-ining the I- V characteristics of C M - C N T F E T devices and the development of a full 3D elec-trostatics solution for accurately modeling experimental C N T F E T architectures. Obtaining non-equilibrium solutions for C M - C N T F E T devices wi l l require a more robust D O S represen-tation that can accurately accommodate the very steep band bending observed in these devices. Accurate potential barrier profiles are critical for calculating currents due to carrier tunneling through these barriers. The tunneling calculation would also have to be extended to include the possibility of inter-band tunneling. Finally, the method of quasi-equilibrium carrier statistics would also have to be expanded in order to account for the effects of multiple potential barriers 87 Chapter 5. Conclusions encountered in CM-CNTFETs. The FEM solver used in the numerical CNTFET model is capable of calculating 3D electrostatics solutions, however a new method for terminating the model space of open-bounded CNTFET devices will be required. The method of conformal mapping is inherently limited to 2D model spaces and cannot be extended to three dimensions. An accurate method for applying artificial boundary conditions will have to be developed. 88 Bibliography [1] S. Iijima. Helical microtubules of graphitic carbon. Nature, 354:56-58, 1991. [2] A . Javey, H . K i m , M . Brink, Q. Wang, A . Ural , J . Guo, P. Mclntyre, P. McEuen, M . Lundstrom, and H . Dai . High -K dielectrics for advanced carbon-nanotube transistors and logic gates. Nature Materials, 1:241-246, 2002. [3] B . Q . Wei, R. Vajtai, and P . M . Ajayan. Reliability and current carrying capacity of carbon nanotubes. Appl. Phys. Lett, 79(8):1172-1174, 2001. [4] R. Martel . Nanotube electronics: high-performance transistors. Nature Materials, 1:203-204, 2002. [5] S.J. Tans, A . R . M . Verschueren, and C. Dekker. Room-temperature transistor based on a single carbon nanotube. Nature, 393:49-52, 1998. [6] R. Martel , T . Schmidt, H .R . Shea, and P h . Avouris. Single- and multi-wall carbon nan-otube field-effect transistors. Appl. Phys. Lett, 73(17):2447-2449, 1998. [7] H . T . Soh, C . F . Quate, A . F . Morpurgo, C M . Marcus, J . Kong, and H . Dai . Integrated nan-otube circuits: Controlled growth and ohmic contacting of single-walled carbon nanotubes. Appl. Phys. Lett, 75(5):627-629, 1999. [8] R. Martel , V . Derycke, J . Appenzeller, S. W i n d , and P h . Avouris. Carbon nanotube field-effect transistors and logic circuits. In Proc. Design Automation Conference, pages 94-98, June 2002. [9] J . Guo, M . Lundstrom, and S. Datta. Performance projections for ballistic carbon nanotube field-effect transistors. Appl. Phys. Lett, 80(17):3192-3194, 2002. [10] L . C . Castro, D . L . John, and D . L . Pulfrey. Towards a compact model for Schottky-barrier nanotube F E T s . In M . Gal , editor, Proc. IEEE COMMAD, pages 303-306, December 2002. [11] D . John. Towards a compact model for carbon nanotube field-effect transistors. Unpub-lished, A p r i l 2002. [12] A . A . Grinberg, M . S . Shur, R . J . Fischer, and H . Morkoc. A n investigation of the effect of graded-layers and tunneling on the performance of A l G a A s / G a A s heteroj unction bipolar transistors. IEEE Trans. Electron Devices, ED-31:1758-1765, 1984. [13] M . S . Lundstrom. A n Ebers-Moll model for the heterostructure bipolar transistor. Solid-State Electron., 29:1173-1179, 1986. [14] D . L . Pulfrey and S. Searles. Electron quasi-Fermi level splitting at the base-emitter junc-tion of A l G a A s / G a A s H B T ' s . IEEE Trans. Electron Devices, 40(6):1183-1185, 1993. [15] T . Durkop, T . Brintlinger, and M . S . Fuhrer. Nanotubes are high mobility semiconductors. In Structural and Electronic Properties of Molecular Nanostructures—AIP Conference Pro-ceedings, pages 242-246, 2002. 89 Bibliography [16] R. Saito, G . Dresselhaus, and M . S . Dresselhaus. Physical Properties of Carbon Nanotubes. Imperial College Press, London, 1st edition, 1998. Weblab viewer lite 4.0. http://www.msi.com, 2000. J .T . Frey and D . J . Doren. Tubegen 3.1 web-interface. http://deaddog.duch.udel.edu/~frey/research/tubegenonline.html, 2003. K . Esfarjani, A . A . Farajian, Y . Hashi, and Y . Kawazoe. Electronic, transport and mechan-ical properties of carbon nanotubes. In Y . Kawazoe, T . Kondow, and K . Ohno, editors, Clusters and Nanomaterials—Theory and Experiment, Springer Series in Cluster Physics, pages 187-220. Springer-Verlag, Berlin, 2002. N . W . Ashcroft and N . D . Mermin. Solid State Physics. Harcourt College Publishers, New York, 1st edition, 1976. J .W. Mintmire and C . T . White. Universal density of states for carbon nanotubes. Phys. Rev. Lett, 81(12):2506-2509, 1998. ' S.J. W i n d , J . Appenzeller, and P h . Avouris. Lateral scaling in carbon-nanotube field-effect transistors. Phys. Rev. Lett, 91 (5):058301-1-058301-4, 2003. J . Appenzeller, J . Knoch, R. Martel , V . Derycke, S. W i n d , and P h . Avouris. Short-channel like effects in schottky barrier carbon nanotube field-effect transistors. In IEDM Tech. Digest, pages 285-288, 2002. J .P. Clifford, D . L . John, L . C . Castro, and D . L . Pulfrey. Electrostatics of partially gated carbon nanotube F E T s . IEEE Trans. Nanotechnol, 2003. Submitted. A . Javey, J . Guo, Q. Wang, M . Lundstrom, and H . Dai . Ballistic carbon nanotube field-effect transistors. Nature, 424:654-657, 2003. D . L . John, L . C . Castro, J.P. Clifford, and D . L . Pulfrey. Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors. IEEE Trans. Nanotechnol., 2(3):175-180, 2003. K . Natori. Ballistic metal-oxide-semiconductor field effect transistor. J. Appl. Phys., 76(8):4879-4890, 1994. J . Guo, S. Goasguen, M . Lundstrom, and S. Datta. Metal-insulator-semiconductor elec-trostatics of carbon nanotubes. Appl. Phys. Lett, 81(8):1486-1488, 2002. R. Martel , H.-S.P. Wong, K . Chan, and P h . Avouris. Carbon nanotube field effect tran-sistors for logic applications. In IEDM Tech. Digest, pages 159-162, 2001. J . Appenzeller, J . Knoch, R . Martel , V . Derycke, S. W i n d , and P h . Avouris. Carbon nanotube electronics. IEEE Trans. Nanotechnol, 1(4):184-189, 2002. V . Derycke, R . Martel , J . Appenzeller, and P h . Avouris. Carbon nanotube inter- and intramolecular logic gates. Nano Lett., l(9):453-456, 2001. 90 Bibliography S. Heinze, J . Tersoff, R . Martel , V . Derycke, J . Appenzeller, and P h . Avouris. Carbon nanotubes as Schottky barrier transistors. Phys. Rev. Lett., 89(10):106801-l-106801-4, 2002. J . A . Misewich, R. Martel , P h . Avouris, J .C . Tsang, S. Heinze, and J . Tersoff. Electrically induced optical emission from a carbon nanotube F E T . Science, 300:783-786, 2003. F . Leonard and J . Tersoff. Role of Fermi-level pinning in nanotube Schottky diodes. Phys. Rev. Lett., 84(20):4693-4696, 2000. F E M L A B . http://www.femlab.com, 2002. C O M S O L A B , Stockholm. FEMLAB Reference Manual, version 2.3 edition, 2002. D . J . Griffiths. Introduction to Quantum Mechanics. Prentice Hal l , New Jersey, 1994. D . L . Pulfrey. Modeling high-performance H B T s . In P. Robl in and H . Rohdin, editors, High-Speed Heterostructure Devices, chapter 18. Cambridge University Press, 2002. S.S. Perlman and D . L . Feucht. p-n heterojunctions. Solid-State Electron., 7:911-923, 1964. D. K . Ferry and S . M . Goodnick. Transport in Nanostructures. Cambridge University Press, New York, 1997. J.P. Clifford, D . L . John, and D . L . Pulfrey. Bipolar conduction and drain-induced barrier thinning in carbon nanotube F E T s . IEEE Trans. Nanotechnol, 2(3): 181-185, 2003. C. R. I . Emson. Methods for the solution of open-boundary electromagnetic-field problems. IEE Proc— Sci. Meas. Technoi, 135(3):151-158, 1988. E . M . Freeman and D . M . Lowther. A novel mapping technique for open boundary finite element solutions to Poisson's equation. IEEE Trans. Magn., 24(6):2934-2936, 1988. D . John. Invariance of Laplace's equation under a conformal mapping. Unpublished, January 2003. F . Leonard and J . Tersoff. Dielectric response of semiconducting carbon nanotubes. Appl. Phys. Lett, 81(25):4835-4837, 2002. F . Leonard and J . Tersoff. Negative differential resistance in nanotube devices. Phys. Rev. Lett, 85(22):4767-4770, 2000. V . Derycke, R. Martel , J . Appenzeller, and P h . Avouris. Controlling doping and carrier injection in carbon nanotube transistors. Appl. Phys. Lett., 80(15):2773-2775, 2002. R. Martel , V . Derycke, C . Lavoie, J . Appenzeller, K . K . Chan, J . Tersoff, and P h . Avouris. Ambipolar electrical transport in semiconducting single-wall carbon nanotubes. Phys. Rev. Lett, 87(25):256805-l-256805-4, 2001. S.J. W i n d , J . Appenzeller, R. Martel , V . Derycke, and P h . Avouris. Vertical scaling of carbon nanotube field-effect transistors using top gate electrodes. Appl. Phys. Lett, 80(20):3817-3819, 2002. 91 Bibliography [50] S. Suzuki, C . Bower, Y . Watanabe, and O. Zhou. Work functions and valence band states of pristine and Cs-intercalated single-walled carbon nanotube bundles. Appl. Phys. Lett., 76(26):4007-4009, 2000. [51] A . Bachtold, P. Hadley, T . Nakanishi, and C. Dekker. Logic circuits wi th carbon nanotube transistors. Science, 294:1317-1320, 2001. 92 Appendix A Compact Expressions Based on the Maxwell-Boltzmann Distribution Function A . l Surface Charge Density The Mintmire and White density of states (DOS) [21] for C N T s can be combined with the Maxwell-Boltzmann distribution function to provide a compact expression for charge density on the surface of the C N T as a function of local potential in the C N T (VCNT{Z))-The first step is to convert the D O S per carbon atom per unit energy {p(E)) provided by Mintmire and White into a D O S per unit area per unit energy (g(E)) using the area of the graphene lattice unit cell and the number of carbon atoms per unit cell (2): where acc is the carbon-carbon bond length, V ^ p 7 r is the tight-binding energy, £ m is the lowest energy of the energy band with index m, and r is the radius of the C N T . The density of states for electrons and holes is equivalent due to the symmetry assumed in the graphene dispersion relation. Using the D O S per unit area per unit energy and a Maxwell-Boltzmann distribution function shifted by VCNT{Z), the electron concentration per unit area is calculated by integrating over g(E) = 2 (A^u)-1 P{E) (A.1) 93 Appendix A. Compact Expressions Based on the Maxwell-Boltzmann Distribution Function the energy range of the conduction bands: rtop-ofjband ptop-Oj Joana ns(z) = / f(E,EF,VCNT(z)) g(E) dE JEc POO JEc fqVCNT(z) + EF-E\ 4 ^ \E\ 6 X P V kBT ) 3accn2r\Vpp7T\ J W ^ l m=-oo V " oo 4 (qVCNT(z) + EF\ ^ r o o f _ ^ 2 ^ £ 2 ^ 3oCc7r 2r|14„„, , „ m=—oo The limits of integration are extended from the top of the conduction band to infinity between the first and second step of this calculation. This can be justified by the fact that the distribution function approaches zero as E —» oo. B y substituting u = \/E2 — 8^, the integral can be solved numerically with greater precision, as the discontinuity occurring when \/E2 — £^ = 0 is eliminated. The hole concentration per unit area is calculated in a similar fashion by integrating over the valence band: t\ 4 f-qVcNT(z)-EF\ ^ [°° -^v? + £l\ Combining these equations yields the equation for net surface charge density, , v , s / \ /~i ( f-qVcNT(z) - EFp\ (qVcNT(z) + EFn\ . Ps(z) = Ps(z) - ns(z) = C i I exp I — ^ 1 - exp I — ) ) , (A.4) where iacc^r\Vpjm\ m ^ J Q —\ kBT Note that the integral terms do not depend on VCNT(Z) and can be calculated separately from the exponential terms which depend on VCNT(Z)- This equation forms a compact expression for the net charge density at a point which is easily incorporated into a numerical Poisson solver. A.2 Solving the Flux Balance Equations to Locate the QFLs Using the Maxwell-Boltzmann distribution function to calculate the electron and hole fluxes described in Section 3.3.2 allows an analytic solution for the location of the Q F L s , as originally 94 Appendix A. Compact Expressions Based on the Maxwell-Boltzrnann Distribution Function proposed by Prof. Pulfrey. For the electron fluxes in the conduction band: 2 roo pj F i = — / T 5 n ( £ ) e x p ( — - ) d E nh J E C KB-I <2 roo pp pi F2 = - / T 5 r i ( £ ) e x p ( - ^ — - )dE •Kh J E R KBI P R / EFU + qVDs. F i — F2 = F3 — F4 F i — F i e x P ( ~ 7 ^ j r ^ = * W kBT ] ~ F i ( F 1 + F 4 ) e x p ( ^ p i ) ) = F 1 + F 4 e x p ( ^ ) / F 1 + F 4 e x p ( ^ ) \ E F N = - f c B r i n ( Fi+pAB )• ( A . 5 ) where E F N is location of the electron Q F L . A similar derivation yields the location of the hole Q F L , E F P . 9 5 Appendix B F E M L A B F E M L A B is a modeling environment for solving partial differential equations (PDEs) using the finite-element method ( F E M ) . It is implemented as a M A T L A B toolbox and extends the capabilities of M A T L A B with a set of data structures and functions designed specifically for F E M modeling in one, two, or three dimensions. A P D E problem can be represented in F E M L A B either by specification in mathematical form or through the use of application modes to abstract the mathematics of the model beneath a physical description. For the C N T F E T model a mathematical form is used since it provides complete control over how the C N T F E T is represented and how the P D E governing its oper-ation is solved. The model can be constructed and solved using the F E M L A B graphical user interface (GUI) or through M A T L A B m-files (text command files). The use of m-files requires an understanding of model specification at a low-level but provides additional flexibility and efficiency for simulating models with parameterized geometry or bias conditions. This information presented in this section is designed to serve as a guide to anyone attempting to expand the C N T F E T models or develop their own models of semiconductor devices in F E M L A B using m-files. This information is derived from the F E M L A B Reference Manual [36] (provided with the software), the excellent user support provided by support@femlab.com, and through rigorous testing during development of the C N T F E T model. 96 Appendix B. FEMLAB B . l Model Specification and Configuration Solving a P D E problem using a F E M tool requires specification of the physical structure of the model and configuration of the F E M solver. The dimensions of the model are related by the geometric structures used to represent the model features, while materials properties and bias conditions are used to define the P D E and the model boundary conditions. The F E M solver is configured through a definition of the finite elements used, the meshing strategy, and the P D E solver type. B . l . l Environment The model environment describes the geometric space, dependent variables, element type, ele-ment shape, and P D E form of the model. Depending on the number of spatial degrees of freedom required to represent the model, a one, two, or three dimensional environment can be used. For 3D modeling, a Cartesian coordinate system is used with space coordinates (dimensions) x, y, and z. For 2D modeling, a cylindrical coordinate system is used with space coordinates r and z. The azimuthal component (</>) of the coordinate system is the dimension of symmetry and is not included in the space coordinates of the model space. There is only one dependent variable, the local potential (V) , which is a function of the space coordinates. The element type refers to the shape function used to represent the dependent variable in an element. For this work a second-order Lagrange shape function object is used, allowing two degrees of freedom in each element. Thus the local potential is represented by a quadratic function in each element of the model. The Lagrange shape function also allows discontinuities in the first derivative of the dependent variable between elements. This feature is necessary to correctly model the discontinuity of the electric field (the first derivative of local potential) at the C N T / dielectric boundary where there is a discontinuity in permittivity. A second order geometry shape is used to give the elements in the mesh two degrees of freedom 97 « Appendix B. FEMLAB for conforming to the physical geometry of the model. This allows the elements to have curved edges where necessary. This feature is especially important when modeling curved structures such as cylindrical CNTs. It is common practice to use geometry shapes with the same order as the element type, giving rise to iso-parametric elements. The PDE can be specified in coefficient or general form. The coefficient form incorporates physical properties of the model more easily, however, the general form is preferable for solving a nonlinear PDE, such as that encountered in the CNTFET. To resolve this dilemma, FEMLAB allows PDEs to be entered in one form and converted to the other before solving. B . l . 2 G e o m e t r y The model geometry is defined using fundamental geometric objects which are combined to define the various subdomains of the model space. The CNT is represented by a cylinder with radius rcNT and a length equal to that of the CNTFET channel length. The dielectric gate insulator is represented by a second cylinder, concentric with the CNT, with radius rgate and the same length. Only the CNT and the dielectric regions are included in the CNTFET geometry. The contacts are outside the modeled geometry and are represented by boundary conditions. B . l . 3 B o u n d a r y C o n d i t i o n s All external boundaries must be specified using either Neumann or Dirichlet boundary condi-tions (BCs). Boundaries with fixed values, such as those corresponding to electrical contacts, are specified using the Dirichlet BC. V{P, <t>, L) = Vsource V{p, <f>, 0) = Vdrain V(rgate,<f>,z) = Vgate Boundaries which form surfaces of symmetry are specified using the Neumann BC. The bound-ary between the surface of the CNT and the dielectric is also specified using a Neumann BC 98 Appendix B. FEMLAB which describes the surface charge as a function of local potential. e 0 V V = ps{V,EFn,EFp) er,oie0VV = ps(V,EFn,EFp) The condition on this internal boundary must be satisfied on both sides of the boundary, hence the two equations with different permittivity coefficients. It is assumed that F E M L A B uses a matching condition to insure both of the equations for the boundary are satisfied when a solution to the P D E is calculated. The net charge on the surface of the C N T (ps) is a function of the local potential (V) making the solution of Poisson's equation nonlinear. B.1.4 P D E P D E s are associated with subdomains in the model space. In the case of the C N T F E T there are two subdomains: the region within the C N T and the dielectric region between the C N T and the gate electrode. e 0 V 2 V = 0 er,ox£ 0 V 2 V = 0 The subdomain within the C N T uses a permittivity equal to that of free space (e 0). It is assumed that there is no material within the walls of the C N T molecule, hence this region is modeled as free space. The subdomain corresponding to the dielectric uses a permittivity equal to the relative permittivity of the dielectric material multiplied by the permittivity of free space(e r j 0 x e 0 ). There is no free charge in either of the subdomains, hence the zero source term on the right hand side of both equations. Note that although both source terms in the subdomain P D E s are constants, the solution is still nonlinear due to the dependence of the charge on the surface of the C N T on the local potential. In F E M L A B , the initial value of the solution component is specified in the P D E components. A reasonable initial guess for the local potential in the C N T F E T is the gate potential (Vgate). This value is used in all simulations. 99 Appendix B. FEMLAB B . l . 5 Mesh Development of the mesh consists of three steps: initialization, refinement, and extension. Initialization and extension of the mesh must be performed before the P D E can be solved, as the mesh forms the framework for the solution. Refinement is an optional step used to increase the resolution of the mesh. When the mesh is initialized, parameters which control the mesh geometry can be specified. These typically include the mesh growth rate, quality improvement through jiggling, maximum element size, and the size of elements along curved model features. Another parameter of particular importance for modeling structures with high aspect ratios, such as those found in C N T s , is hpnt. This parameter controls the number of points placed on edges of objects for resolving geometry during meshing. This value must be increased from its default setting of 10 to roughly 200 in order to allow good quality meshing of C N T s longer than 1000 nm. Mesh refinement increase the number of mesh elements. In 2D this is typically done by dividing each triangle into four smaller triangles of the same shape. In 3D each triangle edge is bisected to form the smaller triangles Extending the mesh translates the complete P D E problem description into element syntax, a form suitable for use by the solver. This process also partitions the domains into mesh domain groups and organizes the degrees of freedom of the mesh nodes. B.1.6 Initial Values Providing a good initial value of the solution in nonlinear problems decreases the time needed by the solver to find the actual solution. Initial values defined in the P D E s corresponding to subdomains are explicitly mapped to the mesh through calculation of the init ial solution vector. B . l . 7 Solver F E M L A B provides several stationary nonlinear P D E solvers, femnlin and femiter are non-adaptive algorithms which have shown consistent success in solving the nonlinear P D E associ-ated with the C N T F E T . Measurements of solution time in 3D models show that the iterative 100 Appendix B. FEMLAB solver (femiter) performs better than the nonlinear stationary solver (femnlin). There was no apparent advantage with either solver in the 2d models. Both solvers yielded identical solu-tions, the only difference being the time required for convergence. The results of these simple, non-adaptive solutions show excellent correlation with analytically derived solutions, and no manipulation of the default solver parameters is required. Examinations of the adaptive and multigrid solvers wil l be undertaken in the future to test the applicability of these algorithms. B.2 3D Solutions The simplest way to define the C N T F E T structure is to use the three dimensional Cartesian coordinates system and specify the complete C N T F E T device in three dimensions. This ge-ometry consists of two concentric cylinders, the first defining the surface of the C N T and the second defining the dielectric material and the surfaces of the source, drain, and gate contacts. B.3 2D Solutions While 3D solutions of the C N T F E T afford the least complicated model description, they are inefficient in that they ignore the radial symmetry of the device. In a cylindrical coordinate system, the azimuthal component is excluded from the model space, allowing a 2D abstraction of the cylindrical C N T F E T structure. The 2D geometry model is longitudinal slice through the C N T F E T with rectangles representing the sectional area inside the C N T and the sectional area of the dielectric. The 2D coordinate system in F E M L A B is a Cartesian system. When a cylindrical coordinate system with radial symmetry is projected onto this coordinate system, radial dependencies in the operators must be handled explicitly. The Laplacian operator in a cylindrical coordinate system includes a radial dependency in two of its terms. V 2 y - — + - — + — — + — (Bl) 8r2 r 8r r2 5<p2 8z2 A Laplacian in the 2D Cartesian system, using the coordinates r and z, wi l l only include the 101 Appendix B. FEMLAB first and fourth terms of (B.l). Since there is symmetry with respect to </>, there is no variation of V in that direction and the radially dependent third term is set to zero. However, the radially dependent second term is still an important part of the solution and must be retained. The coefficient form of the stationary PDE in FEMLAB (B.2) provides a mechanism for rep-resenting this term. The )3 coefficient is a l-by-2 matrix corresponding to the components of V in the r and z dimensions. This allows complete specification of radial dependency in the model using a combination of the c (B.3) and (3 (B.4) coefficients. - V • (cVV + aV - 7 ) + / 3 V V + aV = f (B.2) c e (B.3) p = [ i o ] (B.4) 102 

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