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UBC Theses and Dissertations

Modeling of carbon nanotube field-effect transistors Castro, Leonardo de Camargo e 2006

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Modeling of Carbon Nanotube Field-Effect Transistors by Leonardo de Camargo e Castro B.A.Sc. (Electrical Engineering), The University of British Columbia, 2001 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF Doctor of Philosophy in The Faculty of Graduate Studies (Electrical and Computer Engineering) The University Of British Columbia July 2006 © Leonardo de Camargo e Castro 2006 Abstract In this thesis, models are presented for the design and analysis of carbon nanotube field-effect transistors (CNFETs). Such transistors are being seriously considered for applications in the emerging field of nanotechnology. Because of the small size of these devices, and the near-one-dimensional nature of charge transport within them, CNFET modeling demands a rig-orous quantum-mechanical basis. This is achieved in this thesis by using the effective-mass Schrodinger Equation (SE) to compute the electron and hole charges in the nanotube, and by using the Landauer Equation to compute the drain current. A Schrodinger-Poisson (SP) solver is developed to arrive at a self-consistent potential distribution within the device. Normaliza-tion of the wavefunction in SE is achieved by equating the probability density current with the current predicted by the Landauer Equation. The scattering matrix solution is employed to compute the wavefunction, and an adaptive integration scheme to obtain the charge. Overall convergence is sought via the Picard or Gummel iterative schemes. An AC small-signal circuit model, employing the DC results from the SP solver, is also constructed to obtain estimates of the high-frequency capabilities of the transistors. The DC results predict the unusual ambipolar behaviour of CNFETs reported in the literature, and explore the possibilities of using work-function engineering to tailor I- V characteristics for different device applications. The model qualitatively agrees with some experimental results in the literature, and gives confidence that the performance of coaxial devices, when they become available, will be well predicted by the models. In the AC regime, it was found that under somewhat ideal operating conditions the operating limit of these devices might just reach into the 1-10 THz regime. In addition to the development of rigorous modeling procedures for CNFETs, a preliminary compact model is developed, in which some of the essence of the detailed model is distilled into a set of simpler equations, which may prove useful in guiding device design towards CNFETs for applications in nanoelectronics. ii Table of Contents Abstract ii Table of Contents iii List of Figures vi List of Symbols viii Acknowledgments ix Co-Authorship Statement x Chapter 1. Introduction 1 1.1 Carbon Nanotubes 2 1.1.1 Crystal Structure 3 1.1.2 Electronic Properties 5 1.1.3 Transport 10 1.2 Carbon Nanotube Field-Effect Transistors 12 1.2.1 Planar Devices 13 1.2.2 Coaxial Devices 15 1.3 Modeling Coaxial CNFETs 16 1.3.1 Electrostatics 18 1.3.2 Charge 24 1.3.3 Transport 27 1.4 Thesis Outline 28 1.5 Specific Contributions 28 References 29 Chapter 2. Towards a Compact Mode l for Schottky-Barrier C N F E T s 36 2.1 Introduction 36 2.2 The Model 38 2.3 Results and Discussion 39 2.4 Conclusions 42 References 42 Chapter 3. Electrostatics of Coaxial Schottky-Barrier C N F E T s 44 3.1 Introduction 44 3.2 Coaxial Nanotube Electrostatics 46 3.3 Results and Discussion 47 3.4 Conclusions 55 References 55 iii Table of Contents Chapter 4. A n Evaluation of C N F E T D C Performance 58 4.1 Introduction 58 4.2 Fabrication 59 4.3 Theoretical Considerations 59 4.4 DC Modeling 60 4.5 Results and Discussion 64 4.5.1 Ambipolarity 64 4.5.2 Conductance 65 4.5.3 Subthreshold Slope 66 4.5.4 ON Current 67 4.5.5 Transconductance 70 4.6 Conclusions 70 References 71 Chapter 5. A Schrodinger-Poisson Solver for Model ing C N F E T s 75 5.1 Introduction 75 5.2 Solution Method 76 5.2.1 Potential 77 5.2.2 Charge 78 5.3 Results 80 5.4 Conclusions 82 References : 83 Chapter 6. A n Improved Evaluation of the D C Performance of C N F E T s 85 6.1 Introduction 85 6.2 Correspondence of the Compact and Quantum Models 86 6.3 Quantum-Mechanical Reflection for the Thermionic Case 89 6.4 Results and Discussion 90 6.5 Conclusions 93 References 94 Chapter 7. Quantum Capacitance in Nanoscale Device Model ing 96 7.1 Introduction 96 7.2 Equilibrium Quantum Capacitance 97 7.2.1 Two Dimensions 97 7.2.2 • One Dimension 98 7.3 General Considerations 99 7.4 Application: CNFETs 102 7.5 Conclusions 107 References 108 Chapter 8. Method for Predicting ft for C N F E T s 110 8.1 Introduction 110 8.2 The Small-Signal Model I l l 8.2.1 Equivalent circuit I l l 8.2.2 Model parameters 113 iv Table of Contents 8.3 Results and Discussion 114 8.4 Conclusions 120 References 120 Chapter 9. High-frequency Capability of Schottky-Barrier C N F E T s 123 9.1 Introduction 123 9.2 Model 124 9.3 Results and Discussion 124 9.4 Conclusions 127 References 127 Chapter 10. Extrapolated / m a x for C N F E T s 129 10.1 Introduction 129 10.2 Modeling Procedures 130 10.3 Results and Discussion 133 10.4 Conclusions : 137 References 138 Chapter 11. Conclusion and Recommendations for Future Work 140 Appendix A . Complete Schrodinger-Poisson Mode l 145 A . l Sample Results 153 References 156 v List of Figures 1.1 Hybridized carbon atom and graphene lattice structure 3 1.2 Carbon nanotube lattice structure 4 1.3 Graphene energy dispersion relation 7 1.4 Nanotube energy dispersion relation, density of states, and carrier velocity 8 1.5 Nanotube properties for various tube radii 9 1.6 Examples of planar CNFETs 14 1.7 Examples of electrolyte-gated CNFETs 15 1.8 Coaxial CNFET structure 18 1.9 Solution to Laplace's equation in 2D cylindrical coordinates 21 1.10 Electric field on nanotube surface 22 1.11 CNFET band diagrams for Laplace solutions 24 2.1 Model geometry: closed cylindrical structure 37 2.2 Conduction energy band diagram for various bias conditions 40 2.3 Electron distribution in the mid-length region of nanotube 40 2.4 Drain current-voltage characteristics 41 2.5 Comparison of drain i - V characteristics 41 3.1 Equilibrium energy band diagram: (16,0) tube, tj n s = 5.6 nm and ei n s = 3.9 48 3.2 Equilibrium energy band diagram: (16,0) tube, t[ns = 30-9 nm and ei n s = 3.9 49 3.3 Equilibrium energy band diagram: (16,0) tube, i ; n s = 5.6 nm and ei n s = 19.5 50 3.4 Sub-threshold current: (16,0) tube, i j n s = 5.6 nm and e;ns = 3.9 52 3.5 Non-equilibrium energy band diagram: (16,0) tube, t;ns = 5.6 nm and ei h s = 3.9 . . . . 53 3.6 Drain I- V characteristics 54 4.1 Coaxial CNFET model geometry 61 4.2 Band diagram illustrating CNFET ambipolarity 65 4.3 ID-VGS for various contact work functions 66 4.4 Band diagrams for various contact work functions 66 4.5 Ratio of equilibrium quantum capacitance to insulator capacitance 67 4.6 ID and (b) gm as a function of gate-source voltage 68 4.7 ID-VDS for various contact work functions 69 4.8 Band diagrams for various gate work functions 70 5.1 CNFET net carrier density as a function of position and V~DS 80 5.2 Conduction band edges for Vos — 0 and 0.4 V 80 5.3 Conduction band edges and transmission probabilities for electrons 81 5.4 Energy distribution of mid-channel, source-originated electron concentration 82 6.1 Drain current versus gate-source voltage for various models 91 6.2 Drain current and transconductance for various models 92 6.3 Transmission probabilities of source-injected electrons 92 6.4 Drain current versus drain-source voltage for various models 93 vi List of Figures 7.1 Equilibrium ID quantum capacitance 101 7.2 Non-equilibrium ID quantum capacitance 102 7.3 Energy band diagram comparison for Laplace and Schrodinger-Poisson solutions .. 103 7.4 CNFET quantum capacitance as a function of gate- and drain-source voltages . . . . 105 7.5 CNFET transconductance and its constituent elements 106 8.1 Small-signal equivalent circuit for ft calculation 112 8.2 Charge supply to and through the source and drain electrodes 113 8.3 Charge density in CNFET as a function of position and energy 115 8.4 Components of the source and drain capacitances 116 8.5 ft and its components 117 8.6 Bias and temperature dependences of capacitance and transconductance 118 9.1 Structure of the modeled CNFET 124 9.2 Small-signal equivalent circuit 125 9.3 Capacitances and transconductance for the model device 126 9.4 Charge density versus energy and position 126 9.5 Extrapolated figures of merit for various contact resistances 127 10.1 Unilateral power-gain for Device 1 134 10.2 reff estimates for Device 1 135 10.3 / m ax estimates for Device 2 136 10.4 Error in / m a x prediction for Device 1 137 10.5 / m a x for improved Device 1 137 A . l Simulation space for CNFET with metal contacts 147 A.2 Complex bands in semiconducting nanotube 148 A.3 Dispersion relation and density of states in ID metal contact 149 A.4 Agreement with experimental data 154 A.5 Agreement with atomistic simulations 155 vii List of Symbols k Wavevector E Energy V Electric potential . m Carrier effective mass T Temperature z Position along nanotube axis of symmetry p Radial distance from nanotube axis of symmetry Xt Electron affinity of semiconducting nanotube (j> Workfunction Eg Energy bandgap Ec Conduction band edge Ev Valence band edge EF, fJ> Fermi energy level Eo Charge neutrality level A Distance between sub-band bottom and charge neutrality level Evac Vacuum energy level Lt Nanotube length Lu Gate underlap Lg Gate length Lc Source/drain contact length (when assumed equal) Rt Nanotube radius Rg Cylindrical gate radius Insulator thickness tg Gate thickness tc Source/drain contact thickness (when assumed equal) et Nanotube relative permittivity ei n s Insulator relative permittivity Q Charge, charge density n, p Electron and hole carrier densities Cms Insulator capacitance CQ Quantum capacitance (also known as semiconductor capacitance) ip Wavefunction / Fermi function T Transmission function g Density of states Q Local density of states C Linear operator in Poisson's equation Ti Hamiltonian operator in Schrodinger's equation q Magnitude of electronic charge HB Boltzmann's constant h Planck's constant (h is sometimes labeled Dirac's constant) eo Permittivity of free space viii Acknowledgments I would like to acknowledge my colleagues Dylan McGui re and Sasa Rist ic , and the members of my supervisory committee Anthony Peirce and John Madden, for many fruitful discussions. I would also like to thank David John of the UBC Nanoelectronics Group as our collaboration was invaluable in my learning about and understanding carbon nanotube transistors. It is with great pleasure that I thank my mentor Dave Pulfrey, who provided me with many opportunities over the last few years of graduate studies, was always a source of inspiration, support and experience, and turned out to be much more than just a thesis supervisor. Finally, this thesis is dedicated to my family, Er ika , Fernando and Olavo, as it could not have been realized without their unwavering support and encouragement. ix Co-Authorship Statement The core chapters of this thesis are publications based on work performed with other members of the UBC Nanoelectronics Group during the period 2002-2005. This statement acknowledges the role of this thesis' author in their creation. Most of the derivations in Chap. 3 and Chap. 7 were the work of David L. John. Chap. 5 and Chap. 6 had equal contributions from this author and David L. John. The author of this thesis was responsible for all the remaining research presented herein, with the exception of a few equations present in Chap. 4 and Chap. 8, which are due to co-authors. As far as manuscript preparation is concerned, all the publications included here were collaborative efforts by their co-authors, and the author of this thesis had a major role in the production of all but Chap. 3 and Chap. 7, wherein he was involved mostly in editing. Nearly all the data and figures in this thesis were prepared by its author. Chapter 1 Introduction The last few years witnessed a dramatic increase in nanotechnology research. Among others, one of the most exciting fields to emerge is nanoelectronics, where a myriad of possibilities are appearing in the form of sensors, actuators, and transistors, each characterized by feature sizes of the order of a few nanometres. All this innovation has been fueled by the discovery of new materials and the invention of manu-facturing methods that allow design and development at such a minute scale. Carbon nanotubes are at the forefront of these new materials, due to the unique mechanical and electronic proper-ties that give them, for example, exceptional strength and conductivity. One exciting possibility is the creation of nanometre-scale transistors, perhaps to be embedded, in the future, inside complex and miniscule electronic circuits that will make today's chips seem enormous in com-parison. Moreover, these nanotubes exhibit a tremendous current-carrying ability, potentially allowing for increased miniaturization of high-speed and high-power circuits. Although some devices have already been produced, the technology is still in its infancy when compared to, for instance, that of bulk-silicon MOSFETs. This thesis is a report on studies performed during the period 2002-2005 with the UBC Na-noelectronics Group, with the aim of understanding and obtaining performance predictions for carbon nanotube field-effect transistors (CNFETs). During the time this research was being conducted, there were few published works illustrating certain phenomena predicted herein. During the course of this work, and the writing of this manuscript, several researchers from many institutions, both private and academic, reported results from experimental devices that 1 Chapter 1. Introduction could be explained by the models presented herein. The remainder of this chapter will cover some of the CN properties that are relevant to this work on transistor modeling, will introduce the Schottky-barrier carbon nanotube field-effect transistor (SB-CNFET), and will describe the problem to be modeled. Finally, an outline of the thesis will be presented. 1.1 Carbon Nanotubes Carbon nanotube (CN) molecules have a cylindrical structure and are formed by one or more concentric, crystalline layers of carbon atoms. These atoms are assembled in hexagonal-lattice graphene sheets, which are rolled up into seamless tubes and named according to the number of concentric sheets as being either multi- or single-wall nanotubes. Both kinds were originally observed experimentally, via transmission electron microscopy, decades ago and work in the field has subsequently increased dramatically [1,2]. These molecules exhibit unique physical properties and, while this thesis mostly focuses on em-ploying their electronic characteristics for nanoelectronic applications, it is important to note that CNs are being studied in a variety of fields that make use of other properties. Nanotubes appear to be paving the way for myriad possibilities in the growing nanotechnology and emerg-ing biotechnology industries, particularly in nanoelectronics in the form of sensors, transistors, and nanowires. An in-depth description of CNs and their properties may be found, for example, in Ref. [3] and Ref. [4]. It is also noteworthy that carbon is not the only chemical element to form nanotubes—for instance, inorganic BN, WS2, and V2O5 nanotubes have been reported in the literature [5]. Since we will be dealing, in subsequent chapters, with carrier density and transport in carbon nanotubes, it is necessary to understand the basic electronic properties of the material. In particular, we are concerned with the energy dispersion (e-k) relation, density of states (DOS) calculations, and scattering mechanisms. 2 Chapter 1. Introduction 1.1.1 Crystal Structure Carbon is found in nature most commonly as graphene or diamond, which are crystal formations of covalently-bonded carbon atoms. In its ground state, carbon has an electron configuration of l s 2 2 s 2 2 p 2 (6 electrons i n 3 orbitals). Covalent bonding occurs v i a hybridization of the two outermost shells (4 electrons), and in graphene this takes shape via sp 2 orbitals, as illustrated in F i g . 1.1A. Figure 1.1: Pic tor ia l representation of (A) sp 2 -hybridized carbon and (B) graphene lattice structure. Note that the orbitals are for illustration only, and are neither rigorously-derived probability densities nor drawn to scale. A carbon atom in graphene wil l assemble in a single-sheet hexagonal lattice resembling the surface of a honeycomb, as illustrated in F ig . L I B . This is also known as a trigonal-planar cr-bond framework, wi th an inter-atomic spacing, a c c , of approximately 1.42 A along the bonds that are separated by 120 degrees. The 2p electrons from all the atoms on the sheet constitute a "cloud" of delocalized 7r-orbitals surrounding the carbon cores, and these valence electrons, once excited, are responsible for conduction in graphene. Note that in F i g . L I B , the delocal-ized orbitals are illustrated as individual lobes connected by hexagonal rings above and below the sheet—a proper derivation of probability clouds would show that in reality the electrons form thicker, donut-shaped rings above the hexagonal lattice, and that adjacent donuts merge, thereby allowing electrons to move about the entire two-dimensional plane. Furthermore, mul-tiple sheets of graphene may assemble in stacks, whereby two adjacent sheets are held together 3 Chapter 1. Introduction weakly by dispersion forces1 and have an inter-layer spacing of about 3.35 A. While the strength of the cr-bonds is responsible for some incredible mechanical properties of carbon nanotubes, the weak dispersion forces are the reason sheets of graphene readily slide off each other, giving the softness of graphite in pencils. Figure 1.2: Pictorial representation of (A) unrolled and (B) rolled carbon nanotube lattice structures. The latter shows a (16,0) tube. Single-wall nanotubes are characterized by a wrapping (or chiral) vector Ch — nidi + 71262, where [a\, S2] are lattice unit vectors separated by 60 degrees and the indices (rii, 712) are positive integers (0 < ri2 < n{) that specify the chirality of the tube [6], as shown in Fig. 1.2A. The chiral vector begins and ends at equivalent lattice points, so that the particular (n\, 712) tube is formed by rolling up the vector so that its head and tail join, forming a ring around the tube. The length of Ch is thus the circumference of the tube, and the radius is given by the formula Rt = \Ch\/(2n) — aC(*J?>(n\ + + n\ri2) / {^), where, for example, Rt ~ 0.63nm for a (16,0) nanotube (see Fig. 1.2B). The smallest possible radius, a limit imposed by the requirement that the energy of the system in tube form be lower than that of the unrolled graphene equivalent, is thought to be ~ 2 A [7], whereas the upper limit on radius is in the several nanometre range. Depending on their (ni, 712) indices, nanotubes are placed in one of three groups, which are named according to the shape of the cross-section established by the Dispersion forces, also known as London forces, are intermolecular attractive forces caused by instan-taneous dipoles created by electron motion about the nuclei; they are also a subset of van der Waals forces. 4 Chapter 1. Introduction chiral vector slicing across the hexagonal pattern: armchair (ni = ri2 and £ = 30°), zig-zag (ri2 = 0 and £ = 0°), and chiral (all other cases), where £ is the angle between Ch and di [6]. The indices also serve to quickly identify the conduction properties of a nanotube—when [n\ — 712) is a multiple of 3, the nanotube is metallic, otherwise it is semiconducting. Finally, to appreciate the size of these molecules, it is convenient to keep in mind the number of atoms composing a given tube. The number of atoms per nanometre-length on a single-wall nanotube can be estimated by the formula A'atoms « 2- — - — « 24QRt, Ahex J->t 3v3a^c M where A denotes area, and Rt and Lt are, respectively, the tube radius and length in nanometres. Since a typical tube used in the devices examined in this thesis will have dimensions of Rt ~ 0.63 nm, we expect to have an atom density of roughly 150 n m - 1 contributing to the conducting "cloud". 1.1.2 Electronic Properties Owing to the small diameter of carbon nanotubes, it is necessary to account for the quantiza-tion of wavevectors in the circumferential direction. Moreover, the thinness of the nanotube's cylindrical shell obviously yields an even shorter length of confinement in the radial direction, thus making the material virtually one-dimensional as far as electron transport is concerned. Many published works to date have corroborated this claim with experimental evidence from device studies [8]. In order to examine the band structure and conductivity properties of the nanotube, it is nec-essary to derive its e-k relation. This is done by starting from the equivalent relation of a two-dimensional graphene lattice (a function of two wavevectors), and introducing a quantization of wavevectors in the direction of the chiral vector Ch via the imposition of a periodic bound-ary condition. A detailed derivation of the energy dispersion relations using a tight-binding approximation is presented in Ref. [6], and we only outline its major points here. It should be noted that this approach ceases to be valid at the onset of curvature effects, i.e., for tubes 5 Chapter 1. Introduction of radius smaller than 0.5 nm [7,9,10]. These effects are a manifestation of re-hybridization (mixing) of the cr and 7r orbitals due to their proximity in small-diameter tubes. This impacts the determination of bandgap (and thus the conduction properties) and density of states, and, as such, the tubes in this work are kept to radii that try to avoid significant contribution of these second-order effects. The dispersion relation for graphene, obtained by the Slater-Koster tight-binding scheme, con-sidering only 7r-orbitals, and following the lattice vector conventions given in Ref. [6] is: ^graphene{^ xi ky) ~ where the positive and negative terms correspond to the symmetrical bonding and anti-bonding energy bands, respectively, the fc's are wavevectors, and t is the transfer integral (or nearest-neighbour parameter), the value of which is typically taken to have magnitude 2.8eV [11]. A plot of Eq. (1.1), representing a surface of allowed energies for the two-dimensional wavevector, is illustrated in Fig. 1.3, where the high-symmetry points are indicated by capital letters (K, M, and T). The K-points are degenerate near the Fermi energy (E = 0 in the plot) and indicate the zero-gap characteristic of conducting graphene. Near these K-points, as k —• 0, the dispersion relation is approximately cone-shaped and from the constant slope the Fermi velocity is given by [6] 3 M VF - 2 ^ a c c H • We now seek an expression for the nanotube dispersion relation, which is obtained by taking slices of the surface above, each cut being determined by the circumferential quantization. Again following the lattice vector definitions in Ref. [6], we switch the wavevector notation (kx, ky) —> (kz, p), where the subscript z denotes the direction of transport, and p is an integer representing the quantized modes obtained by imposing a periodic boundary condition in the circumferential (perpendicular to transport) direction. The nanotube dispersion relation is then given by Et{kz, p) = ±t (l + 4cos7i cos72 + 4cos2 w)1^2 , 1 + 4 cos 3kx&c cos \/3kva, + 4 cos2 \/3 ky (1.1) 6 Chapter 1. Introduction Anti-bonding Bonding Figure 1.3: Energy dispersion relation for graphene, from nearest-neighbour tight-binding cal-culation, using Slater-Koster approximation. T, M , and K are high-symmetry points, where the K-points lie on the plane of E = 0 (the Fermi level). where the cosine arguments are given by 3acc ri2 — n\ TT n\ + 3ri2 71 = —j kz + j P 3acc n,2 + n\ n 3n i — n-i 72 = — : kz + * P' 4 r\ 2 r/z in which r/2 = n\ + n\ + n\Ti2, -TT < {'iaccr}/dg)kz < +7r, p = 0...(2rj/dg - 1), and dg = gcd(2ni - n-2,2n2 — n i ) . The one-dimensional density of states, g(E), is obtained v ia a summation over all wavevectors. This summation can be split between transverse and longitudinal components, and we convert the latter summation into an integral over energy [12, p.52] (note that quantization due to the tube length is neglected here): G{E) = J: Ew=E(2' h / *) = E (| / I H • M p f c 2 p p ' where the factors of 2 in the numerator account for spin degeneracy and include ±fc-states (dEt/dk is an even function). F ig . 1.4 shows the band structure and density of states for the lowest, doubly-degenerate [6], bands of two different nanotubes. Note also that a simpler alter-native to the tight-binding calculations was derived analytically, under some approximations, by Mintmire et al. [13] and allows one to calculate the density of states with a good fit of the Chapter 1. Introduction bandgap for most cases, but a progressively worse fit to the DOS as one moves up in energy and includes more bands. Wavevector Density of States Velocity (m/s) x 10 Figure 1.4: Energy dispersion relation, density of states, and carrier velocity plots for: (16,0) (solid) and (10,0) (dotted lines) carbon nanotubes, illustrating the lowest, doubly-degenerate [6], bands. The energies are referenced to the Fermi level (E = 0) and are thus symmetrical about the x-axis. The carrier velocity is computed from v(E) — l/h(dE/dk). Knowledge of the carbon nanotube dispersion relation allows one to identify whether a tube produces metallic or semiconducting behaviour, and in the latter case, determine the conduction and valence band edges and thus the bandgap. Typical values for the bandgap are in the range of tenths of an electron-volt to a few electron-volts, and for the previously mentioned (16,0) tube it is ~ 0.62 eV (see Fig. 1.4). For transistors, it is desirable to have a bandgap large enough to suppress excessive thermal generation of carriers, i.e., beyond what the gate is able to control. However, as will be shown later, a smaller bandgap allows for greater carrier densities on the tube under certain conditions, and thus higher currents. It is noteworthy that the bandgap can be found to fit the expression [6]: rp N a cc The bandgap is thus a geometrically-tunable property, and given that we can make devices by choosing nanotubes by their approximate radius (presently via scanning tunneling microscopy), we may be able to exploit this tunability in nanoelectronics. The bandgap and other parameters of some tubes are illustrated in Fig. 1.5. Also shown is the intrinsic electron concentration, m, typically employed in describing bulk semiconductors; although it may not be as useful of a 8 Chapter 1. Introduction measure in quantum devices involving quasi-lD carbon nanotubes, it is worthwhile mentioning. Nanotube radius (nm) Nanotube radius (nm) Figure 1.5: Nanotube properties for various tube radii (larger chiral numbers correspond to larger radii): (a) intrinsic electron concentration (at T=300K) and energy gap; (b) carrier effective mass of lowest subband, normalized to the free electron mass mo = 9.11 • I O - 3 1 kg. During the early stages of research in nanotube transistor fabrication, the question of doping was one of much controversy. Although many experimental works have shown nanotubes to behave as p-type semiconductors, i.e., with holes as majority carriers, it is now thought that such behaviour is caused by inadvertent doping of the contacts or surrounding materials. However, a more recent theoretical explanation for the effect of hole self-doping has been proposed by Rakitin et al., where it is claimed that a-it hybridization due to curvature in small-radius tubes causes a charge transfer from the portion of the 7r-orbitals lying inside the tube to the a-orbitals lying in-plane with the nanotube surface [14]. However, given that this study has not been corroborated experimentally, and that effects from external dopants (e.g., due to charge trapped in oxide layers) and contacts are more significant, herein we treat all tubes as being intrinsic. Finally, we deal with the elusive issue of obtaining the work function and electron affinity of carbon nanotubes. Given the uncertainty in determining these parameters, we have employed a value of 4.5 eV in most chapters in this thesis, in agreement with Ref. [15]. As will be made clear throughout this work, since both the nanotube and metal-contact work functions 9 Chapter 1. Introduction are difficult to measure, they are simply taken as inputs to the model. Still, here we note that subsequent to performing the work in the aforementioned chapters, we became aware of experimental results published by-other research groups: Suzuki et al. found the work function to be 4.8eV, which is about 0.1-0.2eV larger than graphene [16], and is in agreement with measurements by Chen et al. [17]; Kazaoui et al. gave an experiment-based estimate for the electron affinity and first ionization energy to be, respectively, 4.8 eV and 5.4 eV [18]. Moreover, a recent work by Chen et al. on carbon nanotube transistors correlated the Schottky barrier height at the metal-nanotube interfaces with the tube diameter [19], and from this we inferred values for the work functions in Chap. 10. 1.1.3 Transport From early carbon nanotube experiments, owing to their molecular uniformity and quasi-one-dimensional nature, it was expected that they would exhibit ballistic transport properties, and early theoretical predictions stated they would be ballistic for most radii encountered in experiments [20]. Research of such characteristics delves into the realm of mesoscopic systems, those being materials or devices that are small enough (between 1 and 100 nm) to have their behaviour governed by interactions at the scale of the electron wavefunction, yet large enough that we need not deal with microscopic (at the atomic scale) phenomena. Carbon nanotubes (even samples that are a few microns in length), modern short-channel MOSFETs with two-dimensional electron gas (2DEG) inversion channels, and quantum-well lasers are all examples of devices that may be labeled as mesoscopic. Such systems have been the focus of much study in recent history, driven by an attempt to bridge the knowledge gap between bulk semiconductor technologies and nanoscale devices. The reader may refer to the material in Refs. [12,21] for in-depth information on the subject. Also, it must be noted that although CNs exhibit low-temperature transport effects such as Coulomb blockade, these are not dealt with in this work as they are not pertinent to room-temperature transistor operation. A mesoscopic system may be subject to a variety of scattering mechanisms. Electron-electron interactions have been studied in metallic carbon nanotubes, showing that a transition from a 10 Chapter 1. Introduction Fermi Liquid to a Luttinger Liquid2, as expected for a one-dimensional conductor, can play a role in the transport properties of CNs [22]. However, further research is still required to de-termine experimentally their role in nanotube devices. Backscattering due to electron-phonon interactions is another phenomenon that has been demonstrated in single-wall carbon nanotube samples at biases of several volts [23]. Still, this is only manifested in relatively low-energy elec-trons, and recent experimental CN-based transistor studies have indicated that no backscatter-ing occurs for operation under a bias of about one volt and device length of several-hundred nanometres [24]. A variety of studies have also reported values for mobility, mostly derived from conductance experiments in transistors, typically in the range of 103 ~ 104cm2/V-s [25,26]. Theoretical predictions, accounting for electron-phonon scattering and using Monte Carlo techniques for electron transport simulations, have also yielded a mobility of ~ 104 cm2/V-s in semiconducting tubes of radii up to ~ 2 nm [27]. Yet another report yielded conductance measurements indi-cating that metallic nanotubes are indeed ballistic conductors up to several microns in length, while semiconducting ones have strong barriers along the tube impeding transport [28]. Further studies have focused on the dependence of transport properties on mechanical deformations and defects [29-31]. Recent experiments with CN transistors, devised for the purpose of studying transport prop-erties, have concluded that for these devices transport appears to be ballistic in nature [8,24]. As such, we assume in this work that, under the simulated conditions, inelastic scattering pro-cesses are negligible and thus we are dealing with an effectively ballistic transport regime. In this situation, we adopt the single-particle approach to transport modeling, which is based on the Landauer-Biittiker formalism [21], and still allow for elastic scattering from macroscopic potentials. As a final note, we point out that the current-carrying capacity of multi-wall nanotubes has 2 A Fermi liquid is a population of electrons whose interactions do not significantly alter their energy distribution near the Fermi level, and thus they are governed by the Fermi function. A Luttinger Liquid is one in which electron-electron interactions give rise to exotic properties. 11 Chapter 1. Introduction been demonstrated to be more than 10 9 A/cm 2 , without degradation (such as that due to electromigration) after several weeks or during operation well above room-temperature [32]. This is a promising characteristic for fabricating devices designed for high-power circuits. In a device context, subsequent chapters will show that a CNFET with a 1 nm-diameter single-wall nanotube can reach currents of at least I O J U A , which normalized to its circular cross-section corresponds to a current density of order 10 8 A/cm 2 . 1.2 Carbon Nanotube Field-Effect Transistors Following the discussion on the properties of carbon nanotubes, we now give an overview of an important application and the topic of this thesis: the carbon nanotube field-effect transis-tor. This three-terminal device consists of a semiconducting nanotube bridging two contacts (source and drain) and acting as a carrier channel, which is turned on or off electrostatically via the third contact (gate). Presently, there are various groups pursuing the fabrication of such devices in several variations, achieving increasing success in pushing performance limits, while encountering myriad problems, as expected for any technology in its infancy. While the ease of manufacturing has improved significantly since their first conception in 1998, CNFETs still have a long way to go before large-scale integration and commercial use become viable. Furthermore, as these transistors evolve at every research step, the specifics of their workings become clearer, and given that the aim of this thesis is to present a working model of CNFETs, it is reassuring to see some of the findings presented herein being proven by recently released experimental data. As regards the CNFET's principle of operation, we briefly introduce two distinct methods by which the behaviour of these devices can be explained. Primarily, the typical CNFET is a Schottky-barrier device, i.e., one whose performance is determined by contact resistance rather than channel conductance, owing to the presence of tunneling barriers at both or either of the source and drain contacts. These barriers occur due to Fermi-level alignment at the metal-semiconductor junction, and are further modulated by any band bending imposed by the gate electrostatics. Moreover, in some devices, the work-function-induced barriers at the end 12 Chapter 1. Introduction contacts can be made virtually transparent either by selecting an appropriate metallization or by electrostatically forcing via a separate virtual-gate terminal (see, for example, Ref. [24]). These devices, sometimes labeled as bulk-modulated transistors, operate differently in that a thicker (non-tunneling) barrier, between the source contact and the mid-length region of the device, is modulated by the gate-source voltage. This operation is akin to that of a ballistic MOSFET, and effectively amounts to a channel modulation, by the gate, of a barrier to thermionically-emitted carriers, injected ballistically from the end contacts. We now provide a brief description of typical CNFET geometries, which are grouped in two major categories, planar and coaxial. The specifics of nanotube growth and transistor fabri-cation issues, albeit of tremendous importance for this emerging field of nanoscale transistors, are beyond the scope of this work. The reader may refer to numerous journal papers on the subject for more information, or for a fairly current summary, to Refs. [33,34]. 1.2.1 P l a n a r Dev ice s Planar CNFETs constitute the majority of devices fabricated to date, mostly due to their relative simplicity and moderate compatibility with existing manufacturing technologies. The nanotube and the metallic source/drain contacts are arranged on an insulated substrate, with either the nanotube being draped over the pre-patterned contacts, or with the contacts being patterned over the nanotube. In the latter case, the nanotubes are usually dispersed in a solution and transferred to a substrate containing pre-arranged electrodes; transistors are formed by trial and error. Manipulation of an individual nanotube has also been achieved by using the tip of an atomic force microscope (AFM) to nudge it around the substrate; due to its strong, but flexible, covalent bonds, this is possible to do without damaging the molecule. In the case where the electrodes are placed over the tube, manipulation of the CN is not required and alignment markers, pre-arranged on the substrate, allow accurate positioning of the contacts once the nanotube is located via examination by a scanning tunneling microscope (STM). The gate electrode is almost always on the back side of the insulated substrate, or alternatively is patterned on top of an oxide-covered nanotube. 13 Chapter 1. Introduction 1 ym Figure 1.6: Examples of planar CNFETs: (A) Ref. [15], (B) Ref. [35], (C) Ref. [36], (D) Ref. [8], and (E) Ref. [24]. The first CNFET devices were reported in 1998, and involved the simplest possible fabrica-tion. They consisted of highly-doped Si back gates, coated with thick Si02, and patterned source/drain metal contacts, either using Au or Pt, as shown in Fig. 1.6A [15,37]. Experimen-tations with different metals such as Ti, Ni, Al , and Pd have since been carried out by several groups, primarily to manipulate the work function difference between the end contacts and the nanotube3. Subsequent work also produced a device that replaced the back gate with an elec-trode placed over the substrate, perpendicular to the source and drain contacts, as illustrated in Fig. 1.6B [35]. Here, the nanotube was separated from this gate electrode by a thin insulating layer of AI2O3, with the source/drain electrode strips placed over the tube ends for reduced 3Recent ab initio theoretical studies comparing the interfaces between different bulk metals and metallic nanotubes, studying both end- and side-contacted tubes, concluded that T i contacts yield superior conductance over their A u and A l counterparts [38,39]. 14 Chapter 1. Introduction contact resistance. F ig . 1.6C shows a further improvement in C N F E T s through the placement of the gate electrode over the semiconducting nanotube, thus improving the channel electrostatics v ia the thin gate oxide [36,40]. Moreover, the T i source/drain metalizations in this device form t i tanium carbide abrupt junctions with the nanotube, yielding increased conductance [41]. Another attempt to obtain better gate electrostatics involved materials wi th high dielectric constants, such as zirconia (ZnO^) and hafnia (Hf02), being used as gate insulators [42]. F ig . 1.6D illustrates a device built wi th P d source/drain contacts in order to exploit the sensitivity of this material's work function to hydrogen [8]. A multi-gate device, as shown in F ig . 1.6E, has recently been reported, whereby parallel top gates are used to independently control the electrostatics of different sections of the channel, thus facilitating a study of the transport characteristics of the nanotube channel [24]. Most recently, a device with excellent D C characteristics was fabricated with P d end contacts, A l gate, and hafnia for the insulator [43]. 1.2.2 C o a x i a l Dev ice s Although yet to be fabricated in its ideal form, coaxial devices are of special interest because their geometry allows for better electrostatics than their planar counterparts. Capital izing on the inherent cylindrical shape of nanotubes, these devices would exhibit wrap-around gates that maximize capacitive coupling between the gate electrode and the nanotube channel. Figure 1.7: Examples of electrolyte-gated C N F E T s : (A) Ref. [44], (B) Ref. [45]. Presently, the closest approximation to this geometry has been the development of electrolyte-gated devices. Kruger et al. reported the first such device, shown in F ig . 1.7A, using a multi-wall 15 Chapter 1. Introduction nanotube for the channel [44]. Two gates can be activated: a highly-doped Si back gate similar to planar devices; and an electrolyte gate, formed by a droplet of LiC104 electrolyte contacted by a thin platinum wire. Fig. 1.7B illustrates an improved version of this device, this time using single-wall carbon nanotubes and NaCl for the electrolyte, and yielding current-voltage characteristics that match those of modern Si MOSFETs [45]. Alternative structures for CN devices that place the tube vertically with respect to the sub-strate have already been used for field-emission applications.Coaxial CNFETs could perhaps be fashioned by placing nanotubes inside the cavities of a porous material such as alumina, surrounding them by an electrolyte solution for gating of individual devices. Carbon nanotube transistors are not, however, the only devices in which an increased channel coupling is being sought. Other Si technologies, such as the FinFET and the tri-gate MOSFET are presently attempting to do this, and "wrap-gated" InAs-nanowire transistors have already been successfully prototyped [46]. 1.3 Modeling Coaxial C N F E T s In this section we will describe the problem being modeled, creating the framework for the CNFET models presented in later chapters. To begin with, modeling of the CNFET requires an understanding of the electrostatics in the device, i.e., the relationship between the potential and charge therein. Furthermore, appropriate treatment of carrier transport in the nanotube is necessary for the determination of the current-voltage characteristics. Evidently due to the nanometre scale dealt with here, we must look at quantum phenomena and their influence on device performance. While this work does not delve into all quantum physical issues, it attempts to balance the incorporation of key phenomena with some simplicity of implementation. This is done primarily because quantum phenomena in carbon nanotube transistors are still a freshly debated topic, but also for the sake of avoiding elaborate ab initio calculations, achieving reasonable simulation times, and arriving at some conclusions regarding performance trends. As in other field-effect transistors, the CNFET relies on one of its three terminals, the gate, 16 Chapter 1. Introduction to modulate the carrier concentration in the device channel by applying a field perpendicular to the charge flow between the other two contacts, the source and the drain. In a typical MOSFET, for example, the gate lies squarely on top of the substrate, wherein a very thin layer of mobile charge is induced by the gate contact, via capacitive coupling through a thin oxide. The volume occupied by this charge then constitutes the channel, enabling charge flow from the source terminal to the drain terminal. A CNFET, whether planar or coaxial, relies on similar principles, while being governed by additional phenomena such as ID density of states (DOS) and ballistic transport, which we have already presented in Sect. 1.1 and must now deal with in the device model context. The coaxial geometry maximizes the capacitive coupling between the gate electrode and the nanotube surface, thereby inducing more channel charge at a given bias than other geometries. This improved coupling is desirable in mitigating the short-channel effects that plague tech-nologies like CMOS as they downsize device features. It is also of importance to low-voltage applications—a dominating trend in the semiconductor industry—and to allow, potentially, for easier integration with modern implementations of existing technologies such as CMOS. Besides the wraparound gate, special attention must also be paid to the geometry of the end contacts, since these play a role in determining the dimensions of the Schottky barriers that are present in the channel near the device ends and have a direct effect on current modulation. We here-after deal specifically with the coaxial geometry of the CNFET shown in Fig. 1.8, but we note that most concepts and results discussed in this work are still transferable to planar devices, at least in a qualitative sense. The key device dimensions are: the gate inner radius, Rg, and thickness, tg\ the nanotube radius, Rt, and length Lt; the insulator thickness £j n s = Rg-Rf, the end-contact radius, tc (the source and drain may sometimes be of different sizes), and length, Lc; and the gate-underlap Lu. Note that in the following section, and in Chapters 2 to 7 a closed-cylinder structure was employed for simplicity in treating the electrostatics, wherein Lc = tg = Lu = 0, tc = Rg and Lg = Lt. 17 Chapter 1. Introduction Figure 1.8: Coaxial CNFET structure. The insulator fills the entire simulation space not occupied by metal or the nanotube. 1.3.1 Electrostatics In the system of Fig. 1.8, we solve Poisson's equation, restricted to just two polar-coordinate dimensions due to the device symmetry in the azimuthal direction, as given by where V(p, z) is the potential within the device cylinder, Qv is the volumetric charge density, e is the permittivity, and the de/dp term allows for continuous changes in the dielectric constant. In the case of abrupt dielectric heterointerfaces, as outlined in subsequent chapters, a jump condition on the electric flux is employed. Under certain conditions, Poisson's equation can be solved analytically using, for example, a Green's function approach [47], or more generally and with limited accuracy using, for example, a finite-difference numerical algorithm. The Dirichlet boundary conditions presented by the three terminals, referenced to the source potential, are given by Vs - -4>s/q Vb = VDs-4>D/q VG = VGs-<t>G/q, 18 Chapter 1. Introduction where the cp terms represent the work function of each electrode, and VQS a n a " VDS are, respec-tively, the gate- and drain-source voltages. These conditions hold in the absence of Fermi-level pinning [48]. We also do not include any series resistance in the contacts4, i.e., the voltages VQS and VDS are measured at the contact surfaces in Fig. 1.8 and not in the external cir-cuit. The remaining (open) boundaries of the system are assigned a null-Neumann condition; this assumption is valid with appropriate choices of geometry, which have been ensured in the simulations herein [49,50]. Aside from the electric potential boundary conditions imposed by the contacts and required for solving Poisson's equation inside the closed cylindrical gate, we must also deal with disconti-nuities in the electrical properties at each interface. The CNFET is composed of at least three different materials: metal contacts, semiconducting nanotube, and insulating oxide. At the interface between any two different materials there will likely be a discontinuity in one or more of the properties, thus affecting the potential profile in the device. Alternative prototype de-vices, such as the electrolyte-gated CNFET, may introduce different types of heterointerfaces, but the device performance would nonetheless be sensitive to the aforementioned or similar discontinuities. The major material properties that need to be considered at the heterointerfaces are elec-tron affinity, work function, and relative permittivity. In the coaxial device, the interfaces we are concerned with are grouped in three categories: (1) metal-semiconductor (source-tube, drain-tube); (2) metal-insulator (source-oxide, drain-oxide, gate-oxide); and (3) semiconductor-insulator (tube-oxide). In a bias-free, intrinsic-nanotube CNFET, the first category would correspond to the presence of Schottky barriers at the endpoints of the tube, near the source and drain contacts. The potential at either endpoint depends on the work function of the metal contact, cp, and on the electron affinity, x, of the nanotube itself.5 Since the Fermi level of an intrinsic nanotube 4This statement pertains to DC modeling—in the AC analysis described in Chapters 9 and 10 of this work, the contact series resistances are included in the small-signal model of the CNFET. 5In a semiconductor, the work function <p — x + (Ec — Ep). 19 Chapter 1. Introduction lies exactly at midgap6, and in the cases where the metal and nanotube work functions are not identical, a Schottky barrier will be present near the contacts to maintain a flat Fermi level throughout the bias-free device (see, for example, Ref. [51]). Depending on the relative magnitude of the material properties, this barrier may be deemed positive or negative with regards to the carrier flow, and as will be shown below, the shape of a positive Schottky-barrier is critical in modulating the drain current. The presence of Schottky barriers in carbon nanotube transistors has already been explored thoroughly in various works [8,48,52-55]. The remaining heterointerfaces concern oxide-mismatch surfaces and are of importance to the solution of Poisson's equation, since they directly influence the permittivity-sensitive terms of Eq. (1.3). The CNFET insulator is typically homogenous, and previously fabricated devices have typically used Si0 2 (er « 3.9) [56], Zr0 2 (er ~ 25) [42] or Hf0 2 (er « 16) [57]. High-permittivity dielectrics are preferable, since they allow for better electrostatic coupling of the gate to the channel; employment of non-SiC>2 materials with Si technologies typically yields surface roughness problems that are detrimental to carrier mobility. The relative permittivity of the nanotube is taken to be unity [58], and the tube-insulator interface creates field-fringing effects and a slight variation in the potential profile on the tube surface that cannot be neglected. It must be noted that, although the solution of Eq. (1.3) encompasses the entire volume of the device, we are primarily concerned with the longitudinal potential profile on the surface of the tube, Vcs{.z)> since knowledge of band bending there is required for determination of the Schot-tky barriers and subsequent transport calculations. Furthermore, the key challenge in modeling the electrostatics lies in stipulating the charge distribution on the nanotube, as required on the right-hand-side of Poisson's equation. Prior to doing so, we further explore the electrostatics of the CNFET by analyzing solutions to Laplace's equation, i.e., in the absence of carriers on the tube (Qv = 0 in Eq. (1.3)). This is equivalent to observing the potential within the cylindrical system in the absence of the carbon nanotube. Since knowledge of local electrostatic potential allows us to determine any band bending imposed by the contacts, we subsequently derive 6 This is under the assumption of symmetrical density of states, as presented in Section 1.1.2. 20 Chapter 1. Introduction qualitative band diagrams that serve to illustrate some important behaviours of the devices. We begin by taking an arbitrary pair of voltages, and F ig . 1.9A illustrates the solution for the bias conditions of VQS = 0.3 V and VDS = 0.5 V , and device dimensions Rg = 1 nm and Lt = 50 nm. Note that the potential rises towards the drain end of the channel (z = Lt), and that in the mid-length region it is held at V « VGS, i-e-, the mid-length region of the device is entirely controlled by the gate electrode, tracking its voltage identically. 0 P (nm) 1 0 20 30 z (nm) 40 50 0.5 0.4 00 I 0.3 .2 c <D 0.2 CL 0.1 10 20 30 z (nm) (B) / ; 40 50 Figure 1.9: Complete solution (A) and cross-section of solution (B) to Laplace's equation in two-dimensional cylindrical coordinates for a cylinder 50 nm in length, 1 nm in diameter, and biased at VGS = 0.3 V and VDS = 0.5 V . F ig . 1.9B, a cross-section of the solution in F ig . 1.9A, taken at p = Rt = 0.5 nm, shows the nanotube surface potential, Vcs{z), the potential where the nanotube surface would be. The exact mathematical behaviour of the decaying end-regions of the plot is determined by the particular geometry, and in the cylindrical case it is specified by Bessel functions [47]. Moreover, in the short-channel l imit these regions wi l l interfere with one another and we cannot make the previous assumption of constant Vcs(Lt/2), as this only applies to device lengths greater than the sum of the decay lengths of both barriers. The mutually-perpendicular electric field components, i.e., radial and longitudinal, are shown in F i g . 1.10, for the same conditions given in F ig . 1.9. Alternate contact geometries wi l l undoubtedly affect these profiles. For instance, decreasing the 21 Chapter 1. Introduction x 10 ,6 x 107 0 > -2 Radial Component -4 -8 0 20 z (nm) 40 0 20 z (nm) 40 Figure 1.10: Electric field correspondent to the structure and potential described in in Fig. 1.9B. gate length, for the same device length, would reduce the field strength near the end electrodes, creating thicker potential barriers. Moreover, reducing the source and drain planar contacts to point (needle) contacts yields field focusing at the end regions and significantly thinner barriers. As previously stated, barring a choice of extreme geometries that are unfavourable to device performance, most of the results discussed here are readily generalized, qualitatively, to other CNFET structures. From the potential profile of Fig. 1.9B, we continue with an illustration of how the CNFET energy band diagram is constructed. Under the approximations of perturbation theory [59,60], which is valid for slow-changing potentials, we assume that relatively small variations in the local electrostatic potential—due to the application of a bias to the CNFET—cause the bands to rigidly shift with the local potential. This implies that the band structure obtained from the E-k relation does not get significantly distorted and that the density of states calculation does not need to be re-computed at each point, but rather just shifted in energy. As such, the conduction and valence band edges, Ec and Ev respectively, can be considered to be pinned to Vcs- This approximation is expected to be valid for transverse electric fields (radial component) Er < 8 x 108 V/m and tubes with radii Rt < 1.5 nm, worsening with larger radii and stronger Moreover, in the absence of mid-channel Ec discontinuities, the solution to Poisson's equation, fields [61]. 22 Chapter 1. Introduction which is always continuous, is here directly related to the amount of band bending of the vacuum level, such that Evac(z) = — qVcs(z)- This band bending also implies that there is an equal amount of bending in the conduction and valence bands, but the exact position of these will also be determined by discontinuities in work functions at the metal-nanotube interfaces. The band diagram is thus obtained via the relations Ec(z) = Evac(z) - x Ev(z) = Ec(z) - Eg , where Ec and Ev are, respective the conduction and valence band edges. Note that the potential at the points z — 0 and z = Lt are determined by both the voltage of the nearest electrode and also the work function difference there, either condition being sufficient to give rise to a Schottky barrier. We now examine the band diagrams derived from Laplace solutions, as shown in Fig. 1.11. The cases illustrated assume, for simplicity, that there are no work function differences between any of the electrodes, and they show four progressive stages, first stepping VQS and then VDS- A S noted before, the extent in the z-direction of the source barrier is independent of VDS, while the drain barrier "length" is a function of both bias voltages. The dependence of the band diagram on the work function difference between the nanotube and metal contacts will be shown later, in Fig. 4.4. Further analysis of these band profiles, in conjunction with the concept of ballistic transport introduced in Sect. 1.1.3, allow us to infer the nature of the nanotube carrier distribution. Due to the small thickness of the barriers, both tunneling and thermionically emitted components are considered. Observing Fig. 1.11C, we note that the asymmetrical source and drain barriers will give rise to different transmission probabilities, at any given energy, thus creating carrier distribution functions that are distorted from their equilibrium (Fermi) forms. At high VGS, where the conduction band edge at mid-length, Ec(Lt/2), dips significantly below the source Fermi level, the effect of hot carrier injection into the channel will dominate transport. It should be evident that these non-equilibrium distributions impose difficulties in the computation of 23 Chapter 1. Introduction (A) 0.5 > D ) L_ CD C UJ -0.5 (C) > 3, >. E> CD C • UJ V G S = 0 ' V D S = 0 10 20 30 40 50 z (nm) 0 10 20 30 40 50 z (nm) (B) > >> E> CP c • UJ (D) > 0.5 0 -0.5 03 C UJ -1.5 10 20 30 40 50 z (nm) V G S = 0 - 5 V ' V D S = 0 - 9 V 10 20 30 40 50 z (nm) Figure 1.11: CNFET band diagrams for Laplace solutions. There is no work function difference between the nanotube and any of the metal contacts. charge in the channel under the conditions of non-zero drain-source voltages. 1.3.2 Charge With the inclusion of the nanotube inside the empty gate cylinder, as in a real device, the potential profile will be affected by the presence of any electron- and hole-charges on the tube surface. Here we present a method for calculating the nanotube carrier concentration, necessary for the charge term in Eq. (1.3), via a ballistic-transport, flux-derived distribution point of view. In the case of short-channel devices, Schrodinger's equation is employed in order to account for quantum-mechanical reflection in the device; for long-channel devices, a compact model for expressing the charge is used by assuming that the mid-length region of the device is charac-terized by incoherent transport. Herein we discuss the latter case only, leaving Schrodinger's equation to subsequent chapters. Under equilibrium conditions, i.e., when VDS = 0 and there is no net charge flow, the carrier 24 Chapter 1. Introduction distributions axe given by the Fermi function. Application of a bias to the gate electrode is manifested as band-bending in the mid-length region of the device, as illustrated in Fig. 1.1 IB, with Ec dropping relative to the fixed Ep for VQS > 0, or rising for VQS < 0. In this situation, a self-consistent solution for the system is obtained by solving Poisson's equation in conjunction with the ID carrier density equation, which accounting for both carrier types is fOO Qv(z) =q(p-n)=q g(E) {/ [E + qVCS(z)] - f [E - qVCS(z)}} dE , (1.4) JO where g(E) is the tight-binding density of states from Sect. 1.1.2, f(E) is the Fermi function, and Vcs{z) is the amount of band-bending. Charge modulation is brought about by shifting the bands relative to the Fermi level, and for example, as the conduction band approaches the Fermi level (dotted line), there is an increase in the local ID electron concentration, with a dependence on AE = Ec — EF- Under a positive gate-source bias regime, this accounts for the accumulation of negative charges at the centre of the tube, attracted by the more positive gate and repelled by the more negative source. Note also that the application of a positive gate-source bias also causes the Schottky barriers, present near the source and drain contacts, to become taller and thinner relative to the mid-length conduction band edge, thereby reducing the restriction on tunneling currents from those terminals; note that these fluxes are balanced under equilibrium conditions. Self-consistency is important because application of Eq. (1.4) with an arbitrary potential profile will yield a net charge Qv(z) that is not consistent with that required by Eq. (1.3). As an example, if we computed the net charge with Eq. (1.4) on a Laplace band diagram under the conditions of VGS > 0 and VDS = u> a n d subsequently solved Eq. (1.3) with this charge, we would find an excess of electrons in the device that would yield a lower electrostatic potential observed in the mid-length region, thus raising Ec there. Recomputing the charge with the new potential profile we would find a discrepancy with the previously determined amount. Similarly, a large hole concentration, induced by the application of a negative gate potential, would cause the local electrostatic potential to increase, pulling the mid-length region of Fig. 1.9B downwards. Thus, it is important to iterate between both equations, using, for example, Newton's method, 25 Chapter 1. Introduction until a self-consistent solution is attained. Since the operation of the CNFET requires the application of biases to both the gate and drain contacts, with the aims of, respectively, inducing mobile charge in the channel and transporting that charge through the device, we now turn to the non-equilibrium situation. Under such conditions, the Fermi level only has meaning in the metal contacts, and the difference between Ep in the end contacts is responsible for the presence of a drain current. Under such conditions, one approach to charge calculation is to apply an equation similar to Eq. (1.4), except that the nanotube distribution functions are postulated to be shifted Fermi functions, having their equiprobable point, labeled as the quasi-Fermi level, lying somewhere in the energy range between the source and drain potentials. While this method yields satisfactory results, it fails to properly account for hot carriers in the channel. In this work, we calculate the nanotube non-equilibrium distributions based on incoming flux from both the source and drain contacts. The electrodes are highly-conductive regions assumed to be in local thermodynamic equilibrium, and thus electrons there are distributed according to the respective Fermi functions. Given that the device operates in the ballistic regime, electrons with sufficient energy to thermionically emit over or tunnel through the Schottky barrier at either end, populate the channel at the same energy as they entered. Thus the bands in the nanotube are populated such that the forward-traveling electron states, +A;-states, are primarily occupied by source-injected electrons, while drain-injected electrons primarily fill the —fc-states. This method therefore amounts to a tunneling-barrier modulation of the Fermi functions present in the source and drain metals, while accounting for backscattering due to macroscopic electrostatic potentials. Finally, CNFETs are different from most other semiconductor devices in that they can demon-strate, when properly engineered, bipolar and ambipolar behaviour—a single device can exhibit transport by either electrons or holes depending on the bias conditions, as well as simultane-ous transport by both carrier types under other conditions. The possibility of ambipolarity is clear in Fig. 1.11D, which reveals how source electrons and drain holes face similar potential 26 Chapter 1. Introduction barriers to injection into the channel. This phenomenon yields unusual current-voltage char-acteristics, which will be explored in later chapters. Recently, experimental observations of ambipolarity have been reported in the literature, manifested via either photoemission [62] or photoabsorption [63] in the infrared spectrum7, and opening up possibilities for applications as light detectors or emitters. 1.3.3 Transport As was described in Sect. 1.1.3, transport is expected to be ballistic for the device lengths and operating bias ranges chosen in this work [8]. Moreover, under the Landauer-Biittiker formalism, we treat the device as being composed of two carrier reservoirs, separated by a ID scattering region described by an energy-dependent transmission probability T{E). Each contact region is described by its own equilibrium carrier statistics f(E), and is assumed to also be one-dimensional. The connection of this contact region to the actual "macroscopic" electrode, wherein we have a 3D electron density, produces a quantized conductance, some-times described as a "mode constriction" resistance, and this phenomenon has been verified experimentally [12,21]. Under this formalism, we modify the standard electron current expres-sion I=—qnv (where the I replaces the usual J because of the ID nature of the system) to account for the energy dependence of its constituents8, and using the relations for electron den-sity n(E)=f(E)g(E)T(E), density-of-states g(E)=(l/ir)(dk/dE), and carrier group velocity v(E)=(l/h)(dE/dk), we find the net current between the source and drain to be where us and firj are, respectively, the source and drain Fermi levels. Note that for each contact we are only concerned with injection into the device, so we consider either +k or —k modes; thus we only use half of the g(E) given in Eq. (1.2), which accounted for both spin-degeneracy i and dtk. 7Due to the bandgap of ~ 0.7 eV being employed in typical devices, photoemission produces light of wavelength A « hc/Eg « 1.8 ^m, lying in the infrared range of the spectrum. 8Recall that, under the elastic conditions here, each energy "level" of transport is independent of all others. 27 Chapter 1. Introduction The transmission coefficient T(E) is a function of virtually all device parameters, but partic-ularly the gate- and drain-source voltages. Because of this, it will be shown in subsequent chapters that it is important to employ a Schrodinger-Poisson model to properly estimate the transmission, and that its behaviour is strongly dependent on the self-consistent computation of the device electrostatics. Conversely, the Fermi functions are solely functions of the contact Fermi energies, which are related to the drain-source voltage only. For example, applying a drain-source voltage lowers \ID in energy and depletes the backward flux, while the forward flux remains constant. This is responsible for the saturation of the current as VDS increases. 1.4 Thesis Outline The remainder of this thesis, prior to the concluding chapter, is a collection of manuscripts published in journals or conference proceedings while this work was being carried out. Chap. 2 presents a preliminary unipolar compact model, which employs simplified, long-channel elec-trostatics and a crude estimate of the transmission function. Chap. 3 shows a detailed study of the equilibrium electrostatics of the device, and, under non-equilibrium, employs the compact model in its bipolar implementation. Chap. 4 gives an overview of the compact model results, with the aim of evaluating the maximum attainable DC performance. Chap. 5 presents the ear-liest version of a Schrodinger-Poisson model and its results. Chap. 6 improves on the original compact model, particularly on the transmission coefficient, and provides a comparison with the more sophisticated model of Chap. 5. Chap. 7 is a study on the concept of quantum capacitance in nanoscale transistors, and its particular application to ID CNFET devices. Finally, a small-signal model, in conjunction with an improved Schrodinger-Poisson model (outlined in detail in Appendix A), is employed in Chapters 8, 9, and 10 to obtain AC performance predictions for CNFETs, namely the fx and / m a x transistor figures-of-merit. 1.5 Specific Contributions This work has contributed several models of varying detail to the growing body of knowledge in CNFETs. In particular, a novel Schrodinger-Poisson (SP) model gave results that were faithful 28 Chapter 1. Introduction to the trends observed in prototype devices and served as the foundation for developing a small-signal methodology from which high-frequency figures-of-merit were estimated. The inclusion of parasitics, previously not considered in the literature, proved to have a substantial influence in these predictions of small-signal AC performance. Also developed was a DC compact model, and its relation to the SP model was mathematically explained. Both showed good agreement in modeling high-performance negative-Schottky-barrier devices, and revealed that earlier DC performance predictions in the literature may have been too optimistic. A detailed analysis of quasi-bound states has also been provided, and this was found to be crucial in explaining device behaviour, particularly as regards quantum capacitance and transconductance. References [1] A. Oberlin, M. Endo, and A. T. Koyama, "Filamentous growth of carbon through benzene decomposition," J. Cryst. Grow., 32(3), 335-349 (1976). [2] Sumio Iijima, "Helical microtubules of graphitic carbon," Nature, 354, 56-58 (1991). [3] M. S. Dresselhaus, G. Dresselhaus, and P. C. 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Martel, S. Wind, and Ph. Avouris, "Field-modulated carrier transport in carbon nanotube transistors," Phys. Rev. Lett., 89(12), 126801-1-126801-4 (2002). 34 Chapter 1. Introduction [55] S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller, and Ph. Avouris, "Carbon nanotubes as Schottky barrier transistors," Phys. Rev. Lett, 89(10), 106801-1-106801-4 (2002). [56] S. Heinze, M. Radosavljevic, J. Tersoff, and Ph. Avouris, "Unexpected scaling of the performance of carbon nanotube Schottky-barrier transistors," Phys. Rev. B, 68, 235418-1-235418-5 (2003). [57] Ali Javey, Ryan Tu, Damon B. Farmer, Jing Guo, Roy G. Gordon, and Hongjie Dai, "High performance n-type carbon nanotube field-effect transistors with chemically doped contacts," Nano Lett, 5(2), 345-348 (2005). [58] Francois Leonard and J. Tersoff, "Dielectric response of semiconducting carbon nan-otubes," Appl. Phys. Lett, 81(25), 4835-4837 (2002). [59] Neil W. Ashcroft and N. 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Thus, the drain current tends to be controlled by modulation of the Schottky-barrier potential profiles at the source and drain ends of the nanotube [4,5]. Present experimental devices are planar in nature [4], but coaxial structures offer better opportunities for modulating the Schottky-barrier properties v ia capacitative coupling between the gate and the contacts [6,7]. In the present work we concentrate on coaxial, Schottky-barrier devices formed wi th intrinsic nanotubes, and seek to develop a compact model for the prediction of the drain / - V characteristics. The basic transistor structure is shown in F ig . 2.1. Under equilibrium conditions, which in these devices means no drain current, i.e., the drain-source voltage VDS = 0, but the gate-source voltage VQS is not necessarily zero, it is straight-forward to compute the charge and potential profile by solving Poisson's equation consistently with the equilibrium charge density on the nanotube [8]. For VGS > 0, as considered in this work, the charge on the tube is negative, and is due to a surfeit of electrons over holes. Here © [2002] IEEE. Reprinted, with permission, from L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a Compact Model For Schottky-Barrier Nanotube FETs," Proc. IEEE Con}, on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). 36 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs Figure 2.1: Schottky-barrier carbon nanotube F E T model geometry. The gate forms the curved surface of the outer cylinder, and the source and drain form the two ends. The semiconducting nanotube is placed coaxially with the outer cylinder. we take the electronic charge to dominate and we neglect the contribution of the holes. Away from equilibrium, when VDS > 0, the induced electron distribution on the tube wil l deviate considerably from a Maxwell ian or Fermi-Diracian form, on account of hot electron injection from the contacts and, in the ballistic case considered here, the lack of opportunity for ther-malizing collisions. This precludes the calculation of the non-equilibrium charge using simple, quasi-Fermi-Dirac statistics. In the present work, we obtain an estimate of the non-equilibrium charge Qc in the mid-length region of the tube, i.e., away from the source and drain potential barriers, by an extension of the method of Guo et al. [9,10]. Qc is related to the mid-length potential on the tube Vcs, which connects, and affects, the potential profiles of the Schottky barriers. These, in turn, affect the tunneling probabilities for electrons entering the tube from the reservoirs of equilibrium charge at the source and drain metallizations. The energy-dependent tunneling probabilities serve to distort the electron distribution in the tube from an equilibrium form. Here, we take the potential profiles at the barriers to have an exponential form, and then solve for Vcs by equating the values of Qc computed by the non-equilibrium-flux approach and an infinite-tube approach [9,10]. This gives a solution to the complete potential profile, and to the tunneling probabilities, which then allows computation of the drain current from Landauer's expression. Our inclusion of the Schottky-barrier nature of the contacts leads to a significantly different saturation current than predicted by the earlier model [9], in which the tunneling barriers were not considered. 37 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs 2.2 The Model At equilibrium, i.e., when VDS = 0, simple electrostatics gives: Qc = -Cins(VGS - VCS), (2.1) where Vcs is the equilibrium potential, with respect to the source, of the carbon nanotube at its mid-length, i.e., away from the influence of the source and drain contacts, and Cms is the insulator capacitance for an infinitely long coaxial system. Out of equilibrium, i.e., when VDS 0) Vcs is influenced by VDS-VCS = Vcs + ctVps Qc = -Cins(VGS-\Vgs), (2.2) where a is a parameter that needs to be determined in order to specify Vcs- F° r VGS > 0, as considered in this work, Qc is a negative, electronic charge. An alternative method for calculating Qc follows from the flux approach, in which electrons in the forward- and backward-directed fluxes are summed [9]. Here, we do not restrict the fluxes within the tube to be hemi-Maxwellian or hemi-Fermi-Diracian in nature, but, instead, we allow the actions of tunneling and repeated reflections between the potential barriers to modify the electron distributions from the equilibrium form that they possess outside the tube, at the actual source and drain metallic contacts. In the following, /s/2 denotes the positive- or forward-directed part of the Fermi-Dirac distribution outside the tube at the source, whereas /p/2 represents the negative- or backward-directed part of the distribution outside the tube at the drain. Thus: + £ 9 ( E ) M ^ (A _ !) r i E } , (2.3) where g(E) is the 1-D density of states computed from the tight-binding approximation [11], the conduction band edge Ec in the mid-length region of the tube is dependent on CUVDS, and the tunneling probabilities at the source and drain, Ts and Tp respectively, are computed using 38 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs the JWKB approximation and are dependent on E, Vcs and aVps- The overall transmission probability T* = TSTD/(TS + TD- TSTD). In our method, we equate Eq. (2.2) and Eq. (2.3) and solve for a, thereby determining Vcs- An iterative procedure is necessary because of the dependence on Vcs of Ts and To, via the potential profile, which are represented by •!(*-«) - il V(z) Vcs V C S 0 < z < a VCS a < z < L - a (2.4) -f(L-a-z) _ i l ,r Vcs - VDS Vcs L — a < z < L, eP-1 where z is the distance from the source and L is the tube length. This representation is based on the observation of the trends in barrier shape under those circumstane for which exact solutions are presently possible, namely: Poisson's equation at equilibrium, and Laplace's equation out of equilibrium. The trends are: a barrier basewidth of a s=s 2Rc, where Re is the radius of the gate (see Fig. 2.1); a barrier height at the source of Vpk = Vcs (see Fig. 2.2); a barrier height at the drain that varies from Vpk = Vcs — VDS when a "spike" is present, through Vpk = 0, to a negative value when VDS > Vcs (see Fig. 2.2); a barrier "concavity" that is captured by B « 3.6 for the device considered here. The barrier profiles are correct inasmuch as they prescribe values for the tunneling probabilities Ts and TD that yield a mid-length charge that is consistent with that predicted by Eq. (2.2). To complete the calculation of the drain current, the Landauer expression is used: ^ = E \ f°° tfsW - f^E' VDS)\ ?* dE, where the sum is over i doubly-degenerate conduction bands, the edges ECti of which are func-tions of a, Vcs and VDS-2.3 Results and Discussion Results are presented for a (16,0) tube, which has a radius of 0.63 nm, and a gate/tube-radius ratio of 10; the source and drain work functions are taken to be equal to that of the intrinsic nanotube. The tube is sufficiently long that there is a "mid-length" region where the tube potential Vcs is flat. In this region, we have found that there is essentially perfect agreement 39 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs between the values of Vcs calculated by the "infinite-tube" method (2.1), and by an exact 2-D solution [8], at least under the equilibrium conditions tested thus far. The energy band diagram for a variety of bias conditions is shown in F ig . 2.2. Note the Vc>5-dependence of Vcs and of the barrier shapes. 0.3 0.2 ^ 0.1 > „ |-0-1 jl -0.2 LU -0.3 -0.4 -0.5 1 6 w o " vcs V i-V, V G S > 0 V D S > V G S 0 < V D S < V G S o I f V G S > 0 V D S = ° 0 20 40 60 80 100 Distance from source contact (nm) Figure 2.2: Conduction energy band diagram for various bias conditions. -0.2 -0.1 0 0.1 0.2 Electron distribution ±k-states 0 " Drain-source voltage Figure 2.3: Electron distribution in the "mid-length" of the tube as a function of VDS for Vbs = 0 .5V. Electrons from the source and drain reservoirs are drawn into the tube by tunneling through, and thermionic emission over, the potential barriers at the contacts. This distorts the injected electron distributions from their equilibrium forms. The total distribution wi th in the tube is determined by the action of reflections at the Schottky barriers on the injected distributions. When VDS — 0 this action produces an equilibrium, Fermi-Dirac distribution, as can be seen in F ig . 2.3. A s VDS increases, there is less injection from the drain, and less reflection from the 40 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs diminishing "spike" at the drain. Thus, the backward-directed part of the distribution starts to disappear, and the forward part assumes a definitely non-equilibrium shape, with a bulge at the kinetic energy corresponding to that of the maximum tunneling flux. V G S = ° - 5 V V G S = 0.4V Drain-source voltage, V 0.4 0.5 DS Figure 2.4: Drain current-voltage characteristics. s 2 c 's Q _ V G S = °-5V — This work - Reference 9 0.1 0.2 0.3 0.4 0.5 Drain-source voltage, V DS Figure 2.5: Comparison of drain current-voltage characteristics at VGS — 0.5 V. The solid line is this work; the dashed line is using the model of Ref. [9]. The drain I- V characteristics are shown in Fig. 2.4. The saturation current at VQS = 0.5 V is around I p,A, which is not inconsistent with values emerging from prototype devices [12]. A revealing comparison with earlier predictions is shown in Fig. 2.5. Note how the present model indicates a considerably larger saturation voltage Vbs,sat- This is because the reflecting-action of the potential "spike" at the tube-drain interface delays the realization of the full saturation current. Also, note how the new model predicts an Io,sat that is about one-order of magnitude less than that of the model of Guo et al. [9]. This is indicative of the importance of accounting for the restrictive action that the Schottky barriers at the source and drain impose on the current. 41 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs 2.4 Conclusions From this work on the modeling of coaxial, carbon nanotube FETs, it can be concluded that the Schottky barriers at the source and drain contacts play a dominant role in determining the I- V characteristics of the transistors. The model presented here represents a significant step towards producing a compact model for these promising new nano-devices. References [1] V. Derycke, R. Martel, J. Appenzeller, and Ph. Avouris, "Carbon nanotube inter- and intramolecular logic gates," Nano Lett, 1(9), 453-456 (2001). [2] Adrian Bachtold, Peter Hadley, Takeshi Nakanishi, and Cees Dekker, "Logic circuits with carbon nanotube transistors," Science, 294, 1317-1320 (2001). [3] Paul L. McEuen, Michael S. Fuhrer, and Hongkun Park, "Single-walled carbon nanotube electronics," IEEE Trans. NanotechnoL, 1(1), 78-85 (2002). [4] J. Appenzeller, J. Knoch, V. Derycke, R. Martel, S. Wind, and Ph. Avouris, "Field-modulated carrier transport in carbon nanotube transistors," Phys. Rev. Lett., 89(12), 126801-1-126801-4 (2002). [5] S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller, and Ph. Avouris, "Carbon nanotubes as Schottky barrier transistors," Phys. Rev. Lett., 89(10), 106801-1-106801-4 (2002). [6] Christopher P. Auth and James D. Plummer, "Scaling theory for cylindrical, fully-depleted, surrounding-gate MOSFET's," IEEE Electron Device Lett, 18(2), 74-76 (1997). [7] Brian Winstead and Umberto Ravaioli, "Simulation of Schottky barrier MOSFET's with a coupled quantum injection/Monte Carlo technique," IEEE Trans. Electron Devices, 47(6), 1241-1246 (2000). 42 Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs [8] D. L. John, Leonardo C. Castro, Jason Clifford, and David L. Pulfrey, "Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors," IEEE Trans. NanotechnoL, 2(3), 175-180 (2003). [9] Jing Guo, Mark Lundstrom, and Supriyo Datta, "Performance projections for ballistic carbon nanotube field-effect transistors," Appl. Phys. Lett, 80(17), 3192-3194 (2002). [10] Jing Guo, Sebastien Goasguen, Mark Lundstrom, and Supriyo Datta, "Metal-insulator-semiconductor electrostatics of carbon nanotubes," Appl. Phys. Lett., 81(8), 1486-1488 (2002). [11] Keivan Esfarjani, Amir A. Farajian, Yuichi Hashi, and Yoshiyuki Kawazoe, "Electronic, transport and mechanical properties of carbon nanotubes," Y. Kawazoe, T. Kondow, and K. Ohno, eds., Clusters and Nanomaterials—Theory and Experiment, Springer Series in Cluster Physics, 187-220 (Springer-Verlag, Berlin, 2002). [12] Sami Rosenblatt, Yuval Yaish, Jiwoong Park, Jeff Gore, Vera Sazonova, and Paul L. McEuen, "High performance electrolyte gated carbon nanotube transistors," Nano Lett, 2(8), 869-872 (2002). 43 Chapter 3 Electrostatics of Coaxial Schottky-Barrier CNFETs 3.1 Introduction Carbon nanotubes (CNs) are being intensively investigated as possible structures from which nanoscale transistors and logic gates might be fabricated [1,2]. In devices where the gate electrode covers the entire length of the nanotube, transistor action is achieved by the mod-ulation, by the gate, of the potential profile at the Schottky-barrier contact appearing at the source-tube interface [3,4], rather than by the modulation of the channel properties, as in a traditional, silicon-like field-effect transistor ( F E T ) [5,6]. Here we concentrate on the coaxial Schottky-barrier carbon nanotube F E T ( S B - C N F E T ) , in which the cylindrical gate surrounds the tube and is insulated from it by a dielectric. The basic structure was shown previously in F ig . 2.1. Whereas planar structures are currently being used experimentally [4], coaxial structures, although much more difficult to fabricate, are likely to exhibit better short-channel performance [7], and, as regards modulating the Schottky-barrier thickness v ia capacitative coupling between the gate and the contact [8], are likely to prove more efficient. As a first step towards providing a model for these new devices, we examine the electrostatics © [2003] IEEE. Reprinted, with permission, from D. L. John, L . C. Castro, J . Clifford, and D. L . Pulfrey, "Electrostatics of Coaxial Schottky-Barrier Nanotube Field-Effect Transistors," IEEE Trans. NanotechnoL, 2(3), 175-180 (2003). Note: the Appendix (An Analytical Solution for the Potential) present in the published version is omitted here, as that was prepared solely by the journal paper's first author. 44 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs of coaxial SB-CNFETs, using both analytical and numerical procedures to obtain the potential profile. Of course, the solution must be consistent with the electron and hole charge induced on the surface of the nanotube, and any inherent charge, such as that due to dopants. Although the latter are easily accommodated, they are not considered here in view of the findings that procedures previously thought to lead to doping of a nanotube are more probably serving to affect the work functions at the metal contacts to the CN [4]. Thus the nanotubes here axe considered to be intrinsic. The electron and hole charge densities can be computed using the nearest-neighbour tight-binding approximation for the nanotube density of states (DOS) [9]. Results are presented here for the equilibrium situation, i.e., the drain-source voltage, Vps, is zero, as this is presently the only case for which the carrier distribution functions are known with certainty. The dependence of the potential profile along the tube on the work functions of the source-, drain- and gate-metallizations, and of the thickness and permittivity of the gate dielectric, is reported. Outside of equilibrium, i.e., for VDS 0, the distribution functions are likely to be highly distorted from their equilibrium shape [10]. This is due to the absence of thermalizing collisions in this one-dimensional (1-D) system, for which there is very little carrier-phonon interaction [11]. Presently, the only way to obtain exact results for V D S 0 is to solve Laplace's equation. This may be appropriate for studying sub-threshold conduction. Such results are presented here and they indicate a sub-threshold slope which depends on dielectric thickness in a different manner from that recently reported for planar-geometry SB-CNFETs [12]. Although, as stressed above, a procedure for obtaining a fully self-consistent solution for the above-threshold case is not yet available, the results presented here for the equilibrium potential profiles can be used to infer the general form of the drain current-voltage (I- V) characteristic. On doing this, the interesting spectre of having simultaneous injection of electrons and holes into the nanotube is raised. The drain characteristics for such a situation are briefly examined using a rudimentary, non-equilibrium, compact model [10], in which the source and drain potential profiles are approximated by exponential expressions that have their basis in the electrostatic solutions presented herein. 45 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs 3.2 Coaxial Nanotube Electrostatics The electrostatic problem reduces to that of a bounded cylinder of length Lt and radius Rg, as shown in Fig. 2.1. In cylindrical coordinates, using the source electrode as reference, the appropriate boundary conditions for the potential V(p, 9, z) are: V(Rg,e,z) = V G S - 4 > G / q v(P,e,o) = -4>s/q V(p,6,L) = V D S - c f > D / q V(p,9,z) = V(p,d + 2K,z), where (f>c, <f>s and <PD are the work functions of the gate-, source- and drain-metallizations, respectively, VGS is the gate-source voltage, and q is the magnitude of the electronic charge. The boundary conditions at z = 0 and z = L are appropriate in the absence of Fermi-level pinning [13]. Note, too, that V(0,9,z) is assumed to be finite. An analytical solution, at least for the case of a homogeneous permittivity within the metallized enclosure, is possible following the methods of [14] and [15]. For the inhomogeneous case of different permittivities for the dielectric and the nanotube, numerical techniques are easier to implement. We have used a standard finite-element package for this purpose1. The net charge density, comprising electrons and holes, is taken to reside on the surface of the CN, and is given by Q{r) = ^ - p S ( P - Rt)Qz(z), where Qz(z) is the net 1-D charge concentration, Rt is the CN radius, and 5(x) is the Dirac delta function. The nanotube charge needs to be computed self-consistently with the potential on the nanotube but, as mentioned in Sect. 3.1, a difficulty arises under non-equilibrium con-ditions because of the present inability to rigorously specify the distribution function for the hot carriers within the tube. However, for equilibrium conditions, there is no such problem and the carrier concentrations are found by allowing the local electrostatic potential to rigidly shift F E M L A B , see http://www.comsol.com 46 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs the CN DOS [14,15]. Using the nearest-neighbour tight-binding approximation [9], the DOS is symmetrical about EF, S O the net 1-D carrier density at some point along the intrinsic tube may be computed as where g(E) is the 1-D tube DOS, the degeneracy in the energy bands is as considered in [16], f(E) is the Fermi-Dirac distribution function, EF is taken to be zero, and fa is the work function of the intrinsic carbon nanotube. For the non-equilibrium case, the only exact solution that can be given presently is for the case of no charge on the nanotube, namely: Qz{z) = 0. A complete solution awaits the formulation of an appropriate solver, likely of the Poisson-Schrodinger variety. 3.3 Results and Discussion Results are presented for (16,0) tubes having a radius of 0.63 nm, a length of 100 nm and a gate work function of 4.5 eV. Various ratios of gate radius to tube radius, relative permittivity of the dielectric, €i n s, and source- and drain-work functions, are considered. The electron affinity for the carbon nanotube is taken to be 4.18eV, based on a wOrk function of 4.5eV [17], and an intrinsic-tube band gap of 0.64 eV. Unless otherwise stated, the relative permittivity of the nanotube, et, is taken to be the same as that of the gate dielectric. The temperature is taken to be 300 K. At equilibrium conditions, and when cj>s = 4>D, the potential profile along the tube will be symmetrical. Thus, only profiles near one contact need be shown. Fig. 3.1 shows the energy band diagrams near the source for Rg/Rt = 10, e;ns = 3.9 and for various (j>s = <PD with VDS — 0 V and VGS = 0.2 and 0.5 V. In Fig. 3.1(a), <ps = 4.5 eV and corresponds to the case of equal work functions for the metal and the nanotube, whereas cps = 4.33 eV [Fig. 3.1(b)] and <j>S — 4.63eV [Fig. 3.1(c)] refer to low- and high-metal work functions, respectively [13]. The potential in the body of the tube, distal from the contacts, depends directly on VGS, leading to potential spikes in the tube at the source and drain of height determined by both <fis,D and g(E) [f(E + qV(z) + <f>t) ~ f(E - qV(z) - fa)\ dE, 47 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs o 20 40 Distance from source (nm) Figure 3.1: Self-consistent equilibrium energy band diagram near the source for a (16,0) tube with a 5.6 nm gate dielectric thickness (Rg/Rt = 10), and 4>s = 4>D set to (a) 4.5 eV, (b) 4.33 eV, and (c) 4.63eV. Data are for VDS = 0V and VQS = 0.2 (dashed line) and 0.5V (solid line). Energies are with respect to the Fermi level (dotted line). VQS- Only in the low-</>s case at low VQS is thermionic, emission likely to make a significant contribution to the source current. In all other cases shown in Fig. 3.1, tunneling of electrons through the spike will dominate. The band diagrams for the same work function cases as used in Fig. 3.1, but for Rg/Rt = 50, are shown in Fig. 3.2. The reduced band bending in the tube at the contacts, due to poorer coupling between the gate and the nanotube, is very evident and will lead to a dramatic decrease in current, except in the low work function case at low VQS where, as mentioned previously, the electron current will be due to thermionic emission and will be determined by the height, and not the shape, of the barrier. The present state of the art as regards gate-dielectric thinness is 2nm [12]. Regarding the permittivity of the dielectric, recent work has reported the use of 48 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs Distance from source (nm) Figure 3.2: Self-consistent equilibrium energy band diagram near the source for a (16,0) tube with a 30.9 nm gate dielectric thickness (Rg/Rt = 50), and the same sequence of work functions as in Fig. 3.1, namely: cps = <f>D set to (a) 4.5eV, (b) 4.33eV, and (c) 4.63eV. Data are for VDS = 0 V and VQS = 0.2 (dashed line) and 0.5 V (solid line). Energies are with respect to the Fermi level (dotted line). zirconia [6], for which e;ns is around five times higher than that used in obtaining the above figures. The effect of such a change in £in s can be seen by comparing Figs. 3.1 and 3.3. At VGS = 0.5 V, the increased capacitative coupling between the gate and the tube drives the mid-tube potential energy to lower values, yet does not change significantly the width of the source barrier at its base. Thus, obviously, an increased current for a given bias will result from using a higher €i n s. At lower VGS,  e-9-, 0.2 V, the increased ei n s makes essentially no difference to the potential profile because, at least for Rg/Rt = 10, there is virtually no charge induced on the tube. From Figs. 3.1, 3.2 and 3.3, it appears that the width of the potential barrier at its base depends strongly on the radius of the gate R g for the contact geometry considered here, as has been remarked upon elsewhere [18], and here we indicate that it also depends barely at 49 Chapter 3. Electrostatics of Coaxial Schottky-Bairier CNFETs o 20 40 Distance from source (nm) Figure 3.3: Self-consistent equilibrium energy band diagram near the source for a (16,0) tube with a 5.6 nm gate dielectric thickness (Rg/Rt = 10), a gate dielectric with permittivity five times higher than used in Fig. 3.1, and the same sequence of work functions as in Fig. 3.1, namely: 4>s = <f>D set to (a) 4.5eV, (b) 4.33eV, and (c) 4.63eV. Data are for VDS = 0V and VGS = 0.2 (dashed line) and 0.5 V (solid line). Energies are with respect to the Fermi level (dotted line). all on VQS-Note that the effect of changing the gate work function from the value of 4.5 eV used here can be readily appreciated from the foregoing figures as an increase in (f>Q of 0.1 eV, for example, has the same effect as a corresponding decrease in VGS. Turning now to the non-equilibrium case, as mentioned above, only the case of zero charge on the nanotube can be solved for exactly, pending the determination of the appropriate distribution functions or wave-functions for the hot carriers. In sub-threshold, the charge accumulation is not significant in the electrostatics solution, thus, we solve Laplace's equation in order to study the behaviour in this regime [12]. The current is computed from the Landauer formula, 50 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs assuming ballistic transport and incorporating the effect of carrier transmission and reflection at the internal source/tube- and drain/tube-barriers in the transmission coefficient, T, computed under the WKB approximation, Some results for various gate/tube dimensions are shown in Fig. 3.4: note that the dielectric thickness t[ns = R g - Rt- The sub-threshold current can be due to either mainly electrons or mainly holes, depending on the bias conditions. For VDS = 0.1 V, as used in obtaining Fig. 3.4, at VGS = 0 there is no band bending at the source end of the tube, but there is a "spike" in the valence band at the drain end of the tube, which permits a hole tunneling current. This current increases as VGS becomes more negative. For positive values of V G S , a spike appears in the conduction band at the source, thereby allowing electron tunneling, but the spike in the valence band at the drain thickens, and so the hole current is reduced. These changes with VGS lead to a minimum in current, which occurs at VGS = V D S / 2 , when the tunneling barriers for source electrons and drain holes are of equal thickness. The magnitudes of the sub-threshold slopes, 151 = |(d log 1 0 / /dVGs) _ 1 | , for both mainly electron conduction (VGS > Vbs/2), and mainly hole conduction (VGS < K D S / 2 ) , are identical, owing to the use here of a symmetrical DOS function for the conduction- and valence-bands, and equal work functions for the source and drain contacts. In analyzing their planar SB-CNFETs, Heinze et al. [12] noted that changing the dielectric thickness is equivalent to rescaling the gate voltage. Thus, they found that S and the tube potential scaled with dielectric thickness in the same manner, i.e., as \Ains- This suggests that we seek a scaling relationship for S in our coaxial devices by examining how the VQS-dependent part of the potential in the vicinity of the source contact varies with t-ms. This can be accomplished by expanding the first term of the first mode of an analytical Laplace solution (Ref. [19], Appendix), i.e., 4(gVbs ~ 0G)IQ {irRtLr 1) nzL; qirl0 {itRgLt 1) - l V ~ 51 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs 1(f5 § 10"7 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Gate-source voltage (V) Figure 3.4: Sub-threshold current for the same tube properties as in Fig. 3.1(a), but with various ratios of gate radius to tube radius, for VDS = 0.1 V. Note: the dielectric thickness is R g — Rt-where Io is the zeroth-order Modified Bessel Function of the First Kind. Thus dV 1 d V G S  a Io (itRgLT 1) and, therefore, we can expect 5 ~ a l 0 ( / 3 t i „ s ) , where a and B are fitting parameters. It is found that a reasonable fit to S, from the data included in Fig. 3.4, results with a « 79mV/decade and B « 0.15 n m - 1 . The fact that a fit can be obtained confirms that the dependence of S on i j n s is related to the specific geometry of the transistor, with a Bessel function being involved in this case because of the cylindrical structure. Electrically, t;ns is related, of course, to the gate capacitance, through which VGS is coupled to the CN potential. Considering now the case of above-threshold conduction, the solution to Laplace's equation for various values of VDS is worth examining as it gives an idea of the evolution of the barrier profiles with drain-source voltage. The exact solutions for the drain end of the tube show that the conduction-band spike diminishes with VDS, and that, on further increasing VDS, a spike occurs in the valence band. The base widths of the potential profiles at both source and drain are found, as in the equilibrium case, to be of the order of R g . This fact has been used, in 52 : 1 r RG/RT = 2 (S - 76 mV/dec) n • RG/RT = 5 (S = 80mV/dec - — Rg/Rj. - 10 (S - 99 mV/dec) RQ/RT - 20 (S = 150 mV/dec) ** - ^ N \ \ \ (16,0) tube / \ v D S = o . iv / ^ \ e'=1 /' / / t '/ r \ * ' • . . \ Hole Majority ^^NvwV^' Electron Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs conjunction with a self-consistent procedure for solving for the potential and the charge in the mid-length, field-free region of the tube, to generate compact expressions which describe the potential along the entire length of the tube as a function of V G S and VDS [ 10 ] . 0 10 90 100 Distance from source (nm) Figure 3 . 5 : Energy band diagrams for the same tube properties as in Fig. 3.1(a) for V G S = 0 . 3 V and various VDS'- 0 V (dotted), 0 . 2 V (dashed), 0 . 5 V (solid). For the equilibrium case the profile is the exact solution of Poisson's equation; for VDS 0 the solutions are from Castro's compact model [ 10 ] , with the base widths of the potential profiles at the source and drain being taken as 2Rg. Results are shown in Fig. 3 . 5 for the SB-CNFET for which the equilibrium band diagram is shown in Fig. 3.1(a). As mentioned, the barrier at the drain for electron flow from the nanotube into the drain diminishes as V D S is increased. This will lead to reduced, reflection of source-injected electrons and an increase, and eventual saturation, of the electron current. Further increase in VDS causes a spike to appear in the valence band profile at the drain. This will allow hole tunneling to occur, and raises the interesting prospect of a hole current issuing from the drain contact and adding to the electron current, thereby leading to a significant increase in the total drain current. When the electron current is small due to the absence of a tunneling barrier at the source, as will occur at low Vgs, the drain current will be due almost entirely to holes, and the drain I- V characteristic will appear near-exponential in shape. An illustrative drain I- V characteristic is shown in Fig. 3 . 6 . This data is generated from the compact model of Castro et al. [ 10 ] , as used to produce the potential profiles in Fig. 3 . 5 , with the base width of the potential barriers being taken as equal to 2Rg. Castro's model is based on that of Guo et al. [5], and improves upon it by: 1 ) introducing Schottky barrier contacts at 5 3 Chapter 3. Electrostatics of Coaxial Schottky-Bairier CNFETs Drain-source voltage (V) Figure 3 . 6 : Drain I- V characteristics, as calculated by the method of Ref. [ 10 ] , for a ( 1 6 , 0 ) tube, Rg/Rt = 1 0 , 4>s = (/>D = 4 . 5 eV and various values of VGS-the source and drain; 2 ) accounting for reflection of carriers in the tube between the source and drain barriers; 3 ) allowing for simultaneous electron and hole flows; and 4 ) not demanding that the charge in the mid-length tube region remain at its equilibrium value. It is an approximate model due to the estimated shape of the barrier profiles, so the current magnitudes given in Fig. 3 . 6 are also only approximate. However, they do confirm the evolution of the drain characteristics, as inferred from the above discussion of the exact potential profiles displayed in the present work. Furthermore, our predictions are consistent with very recent experimental results that show near-exponential drain I- V characteristics for V G S < V D S [20] . As a final comment on the simultaneous presence of electrons and holes in the nanotube, some recombination is to be expected, in which case the drain current will be less than the sum of two noninteracting particle flows, and in practice may not increase as dramatically as indicated here. Experimentally, evidence of recombination within the nanotube of a SB-CNFET has been demonstrated via the measurement of light emission under bias conditions appropriate for the simultaneous injection of holes and electrons [20] . 5 4 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs 3 . 4 Conclusions From this work on the electrostatics of coaxial Schottky-barrier carbon nanotube FETs, the following can be concluded. 1. The potential barriers at the source/nanotube and drain/nanotube interfaces are strongly affected by the work functions of the source, drain and gate, and by the thickness and permittivity of the dielectric that surrounds the nanotube. 2. An analytical solution for the potential distribution in the case of equal permittivities of the gate dielectric and the nanotube gives a good approximation to the numerical solution for the case when the difference in permittivities of the dielectric and tube is taken into account. In other words, the radial field inside the nanotube, for this particular geometry, is not of great importance. 3. The sub-threshold slope approaches the thermionic limit of «60mV/decade as the di-electric thickness is reduced, in a manner consistent with the cylindrical geometry of the device. 4. From the results presented here, trends in the above-threshold drain /- V characteristics can be inferred. The possibility of contributions to the drain current from both electron and hole flow is indicated. References [1] V. Derycke, R. Martel, J. Appenzeller, and Ph. Avouris, "Carbon nanotube inter- and intramolecular logic gates," Nano Lett, 1(9), 453-456 (2001). [2] Adrian Bachtold, Peter Hadley, Takeshi Nakanishi, and Cees Dekker, "Logic circuits with carbon nanotube transistors," Science, 294, 1317-1320 (2001). [3] S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller, and Ph. Avouris, "Carbon nanotubes as Schottky barrier transistors," Phys. Rev. Lett., 89(10), 106801-1-106801-4 (2002). 55 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs [4] J. Appenzeller, J. Knoch, V. Derycke, R. Martel, S. Wind, and Ph. Avouris, "Field-modulated carrier transport in carbon nanotube transistors," Phys. Rev. Lett., 89(12), 126801-1-126801-4 (2002). [5] Jing Guo, Mark Lundstrom, and Supriyo Datta, "Performance projections for ballistic carbon nanotube field-effect transistors," Appl. Phys. Lett, 80(17), 3192-3194 (2002). [6] Ali Javey, Hyoungsub Kim, Markus Brink, Qian Wang, Ant Ural, Jing Guo, Paul Mclntyre, Paul McEuen, Mark Lundstrom, and Hongjie Dai, "High-K dielectrics for advanced carbon-nanotube transistors and logic gates," Nature Mater., 1, 241-246 (2002). [7] Sang-Hyun Oh, Don, Monroe, and J. M. Hergenrother, "Analytic description of short-channel effects in fully-depleted double-gate and cylindrical, surrounding-gate MOSFETs," IEEE Electron Device Lett, 21(9), 445-447 (2000). [8] Brian Winstead and Umberto Ravaioli, "Simulation of Schottky barrier MOSFET's with a coupled quantum injection/Monte Carlo technique," IEEE Trans. Electron Devices, 47(6), 1241-1246 (2000). [9] Keivan Esfarjani, Amir A. Farajian, Yuichi Hashi, and Yoshiyuki Kawazoe, "Electronic, transport and mechanical properties of carbon nanotubes," Y. Kawazoe, T. Kondow, and K. Ohno, eds., Clusters and Nanomaterials—Theory and Experiment, Springer Series in Cluster Physics, 187-220 (Springer-Verlag, Berlin, 2002). [10] L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a compact model for Schottky-barrier nanotube FETs," Proc. IEEE Conf. on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). [11] Paul L. McEuen, Michael S. Fuhrer, and Hongkun Park, "Single-walled carbon nanotube electronics," IEEE Trans. Nanotechnol, 1(1), 78-85 (2002). [12] S. Heinze, M. Radosavljevic, J. Tersoff, and Ph. Avouris, "Unexpected scaling of the performance of carbon nanotube Schottky-barrier transistors," Phys. Rev. B, 68, 235418-1-235418-5 (2003). 56 Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs [13] Francois Leonard and J. Tersoff, "Role of Fermi-level pinning in nanotube Schottky diodes," Phys. Rev. Lett, 84(20), 4693-4696 (2000). [14] Francois Leonard and J. Tersoff, "Novel length scales in nanotube devices," Phys. Rev. Lett, 83(24), 5174-5177 (1999). [15] Arkadi A. Odintsov, "Schottky barriers in carbon nanotube heterojunctions," Phys. Rev. Lett, 85(1), 150-153 (2000). [16] M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes (Academic Press, Toronto, 1996). [17] Sander J. Tans, Alwin R. M. Verschueren, and Cees Dekker, "Room-temperature transistor based on a single carbon nanotube," Nature, 393, 49-52 (1998). [18] Takeshi Nakanishi, Adrian Bachtold, and Cees Dekker, "Transport through the interface between a semiconducting carbon nanotube and a metal electrode," Phys. Rev. B, 66, 073307-1-073307-4 (2002). [19] D. L. John, Leonardo C. Castro, Jason Clifford, and David L. Pulfrey, "Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors," IEEE Trans. NanotechnoL, 2(3), 175-180 (2003). [20] J. A. Misewich, R. Martel, Ph. Avouris, J. C. Tsang, S. Heinze, and J. Tersoff, "Electrically induced optical emission from a carbon nanotube FET," Science, 300, 783-786 (2003). 57 Chapter 4 An Evaluation of C N F E T D C Performance 4.1 Introduction Carbon nanotube molecules can be either metallic or semiconducting, which raises the fas-cinating spectre of filamentary integrated circuits fashioned from nanoscale transistors and interconnects [1]. Such circuits would be of a different form from that of today's silicon circuits, and would surely lead to a host of new applications. The atoms within a carbon nanotube molecule bond covalently in hexagonal rings, and this graphite-like structure has great strength and stability. Electrically, this helps in significantly reducing electromigration, thereby allow-ing high-current operation. Furthermore, carbon nanotubes conduct heat nearly as well as diamond, so extremely high device-packing densities should be possible. When one adds to these advantageous properties the fact that the bandgap of carbon nanotubes can be tuned by varying the tube diameter, then it is clear that carbon nanotubes are worth investigating as a candidate molecule for applications in nanotechnology. The integrity of carbon nanotube molecules may preclude substitutional doping, but adsorbed material [2], and ions within the nanotube [3], can alter the tube's conductivity from its natural intrinsic state. Thus, bipolar junction devices are possible, perhaps operating in single-electron mode with a p-type quantum well between two n-type regions [2]. In this paper, we focus on A version of this chapter has been published. L. C. Castro, D. L. John, and D. L . Pulfrey, "Carbon Nanotube Transistors: A n Evaluation," Proc. SPIE Conf. Device and Process Technologies for MEMS, Microelectronics, and Photonics III, vol. 5276, 1-10 (Perth, Australia, 2004). 58 Chapter 4. An Evaluation of CNFET DC Performance transistors made from intrinsic nanotubes, in which the conductivity is modulated by a gate electrode, i.e., carbon nanotube field-effect transistors (CNFETs). These devices have been under experimental investigation since 1998 [4], and engineering models to aid in their design and analysis are starting to appear [5-11]. Thus, it is timely to assess the performance charac-teristics of CNFETs, at least under DC operation, for which there is considerable experimental data. 4.2 Fabrication Metallic and semiconducting nanotubes are often sorted on the basis of tube diameter, as measured by atomic force microscopy [12], but recent reports of covalent chemical functional-ization should make this task more practical [13]. Successful attempts to merge metallic and semiconducting nanotubes have been reported [14], but this desirable union has not yet been incorporated into experimental CNFETs. Instead, these devices presently rely for their fab-rication on the traditional microelectronic techniques of metal deposition and electron-beam lithography to make and define metal contacts to the nanotube, which forms the channel of the transistor. The nanotubes themselves have either been grown in-situ by CVD from catalytic island sites [15], or dispersed from ropes produced by laser ablation of a catalyst-containing carbon target, and positioned on-chip by atomic force microscopy [16]. Clearly, if there is to be any hope of mass-production of CNFETs, some method of organized growth, or self-assembly is necessary. Progress in this direction includes aligned growth in an electric field [17], and growth of arrays of vertical tubes within alumina nanopores [18]. 4.3 Theoretical Considerations Carbon nanotubes are cylinders of graphene, with a wall thickness of one atomic layer. For small diameter tubes, quantization of the wave vectors in the circumferential (transverse) direction leads to a quasi-one-dimensional energy versus wave vector (e-k) relation, with a continuum of states only in the longitudinal direction. In many cases, this entails the opening of a bandgap, which for a 1 nm-diameter tube, for example, is about 0.8 eV. The band structure is usually 59 Chapter 4. An Evaluation of CNFET DC Performance determined from a tight-binding Hamiltonian, with one p-orbital per carbon atom and a nearest-neighbour matrix element of about 2.8 eV [19]. This approach renders the conduction- and valence-band structures symmetrical. Note that each of the transverse wave vectors corresponds to an allowed mode of propagation, of which there are two in the lowest sub-band, the most relevant in transport calculations. The small number of propagating modes is in contrast to the very large number of modes in bulk terminals to which a device or circuit must ultimately be connected. This mode-constriction leads to a quantized, interfacial conductance, which will exist even if conduction down the nanotube itself is ballistic [20]. In the modeling of carbon nanotube diodes and field-effect transistors, it is becoming customary to locate such interfaces at the end contacts to a semiconducting nanotube, and to regard the nanotube as an object with a characteristic transmission probability T [7,11,21]. The end contacts are viewed as reservoirs of charge, maintained under equilibrium conditions, from which carriers are injected into the nanotube, depending on T and the applied bias, e.g., the drain-source voltage VDS of a CNFET. This Landauer formalism, as described in, e.g., Ref. [20], allows the net electron current for two modes to be expressed as where q is the magnitude of the electronic charge, h is Planck's constant, Em\& is the energy of the conduction band edge in the mid-section of the nanotube, f(E) is the Fermi-Dirac distribution function, and Te refers to electrons. Clearly, the lower limit of integration would need to be changed if source-drain tunneling were an issue. A similar expression can be written for holes, using a transmission probability 7},. Thus, once the T's are known, the problem of the DC drain current is solved. Present CNFETs are usually planar devices, with a single gate situated either above [22], or below [16], the gate dielectric and the nanotube. Double-gated devices [23], and electrolyt-(4.1) 4.4 D C Modeling 60 Chapter 4. An Evaluation of CNFET DC Performance ically gated devices [24], more closely resemble the coaxial structure wi th wrap-around gate that would be ideal, given the cylindrical form of the nanotube. A s we are interested in ultimate-performance assessment, we take a coaxial geometry for modeling purposes, as illus-trated in F ig . 4.1. If the source and drain end contacts are the same radius as the gate, a closed metallic cylinder results, for which there is an analytical solution for the device electro-statics [8]. For smaller diameter end contacts, or for partially gated devices in which the gate electrode does not cover the entire length of the nanotube, an open-boundary problem exists, and numerical methods need to be employed [10,25]. Figure 4.1: Coaxial C N F E T model geometry. The transmission probabilities T are functions of energy and local potential V(p,9,z). The latter is related to charges within the system, such as: electrons, holes and adsorbed ions on the nanotube; charges trapped within the gate dielectric or the nanotube; charges residing in so-called metal-induced-gap-states (MIGS) , or evanescent states wi th in the semiconductor bandgap near the junctions. Here, we consider only electrons and holes, which are taken to reside on the surface of the tube, i.e., Q{T) = ~ 5 { p - R t ) Q z { z ) , where Qz(z) is the net 1-D carrier concentration, Rt is the nanotube radius, and 8(x) is the Dirac delta function. If we allow for a spatially varying permitt ivity in the radial direction, Gauss' Law implies that: (9V_ f l l d e \ dV + d*V_ Q(r) dp 2 \p edpj dp dz 2 e 61 Chapter 4. An Evaluation of CNFET DC Performance where the potential within the device reduces to a function of just two cylindrical coordinates V(p,z), due to the symmetry of the device in the angular direction. The charge Qz{z) is distributed within the allowed 1-D density-of-states (DOS), which is taken to be rigidly shifted by V(p,z). The solution for the potential is relatively straightforward under equilibrium conditions because Fermi-Dirac statistics apply not just in the metal contacts, but everywhere along the tube, and can be used to compute the electron and hole concentrations [8]. In the absence of gate leakage, equilibrium occurs when VDS = 0. Out of equilibrium in the sub-threshold region of operation or even, as it has been claimed, in the turn-on mode [26], Qz{z) may be so small that it can be set to zero, and a solution to Laplace's equation can be used for V(p, z). In the fully-on state, Qz(z) cannot be ignored. In the earliest CNFET model, this charge was taken to maintain its equilibrium value [5], as would be appropriate for a FET working in the traditional, charge-control mode [27]. Because of the large insulator capacitance, and small quantum capacitance, that can be obtained in CNFETs (see next section), a voltage-controlled mode of operation is more relevant to these devices [27], at least when tunneling is not important. In this mode, it is the mid-tube potential energy Emid that is important because it controls the height of the barriers to thermionic emission flow at the end contacts. This potential energy is solved for self-consistently with the mid-length charge in a recent model [7], which has been extended to account for both electron and hole charge and transport. In this model, -Em;d is connected to the end-contact barrier heights using a compact expression for the potential profiles based on solutions to Laplace's equation. Obviously, this will only give approximate results in cases where the actual shape of the barrier is important, but it will give good results in devices where it is only the height of the barrier that is important, as in, for example, metal/nanotube junctions with a negative barrier height for electrons, or doped/intrinsic nanotube junctions. This method was used to obtain the results presented here. Another approach is to employ quasi-equilibrium statistics, with a separate quasi-Fermi level 62 Chapter 4. An Evaluation of CNFET DC Performance for each sub-band. In the case of ballistic transport, the quasi-Fermi levels are flat along the length of the tube, and they split at the end contacts. A flux-balancing approach can then be used to include the current, as well as the charge and the potential, in a self-consistent solution [9]. The shortcoming of any semi-classical model is that, while the non-equilibrium electron and hole concentrations within the allowed bands may be computed, allowance is not made for charge in evanescent states. However, if this charge does not lead to a change in Emtf, then the neglect of this charge will not affect the computation of the salient thermionic emission current. A full, quantum-mechanical, approach is necessary to achieve a complete solution in cases where the charge in evanescent states is important, and where tunneling Schottky barriers are present, and in all devices where resonance and coherency are issues, and also where source-to-drain tunneling is a possibility [11]. Irrespective of the independent method used to compute Qz(z), Poisson's equation needs to be solved subject to the appropriate boundary conditions. Conformal mapping is effective when open boundaries are present [25]. For the metallic electrodes, a simple phenomenological representation, assuming no Fermi-level pinning [21], is V(Rg,z) = V G S - 4 > G / q V(p,0) = -4>s/q V(p,Lt) = VDS-^D/Q, where <pQ, 4>$ and (J>D are the work functions of the gate, source and drain metallizations, re-spectively, and VQS is the gate-source voltage. Recent experimental results, which show a strong dependence of device characteristics on metal work function, indicate that this phenomenolog-ical representation of the contacts may be appropriate [28]. 63 Chapter 4. An Evaluation of CNFET DC Performance 4.5 Results and Discussion In this section we wish to draw attention to the following features of CNFETs: ambipolarity, high conductance, geometry-dependent sub-threshold slope, high ON-current and transconduc-tance. A closed, coaxial geometry is used, with a gate dielectric of thickness 2.5 nm and a relative permittivity of 25, as is appropriate for zirconia, which has been employed in some CNFETs [22]. An intrinsic (16,0) nanotube is used, for which the radius, bandgap and electron affinity are 0.63nm, 0.62eV and 4.2eV, respectively [4]. The tube length is taken to be 20nm, which should ensure that transport is ballistic over the bias range considered [29]. The effect of changing the work functions for all three electrodes is examined. The values chosen are 4.5, 4.2 and 3.9 eV, corresponding, in the case of the source/drain electrodes, to barriers for electrons at the metal/nanotube interfaces that are positive, zero, and negative, respectively. These electron barrier heights are given by the difference between the metal work function and the nanotube electron affinity, i.e., ^Bn — (pM~Xt- All simulations are performed for a temperature T = 300K. 4.5.1 Ambipolarity Ambipolarity refers to the fact that, under certain bias conditions, channel conduction in CN-FETs is due to either electrons or holes [30]. Thus, depending on the bias, a particular CNFET may be either "n-type" or "p-type". The situation is illustrated in Fig. 4.2 for a CNFET with positive barrier end contacts. In this case </>jgn = Eg/2, where EG is the nanotube bandgap. The figure shows that, for a constant V T J S , the conduction can be due to either electrons or holes, depending on the gate bias VGS- In this case, at VQS + = V D S / 2 , where A<f> is the difference in work function between the nanotube and the gate metal, the electron and hole currents are equal, and the total current attains its minimum value [8], as can be seen on the ID-VGS plot in Fig. 4.3. Ambipolarity is an undesirable feature in FETs because it leads to an unwanted OFF current at VGS — 0. Fig. 4.3a shows that, while a negative barrier end contact can reduce the minimum current, it offers no advantage over a zero-barrier end contact as regards reducing 64 Chapter 4. An Evaluation of CNFET DC Performance IOFF- This is because at zero gate bias -Emid is determined only by cpc, and so the same barrier height is presented to the thermionic currents in the two cases (see Fig. 4.4). The OFF current is smaller in the positive-barrier case because of tunneling. Work function engineering of the gate metal can be used to laterally shift the I- V curves so that the minimum current occurs at VGS = 0 (see Fig. 4.3b). Even though the higher E^ that brings this about also reduces the ON current, the ON/OFF ratio is improved. Clearly, there is opportunity for creative work function engineering here, and an ON/OFF ratio of around 103 would appear to be possible. 0.4 0.2 E> -0.2 CD C LU -0.4 -0.6 -0.8 0 5 10 15 20 z(nm) Figure 4.2: Band diagram illustrating ambipolarity in a device with 4>S,D,G — 4.5 eV and VDS = 0.4 V. Hole injection at VQS = 0.05 V (dotted line and arrow); electron and hole injection at VQS — 0.2 V (solid line and arrows); electron injection at VGS = 0.35 V (dashed line and arrow). 4.5.2 Conductance The quantized interfacial conductance, as mentioned earlier, has a maximum value G m a x = Aq 2/h for a nanotube with two transverse modes [20]. Measurements of conductance, G = ID! VDS> can only be expected to approach C? m a x for ballistic transport in nanotubes with end contacts that have neither ohmic- nor tunneling-resistance. Ballistic transport demands measurement at low VDS to avoid exciting optical photons at higher biases, and a nanotube length less than the mean-free-path for acoustical phonon scattering (about 300 nm) [29]. Using devices of about this length with Pd end contacts, which yield low-resistance contacts with near-zero barrier height for holes, impressive values of G « 0 .4G m a x have been reported already [28]. 65 Chapter 4. An Evaluation of CNFET DC Performance -0 .5 -0 .25 0 0.25 0.5 -0 .5 -0.25 0 0.25 0.5 Gate-source voltage (V) Gate-source voltage (eV) Figure 4.3: ID-VGS a t VDS = 0.4 V. (a) <fic = 4.2eV and various (ps,D- 3.9 (solid line); 4.2 (dotted line); 4.5eV (dashed line), (b) <ps,D = 3.9eV and various C/>G: 3.9 (solid line); and 4.37eV (dashed line). z (nm) Figure 4.4: Band diagrams at VQS — 0 and VDS — 0.4 V for the three devices used in Fig. 4.3a'. <f)G = 4.2eV and various 4>S,D: 3.9 (dotted line); 4.2 (solid line); and 4.5eV (dashed line). 4 . 5 . 3 S u b t h r e s h o l d S lope The limiting value for the sub-threshold slope S, in situations where the sub-threshold cur-rent is thermionically determined, is about 60TTIQ mV/decade, where TTIQ = 1 + CQ/C-^ is the "quantum capacitance coefficient", with CQ being the quantum capacitance [5], and CMS the insulator capacitance. The use in CNFETs of high-permittivity dielectrics, such as zirconia [22] 66 Chapter 4. An Evaluation of CNFET DC Performance or aqueous solutions [24], opens up the possibility of attaining values of TUQ approaching unity. Thus, near-minimum values of S would be expected to be approached in CNFETs with negative barrier heights at the end contacts. Higher values can be expected for positive barrier heights due to the presence of tunneling barriers. However, if these barriers are rendered essentially transparent by a suitable gate bias, then values of S in these devices should also approach the thermionic limit. In all cases, because the injecting barrier is modulated by the gate voltage via capacitive coupling, S will show a dependence on the gate/channel geometry [8,23]. For CNFETS with positive barrier-height end contacts and insulator thickness tms = 2 nm, for ex-ample, S — HOmV/decade has been measured for planar CNFETs [23], and S s=s 80mV/decade has been predicted for coaxial devices, in which the capacitive coupling is superior. For a SiC>2 gate oxide with tms = 67 nm, and tubes contacted with palladium, planar devices have been reported with S « 150mv/decade [28]. With a thinner insulator, it is likely that Pd-contacted CNFETs will attain values of S close to the theoretical limit. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Gate-source voltage (V) Figure 4.5: Ratio of equilibrium quantum capacitance to insulator capacitance for an insulator relative permittivity and thickness of 25 and 2.5 nm, respectively, and for all work functions equal to 4.5 eV. 4.5.4 O N Current For maximizing the ON current, tunneling barriers must be avoided, and electron injection from the drain must be suppressed. This situation can be simulated by using a high-enough VDS i n Eq. (4.1) and setting T = 1, which also implies neglecting quantum-mechanical reflection at the 67 Chapter 4. An Evaluation of CNFET DC Performance 0 0.1 0.2 0.3 0.4 Ga te - sou rce voltage (V) 0 0.1 0.2 0.3 0.4 Ga te - sou rce voltage (V) Figure 4.6: (a) ID, and (b) gm, as a function of gate-source voltage. The solid lines are for the "quantum-capacitance limit" from Eqs. (4.2), (4.3) and (4.4). The dashed lines are for a device with CJ>S,D = 3.9eV and 4>G = 4.37eV, i.e., the device that gave the lowest OFF current in Fig. 4.3. metal/nanotube interfaces. Integrating Eq. (4.1) then leads to Ie,m,, = % B T \ n { l + e- E^/^ T h (4-2) where kg is Boltzmann's constant. This equation would be more useful if Emid were converted to an independent parameter, such as VGS- The relationship between these two quantities for an intrinsic nanotube is Eg qVGs + A<ft (4.3) 2 rriQ Note that the relationship would be essentially linear if CQ <§; Cins, i-e-, in the "quantum-capacitance limit" of rriQ —• 1. This inequality can be readily examined in the mid-tube region, where CQ = —q • dQz/dEmid and is easily computed at equilibrium. The result is shown in Fig.4.5 from which it is clear that CQ <^  C i n s only at low bias, i.e., when there is very little charge in the nanotube. However, if one allows this inequality to hold to higher bias [27], then Eq. (4.2)) reduces to a linear form in which the control by VGS is obvious. Such a result is plotted from Eq. (4.2) and Eq. (4.3) in Fig. 4.6a, where the solid line sets an upper limit to the unipolar current. As a reality check, the maximum drain current that has been measured so 68 Chapter 4. An Evaluation of CNFET DC Performance fax in a CNFET is about 25 fiA [28]. Much higher values should be possible with appropriate work function engineering, as shown by the simulation results of Fig. 4.7. Low electron barriers at the end contacts give high currents (see Fig. 4.7a), which can be further enhanced at a given gate bias by reducing the gate work function (see Fig. 4.7b). The effect of 4>Q is examined in the band diagrams of Fig. 4.8, which are for the case of VQS = VDS — 0.4 V. At 4>G = 3.9 eV, Ema is depressed to the extent that a tunneling barrier forms for the low energy electrons in the source. With low </>G, clearly a larger VDS is required to suppress the drain-injected electron current, leading to a higher saturation voltage Vos,Sat- In the example shown in Fig. 4.7b this could be an issue if CNFET logic circuits were constrained to operate at 0.4 V, which is the power-supply voltage specified for lOnm-scale Si MOSFETs [31]. Note that the hole barriers in all cases are too high for ambipolar effects to be seen at the bias values considered here. Thus, the drain characteristics are of the traditional "saturating" variety, with no rapid rise due to hole injection, as has been observed in some experimental data [32], presumably due to the use of positive-barrier end contacts. 0 0.2 0.4 0.6 0 0.2 0.4 0.6 Drain-source voltage (V) Drain-source voltage (V) Figure 4.7: Drain characteristics at VGS = 0.4 V. (a) 4>G = 4.2 eV and various (f>s,D- 3.9 (dashed line); 4.2 (solid line); and 4.5eV (dotted line), (b) 4>S,D = 3.9eV and various </>G: 3.9 (dashed line); 4.2eV (solid line); and 4.5eV (dotted line). In view of the fact that a CNFET with high ON current should also have a low OFF current, we compute ID for the case of 4>G — 4.37 eV and cbs,D = 3.9 eV, which led to the low OFF current shown in Fig. 4.4b. The result is shown in Fig. 4.6a. It can be seen that at VDS = 0.4 V, 69 Chapter 4. An Evaluation of CNFET DC Performance 0.2 0 -0.2 > -0.4 iS -0.8 -1 -1.2 10 z (nm) 15 20 Figure 4.8: Band diagrams for 4>S,D = 3.9 eV, VGS = VDS = 0.4 V and various (f>c- 3.9 (dashed line); 4.2 (solid line); and 4.5eV (dotted line). ID is about 75% of the ultimate value. Finally, it is noteworthy that the highest drain current shown in Fig. 4.7b) is equivalent to a current density (/z?,max/2i?t) of about 70mA//xm! 4.5.5 Transconductance Turning now to the maximum attainable transconductance, this limit can be obtained from the differentiation of Eq. (4.1), which, in the "quantum-capacitance limit", yields 9m, max — h 1 + exp E g / 2 - q V G S - A < l > \ - l (4.4) As Fig. 4.6b reveals, at high VGS, 9m, max attains its limiting value of 4g2//i, which, interestingly, is the same value attainable by G m a x , as noted elsewhere [33]. Taking once more the device with cf>G = 4.37 eV and <f>s,D = 3.9 eV as an example, at VDS = 0.4 V, a value of gm close to 80% of the ultimate value is indicated. 4 . 6 Conclusions From this evaluation of the DC performance of carbon nanotube field-effect transistors, it can be concluded that in n-type devices, for example, the use of negative barrier-height source and drain contacts, and low work function gate metallization, should allow attainment of sub-threshold slopes, conductances, transconductances and ON currents close to the ultimate limits. 70 Chapter 4. An Evaluation of CNFET DC Performance These features, allied to the excellent thermal and mechanical properties of carbon nanotubes, make these molecules strong contenders for implementation in nanoscale integrated circuits. References [1] Philip G. Collins and Phaedon Avouris, "Nanotubes for electronics," Sci. Am., 62-69 (2000). [2] Jing Kong, Jien Cao, Hongjie Dai, and Erik Anderson, "Chemical profiling of single nan-otubes: Intramolecular p-n-p junctions and on-tube single-electron transistors," Appl. Phys. Lett, 80(1), 73-75 (2002). [3] Yoshiyuki Miyamoto, Angel Rubio, X. Blase, Marvin L. Cohen, and Steven G. Louie, "Ionic cohesion and electron doping of thin carbon tubules with alkali atoms," Phys. Rev. Lett, 74(15), 2993-2996 (1995). [4] Sander J. Tans, Alwin R. M. Verschueren, and Cees Dekker, "Room-temperature transistor based on a single carbon nanotube," Nature, 393, 49-52 (1998). [5] Jing Guo, Mark Lundstrom, and Supriyo Datta, "Performance projections for ballistic carbon nanotube field-effect transistors," Appl. Phys. Lett, 80(17), 3192-3194 (2002). [6] Jing Guo, Sebastien Goasguen, Mark Lundstrom, and Supriyo Datta, "Metal-insulator-semiconductor electrostatics of carbon nanotubes," Appl. Phys. Lett., 81(8), 1486-1488 (2002). [7] L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a compact model for Schottky-barrier nanotube FETs," Proc. IEEE Conf. on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). [8] D. L. John, Leonardo C. Castro, Jason Clifford, and David L. Pulfrey, "Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors," IEEE Trans. Nanotechnol, 2(3), 175-180 (2003). 71 Chapter 4. An Evaluation of CNFET DC Performance [9] Jason Clifford, D. L. John, and David L. Pulfrey, "Bipolar conduction and drain-induced barrier thinning in carbon nanotube FETs," IEEE Trans. Nanotechnol., 2(3), 181-185 (2003). [10] Jing Guo, Jing Wang, Eric Polizzi, Supriyo Datta, and Mark Lundstrom, "Electrostatics of nanowire transistors," IEEE Trans. Nanotechnol, 2(4), 329-334 (2003). [11] Jing Guo, Ali Javey, Hongjie Dai, Supriyo Datta, and Mark Lundstrom, "Predicted perfor-mance advantages of carbon nanotube transistors with doped nanotubes as source/drain," (2003). [Online.] Available: http://arxiv.org/pdf/cond-mat/0309039. [12] Ralph Krupke, Frank Hennrich, Hilbert v. Lohneysen, and Manfred M. Kappes, "Sep-aration of metallic from semiconducting single-walled carbon nanotubes," Science, 301 , 344-347 (2003). [13] Michael S. Strano, Christopher A. Dyke, Monica L. Usrey, Paul W. Barone, Matthew J. Allen, Hongwei Shan, Carter Kittrell, Robert H. Hauge, James M. Tour, and Richard E. Smalley, "Electronic structure control of single-walled carbon nanotube functionalization," Science, 301 , 1519-1522 (2003). [14] Cees Dekker, "Carbon nanotubes as molecular quantum wires," Phys. Today, 52(5), 22-28 (1999). [15] Hyongsok T. Soh, Calvin F. Quate, Alberto F. Morpurgo, Charles M. Marcus, Jing Kong, and Hongjie Dai, "Integrated nanotube circuits: Controlled growth and ohmic contacting of single-walled carbon nanotubes," Appl. Phys. Lett, 75(5), 627-629 (1999). [16] R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, "Single- and multi-wall carbon nanotube field-effect transistors," Appl. Phys. Lett, 73(17), 2447-2449 (1998). [17] Ant Ural, Yiming Li, and Hongjie Dai, "Electric-field-aligned growth of single-walled car-bon nanotubes on surfaces," Appl. Phys. Lett, 81(18), 3464-3466 (2002). 72 Chapter 4. An Evaluation of CNFET DC Performance [18] Won Bong Choi, Jae Uk Chu, Kwang Seok Jeong, Eun Ju Bae, Jo-Won Lee, Ju-Jin Kim, and Jeong-0 Lee, "Ultrahigh-density nanotransistors by using selectively grown vertical carbon nanotubes," Appl. Phys. Lett, 79(22), 3696-3698 (2001). [19] Jeroen W. G. Wildoer, Liesbeth C. Venema, Andrew G. Rinzler, Richard E. Smalley, and Cees Dekker, "Electronic structure of atomically resolved carbon nanotubes," Nature, 391, 59-62 (1998). [20] Supriyo Datta, Electronic Transport in Mesoscopic Systems, vol. 3 of Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Cambridge University Press, New York, 1995). [21] Francois Leonard and J. Tersoff, "Role of Fermi-level pinning in nanotube Schottky diodes," Phys. Rev. Lett, 84(20), 4693-4696 (2000). [22] Ali Javey, Hyoungsub Kim, Markus Brink, Qian Wang, Ant Ural, Jing Guo, Paul Mclntyre, Paul McEuen, Mark Lundstrom, and Hongjie Dai, "High-rc dielectrics for advanced carbon-nanotube transistors and logic gates," Nature Mater., 1, 241-246 (2002). [23] S. Heinze, M. Radosavljevic, J. Tersoff, and Ph. Avouris, "Unexpected scaling of the performance of carbon nanotube Schottky-barrier transistors," Phys. Rev. B, 68, 235418-1-235418-5 (2003). [24] Sami Rosenblatt, Yuval Yaish, Jiwoong Park, Jeff Gore, Vera Sazonova, and Paul L. McEuen, "High performance electrolyte gated carbon nanotube transistors," Nano Lett., 2(8), 869-872 (2002). [25] J. P. Clifford, D. L. John, L. C. Castro, and D. L. Pulfrey, "Electrostatics of partially gated carbon nanotube FETs," IEEE Trans. NanotechnoL, 3(2), 281-286 (2004). [26] M. Radosavljevic, S. Heinze, J. Tersoff, and Ph. Avouris, "Drain voltage scaling in carbon nanotube transistors," Appl. Phys. Lett, 83(12), 2435-2437 (2003). [27] Anisur Rahman, Jing Guo, Supriyo Datta, and Mark S. Lundstrom, "Theory of ballistic nanotransistors," IEEE Trans. Electron Devices, 50(9), 1853-1864 (2003). 73 Chapter 4. An Evaluation of CNFET DC Performance [28] Ali Javey, Jing Guo, Qian Wang, Mark Lundstrom, and Hongjie Dai, "Ballistic carbon nanotube field-effect transistors," Nature, 424, 654-657 (2003). [29] Ali Javey, Jing Guo, Magnus Paulsson, Qian Wang, David Mann, Mark Lundstrom, and Hongjie Dai, "High-field, quasi-ballistic transport in short carbon nanotubes," Phys. Rev. Lett, 92(10), 106804-1-106804-4 (2004). [30] Richard Martel, Hon-Sum Philip Wong, Kevin Chan, and Phaedon Avouris, "Carbon nan-otube field effect transistors for logic applications," IEDM Tech. Digest, 159-162 (2001). [31] Semiconductor Industry Association, International Technology Roadmap for Semiconduc-tors (2001). [Online.] Available: http://www.itrs.net. [32] J. A. Misewich, R. Martel, Ph. Avouris, J. C. Tsang, S. Heinze, and J. Tersoff, "Electrically induced optical emission from a carbon nanotube FET," Science, 300, 783-786 (2003). [33] Jing Guo, Supriyo Datta, and Mark Lundstrom, "Assessment of silicon MOS and carbon nanotube FET performance limits using a general theory of ballistic transistors," IEDM Tech. Digest, 711-714 (2002). 74 Chapter 5 A Schrodinger-Poisson Solver for Modeling CNFETs 5.1 Introduction Carbon nanotubes [1] are attracting great interest for their use in nanoscale electronic devices. Recent modeling efforts of carbon nanotube field-effect transistors (CNFETs) have been suc-cessful in examining the subthreshold behaviour of these devices through a simple solution to Laplace's equation [2,3], while the above-threshold behaviour has been modeled using bulk device concepts [4, 5]. Accurate CNFET modeling requires a self-consistent solution of the charge and local electrostatic potential. In order to properly treat such quantum phenomena as tunneling and resonance, the charge is computed via Schrodinger's equation. Owing to the presence of metal-semiconductor interfaces, we also account for the penetration of evanescent wavefunctions from the metal into the energy gap of the nanotube. We deal specifically with the coaxial geometry of the CNFET shown previously in Fig. 4.1. The device consists of a semiconducting carbon nanotube surrounded by insulating material (relative permittivity eins) and a cylindrical, wrap-around gate contact. The source and drain contacts terminate the ends of the device. The device dimensions of note are the gate radius, ©2004 NSTI http://nsti.org. Reprinted and revised, with permission, from D. L . John, L . C. Castro, P. J . S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson Solver for Modeling Carbon Nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Con}, and Trade Show, vol. 3, pp. 65-68, March 7-11, Boston, U.S.A. 75 Chapter 5. A Schrddinger-Poisson Solver for Modeling CNFETs Rg, the nanotube radius, Rt, the insulator thickness t-ms = R g - Rt, and the device length, Lt-In this closed, metallic cylinder system, Poisson's equation, restricted to just two dimensions by azimuthal symmetry, is d 2V l d V d 2V Q dp*  + pdp  + dz 2 ~ e ' ( j where V(p, z) is the potential within the outer cylinder, Q is the charge density, and e is the permittivity. It must be noted that, although the solution of Eq. (5.1) encompasses the entire volume of the device, we are primarily concerned with the longitudinal potential profile along the surface of the tube, hereafter labeled VQS{Z) = V(Rt,z), since knowledge of this potential is required for carrier transport calculations. We treat the nanotube as a quasi-one-dimensional conductor, and the linear carrier density is then computed via the time-independent Schrodinger equation given by g = - ^ ( E - W , (5.2) where tp(z, E) is the wavefunction of a carrier with total energy E and effective mass m, traveling in a region with local effective potential U(z). While Q may include sources such as trapped charge within the dielectric, we neglect any charge other than that of electrons and holes on the nanotube. 5.2 Solution Method We require a solution to Eq. (5.1) with Q = Q{V). Convergence for this non-linear system is achieved with the Picard iterative scheme, whereby iteration fc + 1 is given by Vfc+i = Vfc - ctC^rk , rk = CVk - Q(Vk), where is the residual of the fc-th iteration, 0 < a < 1 is a damping parameter, and C represents the linear, differential operator allowing Eq. (5.1) to be written as LV = Q. 76 Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs 5.2.1 Potential The boundary conditions for V are given by V{Rg,z) = VGs-<pG/q, V(p,0) = -<t>s/q, (5.3) V{p, Lt) = VDS - <t>D/q, f ( 0 , 2 ) = 0 , where <f>G,s,D represent the work functions of the gate, source, and drain metallizations, respec-tively, and VGS and VDS a r e the gate- and drain-source voltages. Due to the discontinuity in e across the nanotube surface, we must also apply the usual matching condition q(p - n) dV dp dV R+ dP \ R - 2nRte0 where p and n are the one-dimensional hole and electron carrier densities, et is the nanotube relative permittivity, and eo is the permittivity of free-space. The solution to Eq. (5.1) was obtained via the finite difference technique, implemented by discretizing the spatial domain and using central differencing to generate a linear system of equations, for some known Q, and subject to the boundary conditions specified by Eq. (5.3). Finite differencing was chosen over an FFT-Green's function approach due to its flexibility in modeling more complex structures. The singularity at p = 0 was addressed by applying l'Hopital's Rule to the offending term, yielding 1 dV d2V p dp ~ dp2 The amount of energy band bending in the vacuum level, along the length of the nanotube, is given by Evac(z) = —qVcs{z), since we assume that the local electrostatic potential rigidly shifts the nanotube band structure. The potential energies seen by electrons and holes in the nanotube are -EPot,e(z) = Ec{z) = Evac(z) -xt, , x (5.4) Epot,h(z) = ~Ev(z) - -(Ec(z) - Eg), where Eg and xt are, respectively, the nanotube band-gap and electron affinity. 77 Chapter 5. A Schrddinger-Poisson Solver for Modeling CNFETs 5.2.2 Charge Having established a solution for the potential and its relation to the energy band structure, we now determine the carrier concentration. In our system, the charge density is given by where 6 is the Dirac delta function in cylindrical coordinates, and p(z) and n(z) are computed via Eq. (5.2), where the nanotube effective mass is obtained from the tight-binding approximation of the band structure, and is the same for both electrons and holes due to symmetry [1]. Only the first, doubly-degenerate band is included in the calculations presented herein. The potential energy, Epot, for each carrier type is specified by Eq. (5.4), given a potential profile V"cs(z). We solve Eq. (5.2) using the scattering-matrix method in which a numerical solution is prop-agated by cascading 2x2 matrices [6]. We find that the use of piecewise constant potentials (plane-wave solutions) are preferable to piecewise linear potentials (Airy function solutions) due to the considerable reduction of simulation time without an appreciable increase in the error. Matching of the wavefunction and its derivative on the boundary between intervals n and n + 1 , assuming a constant effective mass, is performed via the usual relations dlpn _ dlpn+1 dz dz In order to completely specify the wavefunction, we require two boundary conditions. In the contacts, the wavefunction at a given energy is of the form f A s e i k s z + Bse~ iks z , z < 0, [ A D e i k ° z + B D e - i k ° z , z > L t , where ks and kry are the wavevectors in the source and drain contacts, respectively, and As, Bs, AD, and BD are constants. As an example, noting that an analogous calculation may be performed for the drain by exchange of variables, we now illustrate source injection. For this case, BD = 0 for all energies. In addition, we expect that the Landauer equation [7] will hold 78 Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs for the flux, and must be equal to the probability current. For the transmitted wave, this yields ^fsT=Q-kD\AD\\ (5.5) where the pre-factor of 2 accounts for the aforementioned band degeneracy, fs is the Fermi-Dirac carrier distribution in the source, and T is the transmission probability specified by T_kp\Ap\2 ks\As\2 ' Simple manipulation yields the normalization condition I A |2 2m fs |j4sl =^vs- (5-6) At any given energy, multiplication of the unnormalized wavefunction by a constant satisfies Eq. (5.6). Including source and drain injection components, the normalized wavefunctions yield the total carrier densities in the system, poo n{z)= / (k/>e,s|2 + h M | 2 ) dE, -'Sref.e / • O O P(z)= / {Wh,s\2 + Wh,D\ 2) dE, '-Eref.h where the Exe{ terms are taken to be the bottom of the band, for either electrons or holes, in the appropriate metallic contact, and correspond to the bottom of the band in the metal. In practice, the integrals are performed using adaptive Romberg integration, where repeated Richardson extrapolations are performed until a predefined tolerance is reached [8]. We find that an adaptive integration method is a necessity for convergence, in order to properly capture tp, which is typically highly-peaked in energy for propagating modes. Alternatively, one could employ a very fine discretization in energy, however the Romberg method allows for the mesh size to change based on the requirements of the integrand, and results in a much improved simulation time. 79 Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs 5 . 3 Results We now present results for a C N F E T with a (16, 0) nanotube {Rt « 0.63nm; Ea « 0.62eV), Lt = 20 nm, t j n s — 2.5 nm, and C j n s — 25. A l l work functions are taken to be 4.5 eV unless otherwise noted, and \t = 4.2 eV. The nanotube is presumed to have a free-space relative permittivity ej = 1 [9], and Erei was taken to be 5.5 eV below the metal Fermi level, as a rough estimate [10]. Figure 5.1: Net carrier density, p(z) — n(z), for the model device as a function of position and VDS. 0.3 -0.2 0 5 10 15 20 Distance from source (nm) Figure 5.2: Conduction band edges for the model device with VQS = 0.5 V , and VDS = 0 (dashed) and 0.4 V (solid). Energies are with respect to the source Fermi level. 80 Chapter 5. A Schrddinger-Poisson Solver for Modeling CNFETs In equilibrium, i.e., for VDS = 0, we obtain reasonable agreement for the carrier concentrations away from the contacts with that computed using equilibrium statistics [2]. Out of equilibrium, however, interference effects influence the carrier distributions throughout the device. Fig. 5.1 shows the carrier distributions for VQS = 0.5 V as a function of position and VDS, and Fig. 5.2 shows the corresponding band edges for VDS = 0 and 0.4 eV. Under a positive gate bias, the band bending results in an increase in the electron concentration throughout the device as more propagating modes are allowed in the channel. As VDS is increased, this concentration is considerably reduced in the mid-length region. Evanescent modes dominate the carrier concentrations near the end contacts, thus impacting on the local potential. Due to the exponential dependence of the transmission probability on the barrier shapes, the flux is significantly modified if these modes are neglected. 0 10 20 0 0.5 1 z (nm) Probability Figure 5.3: Conduction band edges and transmission probabilities for electrons at VGS = 0.5 V and VDS = 0.4 V: (a) tps = 4>D = 3.9 eV and (b) 4>s — 4>D = 4.5 eV. Energies are with respect to the source Fermi level. We note, also, that it is important to allow for the full inclusion of quantum mechanical reflection for the thermionic component of the flux. Often, carriers above the barrier are assumed to have a transmission probability near unity. However, this approximation does not hold in general, 81 Chapter 5. A Schrddinger-Poisson Solver for Modeling CNFETs as Fig. 5.3 shows, wherein the significant reflection is due to ETef being much lower than the conduction band edge in the nanotube. The effect is most important for devices where the metal-nanotube work function difference yields a negative barrier, shown in Fig. 5.3(a). Here, a classical treatment would considerably overestimate the Landauer flux, a function of T, for energies in the vicinity of the Fermi level. Finally, the present Schrodinger-Poisson method allows for explicit calculation of the carrier distribution functions, as shown in Fig. 5.4. The result is in marked contrast to a previous self-consistent model [5] that utilized quasi-equilibrium distribution functions to calculate the non-equilibrium carrier concentrations. Moreover, while the model provided in Ref. [4] yields similar non-equilibrium carrier distributions, it is not equipped to account for the resonant peaks illustrated here. 0 0.2 0.4 0.6 0.8 1 Normalized n(E) Figure 5.4: Source-originated electron concentration at Lt/2, normalized to its maximum value. VQS = 0.5 V, VDS — 0.4 V, and (j>s = 4>D = 3.9 eV. Energies are with respect to the source Fermi level. 5 . 4 Conclusions From this work on the modeling of CNFETs with a coupled Schrodinger-Poisson solver, we conclude that: 1. equilibrium statistics are not adequate in describing the carrier distributions in energy; 82 Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs 2. consideration of the evanescent modes is crucial for the accurate simulation of devices where transport is dominated by tunneling through the interfacial barriers; 3. for devices dominated by thermionic emission, a full solution of Schrodinger's equation is still required in order to account for significant reflection above the barriers. References [1] R. Saito, T. Takeya, T. Kimura, G. Dresselhaus, and M. S. Dresselhaus, "Raman intensity of single-wall carbon nanotubes," Phys. Rev. B, 57(7), 4145-4153 (1998). [2] D. L. John, Leonardo C. Castro, Jason Clifford, and David L. Pulfrey, "Electrostatics of coaxial Schottky-barrier nanotube field-effect transistors," IEEE Trans. Nanotechnol, 2(3), 175-180 (2003). [3] S. Heinze, M. Radosavljevic, J. Tersoff, and Ph. Avouris, "Unexpected scaling of the performance of carbon nanotube Schottky-barrier transistors," Phys. Rev. B, 68, 235418-1-235418-5 (2003). [4] L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a compact model for Schottky-barrier nanotube FETs," Proc. IEEE Conf. on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). [5] Jason Clifford, D. L. John, and David L. Pulfrey, "Bipolar conduction and drain-induced barrier thinning in carbon nanotube FETs," IEEE Trans. Nanotechnol, 2(3), 181-185 (2003). [6] David Yuk Kei Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B, 38(14), 9945-9951 (1988). [7] David K. Ferry and Stephen M. Goodnick, Transport in Nanostructures (Cambridge Uni-versity Press, New York, 1997). [8] Lee W. Johnson and R. Dean Riess, Numerical Analysis (Addision-Wesley, Don Mills, Ontario, 1977). 83 Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs [9] Frangois Leonard and J. Tersoff, "Dielectric response of semiconducting carbon nan-otubes," Appl. Phys. Lett, 81(25), 4835-4837 (2002). [10] Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976), 1st ed. 84 Chapter 6 An Improved Evaluation of the D C Performance of CNFETs 6.1 Introduction In a recent evaluation of carbon nanotube field-effect transistors (CNFETs), devices were spec-ified that yielded simulated drain currents and transconductances approaching the ultimate limits of a one-dimensional (1-D) ballistic transistor [1]. These devices were coaxial in geom-etry and had a thin, high-permittivity, gate dielectric. The source and drain metallizations to the ends of the nanotube imposed negative Schottky barriers to electron flow, and as such, were predicted to perform much better than CNFETs with positive-barrier end contacts. The designation of barriers as either negative or positive is used here with respect to electrons. With appropriate changes in work functions it applies equally to holes, which are the domi-nant carrier in many experimental devices [2,3]. Recently fabricated short-channel CNFETs have shown that, with suitable design, excellent performance is attainable with negative-barrier metal contacts [4]. The compact non-equilibrium model, from which the results were obtained, allows for quantum-mechanical tunnelling of electrons and holes at appropriate interfaces, using a simplified ex-pression obtained from the JWKB approximation. Consideration of tunnelling is important A version of this chapter has been published. L. C. Castro, D. L. John, and D. L . Pulfrey, "An Improved Evaluation of the D C Performance of Carbon Nanotube Field-Effect Transistors," Smart Mater. Struct, 15, S9-S13 (2006). Online at http://www.iop.org/journals/sms. 85 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs for studying positive-barrier devices, as in the original presentation of the model [5], but, in negative-barrier devices, attention must be paid to thermionic emission of electrons, and to their quantum-mechanical reflection at energies above the barrier height. By assigning a trans-mission probability of unity to all carriers of energy above the barrier, the original compact model (CM1) is likely to severely overestimate the current. In this paper, the need for a more detailed treatment of quantum-mechanical reflection by the compact model is confirmed by examining the correspondence of the latter's method of computing the nanotube charge with that of a solution from Schrodinger's wave equation. Then, we present a derivation of a tractable expression for quantum-mechanical reflection that is incorporated into a new version, CM2, of the compact model. The predictions of the latter for the ON/OFF current ratio, ON current, and transconductance, match more closely the results from a self-consistent Schrodinger-Poisson solver (SP) [6], and indicate that Schottky-barrier CNFETs are likely to operate further from the ultimate limit than previously thought. 6.2 Correspondence of the Compact and Quantum Models In this work, SP is used not only to obtain a better evaluation of the performance of CNFETs, but also to indicate how CM1 may be improved to achieve the same end. To accomplish the latter, it is first necessary to establish that the quantum and compact models correspond at a fundamental level. The basic premise is that there exists a region in the mid-length of the tube in which the potential energy, i? r aiH, is relatively flat, and serves to connect the regions of rapidly varying potential energy near to the end contacts. This condition is primarily dependent on device length, insulator thickness and contact geometries [7]. The decay length for the end potential is of the order of the gate radius [8], which is about 3nm in the example used here. A constant Emid is commensurate with a constant charge in the mid-length region of the nanotube, and it is under these conditions that we seek to prove the correspondence of the CM and SP approaches. It is also assumed that ballistic transport applies, which should not be unreasonable for the tubes of length 20 nm that are considered here [9]. 86 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs In our compact models, the mid-length charge is estimated from a calculation of Emid via a simple capacitance expression, thereby obviating the use of Poisson's equation. Self-consistency is achieved by reconciling the resulting mid-length charge with that computed from the fluxes of electrons and holes through and over the interfacial barriers. In the mid-length region, transport is taken to be phase-incoherent, i.e., the phase of the electron wavefunctions is ignored. At the interface regions, back-scattering is allowed, leading to a composite transmission probability for the device, T* =  T s T d , (6.1) where T$,D are, respectively, the source and drain transmission probabilities. The mid-length electron density is given by -£«** [$ (s->H (5 -0 dE, (6.2) where gio is the 1-D nanotube density-of-states, numerically computed from a tight-binding method, / is the Fermi-Dirac distribution (as is relevant for carriers in the metallized regions), and the subscripts S and D refer, respectively, to source and drain injection. The two terms in the square brackets can be viewed as the components of nm;d arising from electrons injected from the source (first term) and the drain (second term). If we now consider just the lowest band, which is doubly degenerate, and introduce into Eq. (6.2) the effective-mass approximation via / r n . 4 dk 2 / 2m and consider, for brevity, just the component originating at the source, we obtain TS(2 - TD) V2m f°° 1 "mid.S = r - / . = JS TS + T D - TSTD dE. (6.3) Now, from a quantum-mechanical perspective, let the amplitude of a unity-input wavefunc-tion immediately after crossing the source barrier region be given by P. Then, under phase-incoherent transport conditions, and allowing for multiple reflections of these carriers between 87 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs the source and drain barriers, the total probability density in the mid-length region of the chan-nel due to source injection, |'0mid,S'|2j can be found from the infinite geometric series relation, and is given by 2 | P | 2 + | P | 2 ( i - r D ) \Vmid,S\ = T S + T D _ T S T D • \BA) However, the source transmission probability can also be written as rp^ _ Jtrans ^mid ipi2 / ^ -^mid I Jinc ks V E - EierS where the j 's are probability density currents, the k's are wavevectors, and Eiefts is the energy of the conduction band edge of the source metal. Therefore, from Eq. (6.4), TS{2 — TD) \JE - ^ m i d | 2 TS + TD-TSTD ^E-EiefjS Substituting into Eq. (6.3), and including the analogous term for injection from the drain, we get „ „ , , „ = ( w + fei^i2 | d E . (6.5) This is precisely the expression for the electron contribution to the mid-length charge that results from a self-consistent Schrodinger-Poisson solution under the conditions of: a single, doubly degenerate band; a constant effective mass for both nanotube and end-contact metal-lization; a nanotube length that is sufficient for the contribution to the charge at mid-length due to evanescent states to be neglected, and for the transport to be considered phase-incoherent; and a normalization of the carrier density using the Landauer equation, as in Ref. [6]. The effective-mass representation of the band structure is employed in CM2; thus, the correspon-dence of Eq. (6.3) and Eq. (6.5) proves the fundamental equivalence of CM2 and SP under the stated assumptions. As regards the actual numerical equivalence of SP and CM2, it can now be appreciated that this will depend totally on how the transmission probabilities are estimated. Concerning the numerical equivalence of SP and CM1, this will depend also on the agreement between the effective-mass- and density-of-states-representations of the band structure. The agreement is sufficiently good for the single-band case considered here that the numerical difference between Chapter 6. An Improved Evaluation of the DC Performance of CNFETs SP and CM1 is due almost entirely to the difference in estimating the transmission probabilities (T's). In CM1 and CM2, the T's are computed for phase-incoherent transport using the JWKB approximation in the case of tunnelling, and in CM1 are set equal to unity in the case of thermionic emission. In SP, a full Schrodinger calculation, under phase-coherent transport conditions, yields exact values for the T's at all energies. It is not reasonable to expect that there is a compact expression for T in the phase-coherent case, but one may well exist for phase-incoherent transport of thermionically emitted carriers, in which case its incorporation into the compact model should yield a significant improvement. Such an expression, which is derived in the next section, is incorporated into CM2. 6.3 Quantum-Mechanical Reflection for the Thermionic Case Typically, the JWKB approximation is used to compute the tunnelling probability for carriers through a barrier; however, it may also be used to compute the reflection of carriers above the barrier. For this thermionic current component, we assume the usual JWKB form for the wavefunction in three regions: Ae[kconZ + _Be _ i f c c o n Z , z < 0, ^(z) = 1 i (Ce1 So fcbar(z)di + D e - i J o khBI(z)dz\ ^ 0 < z < u;, \ A b a r ( * ) V ' ^ e > f c m i d ( z - ™ ) _|_ Qe-ikmid(z-w) ^ z > w where A to G are constants, w is the barrier width, and fccon, k^iz), and fcmjd are the wavevec-tors in the contact, in the region of the nanotube close to the contact where the potential may change significantly, and in the mid-length region of the nanotube where the potential is rela-tively constant, respectively. Note that only fcbar is a function of z. Taking source injection as an example, we set G = 0, and assume an abrupt change in the band edge when crossing from the source metal into the nanotube. This permits the usual continuity condition for tp and its derivative. For phase-incoherent transport, we can compute the transmission through the regions close to the source and drain contacts separately. If we consider the source barrier, for example, we 89 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs note that k a^r(w) = kmid and k'haT(w) = 0, where the primed notation denotes a derivative with respect to z. This provides a compact expression for the source transmission probability, j, _ 16fcCOnfcbar0 (g g) (^baru)2 + (^^ barO + ^con^baru)2 where the zero subscript indicates that the quantity is evaluated at z — 0. An analogous expression holds for the drain transmission probability. The new compact model (CM2) incorporates Eq. (6.1) and Eq. (6.6), whereas in the original compact model (CM1) T*=l. 6.4 Results and Discussion We model the coaxial geometry CNFET illustrated previously in Fig. 4.1. The device consists of a semiconducting carbon nanotube surrounded by insulating material of relative permittivity ei n s, and a cylindrical, wrap-around gate contact. The source and drain contacts terminate the ends of the device. The device dimensions of note are the device length, Lt, the gate radius, Rg, and the nanotube radius, Rt- Here we take Rg/Rt = 5, and we consider a (16,0) tube with Rt = 0.63 nm and Lt = 20 nm. For the relative permittivities, ei n s = 25, and e<, which is not relevant to CM1 and CM2, is set to unity in SP [10]. For the work functions, 4.5 eV is taken for the nanotube and the gate, and 3.9 eV is taken for the source and drain. This arrangement leads to a negative barrier height of approximately one-half of the bandgap, as used elsewhere in simulations of high-performance CNFETs [1, 11]. All simulations are performed for a temperature of 300 K. The gate characteristics are shown in Fig. 6.1. It can be seen that the improved models do not alter the previous conclusion of Ref. [1] that ON/OFF ratios of around 103 appear possible. Higher values could result from operating at lower VDS [H]- This is because, with a saturating ID-VDS characteristic, selection of V D S at the onset of saturation ensures the highest ON current, yet the low value of VDS delays the onset of hole conduction when VGS is reduced, thus allowing a lower OFF current to be attained. In practical circuitry it would be desirable to use a single power supply, so it is to be hoped that metals of suitable work function exist to give a flat-band voltage such that the minimum in drain current can be engineered to occur at a gate 90 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs bias of V G s = 0 [1,11]. 102 <[ 101 1 10° o . £ -1 2 10 Q 10"2 -0.2 0 0.2 0.4 0.6 V o s M Figure 6.1: Drain current versus gate-source voltage at VDS = 0.4 V for the various models: SP (circles), CM1 (dashed), and CM2 (solid). The results for the ON current and transconductance are shown in Fig. 6.2. The dotted lines are for the ultimate limit, as defined previously [1,12]. The shortfall predicted by CM1 is an indication of how far below this limit CNFETs would perform, even if the transmission proba-bility for all thermionically injected carriers were unity. The further reduction in performance predicted by SP is due mainly to a more realistic representation of this transmission probability, T*(E). The effect is severe and suggests that CNFETs, even with negative barrier heights as extreme as one-half of the bandgap, are unlikely to come close to performing at the ultimate limit. Another revelation of the improved models used in this work is their prediction of a decline in the transconductance, gm, at high gate bias. This is due to the complicated interaction of the charge and VGS with Em\d [13]. In fact, CM1 predicts a similar decline, but at a much higher gate bias than CM2 and SP due to its overestimation of the charge on the nanotube. The actual form of T*(E) is illustrated in Fig. 6.3. Obviously, CM1 does not capture the interference phenomena exhibited in the results of SP by virtue of the latter's consideration of phase-coherent transport. Equally clear is that, unless all the carriers are grouped together at an energy for which T*(E) 91 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs (b) s / / / / / ; •: / :/ / 1 3 A if V G S 0 0 0.3 0.6 Figure 6.2: (a) Drain current and (b) transconductance, as a function of gate-source voltage at VDS = 0.4 V. The dotted lines are for the ultimate limit (see text). Other curves illustrate SP (circles), CM1 (dashed), and CM2 (solid). 0.4 0.6 Probability Figure 6.3: Transmission probabilities, above Em\d, of source-injected electrons for the SP (solid) and CM2 (dashed) models, at VQS = 0.4 V and VDS = 0.4 V. shows a peak close to unity, then CM l's employment of an energy-independent value of T* = 1 will lead to a substantial overestimate of the charge and the current. Evidently, this is happening in the results shown in Fig. 6.2. The employment in CM2 of Eq. (6.6), and its analogue for drain injection, should lead to some improvement because, even though the expression is derived for phase-incoherent transport, it does allow for T*(E) to take on values of less than unity. The ensuing, greatly improved correspondence in the predictions of the current between the compact model and SP is demonstrated in Figs. 6.1, 6.2 and 6.4. It may appear unreasonable to expect that the still-large difference in T*(E) between CM2 and 92 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs 10 I 8 ' /' 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 6.4: Drain current versus drain-source voltage for various CNFET models at VGS = 0.4V: SP (circles), CM1 (dashed), and CM2 (solid). SP should allow such an improved concordance in current. However, as quantities of interest, such as the charge and the current, are computed by performing an integral over energy, some averaging occurs, and, evidently, leads to a mean value for the SP case that is close to that predicted by the phase-incoherent analysis. The structure in the SP results for T*(E) is due to phase coherence, which will become less important for longer devices, so we would expect the new compact model to give even better results for tubes with L t > 20 nm. The converse applies to shorter tubes, when, additionally, issues due to evanescent charge and direct tunnelling between source and drain will need to be considered. Operation at lower gate bias may also lead to the appearance of larger phase-coherence effects, due to the increase in height of the potential barriers at the end contacts. These phenomena will need to be taken into account in further compact modeling of CNFETs. 6.5 Conclusions From this re-evaluation of the DC performance of coaxial carbon nanotube field-effect transistors with negative-barrier contacts, it can be concluded that: 1. ascribing a value of unity to the transmission probability of thermionically injected carriers leads to a significant overestimate of the current and transconductance; 2. inclusion of a short expression for quantum-mechanical reflection into a compact model 93 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs yields much-improved predictions for the current and transconductance, inasmuch as they are in excellent agreement with results from a comprehensive Schrodinger-Poisson solver; 3. accounting for quantum-mechanical reflection indicates that these transistors with met-allized end contacts may not be capable of operating as close to the ultimate limit as previously thought. References [1] L. C. Castro, D. L. John, and D. L. Pulfrey, "Carbon nanotube transistors: An evalu-ation," Proc. SPIE Conf. Device and Process Technologies for MEMS, Microelectronics, and Photonics III, vol. 5276, 1-10 (Perth, Australia, 2004). [2] J. Appenzeller, J. Knoch, and Ph. Avouris, "Carbon nanotube field-effect transistors—an example of an ultra-thin body Schottky barrier device," Proc. TMS 61st Annual Device Research Conference, 167-170 (Salt Lake City, U.S.A., 2003). [3] Ali Javey, Jing Guo, Qian Wang, Mark Lundstrom, and Hongjie Dai, "Ballistic carbon nanotube field-effect transistors," Nature, 424, 654-657 (2003). [4] Ali Javey, Jing Guo, Damon B. Farmer, Qian Wang, Erhan Yenilmez, Roy G. Gordon, Mark Lundstrom, and Hongjie Dai, "Self-aligned ballistic molecular transistors and elec-trically parallel nanotube arrays," Nano Lett, 4(7), 1319-1322 (2004). [5] L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a compact model for Schottky-barrier nanotube FETs," Proc. IEEE Conf. on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). [6] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson solver for modeling carbon nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Conf. and Trade Show, vol. 3, 65-68 (Boston, U.S.A., 2004). [7] J. P. Clifford, D. L. John, L. C. Castro, and D. L. Pulfrey, "Electrostatics of partially gated carbon nanotube FETs," IEEE Trans. Nanotechnol., 3(2), 281-286 (2004). 94 Chapter 6. An Improved Evaluation of the DC Performance of CNFETs [8] Sang-Hyun Oh, Don Monroe, and J. M. Hergenrother, "Analytic description of short-channel effects in fully-depleted double-gate and cylindrical, surrounding-gate MOSFETs," IEEE Electron Device Lett, 21(9), 445-447 (2000). [9] Ali Javey, Jing Guo, Magnus Paulsson, Qian Wang, David Mann, Mark Lundstrom, and Hongjie Dai, "High-field, quasi-ballistic transport in short carbon nanotubes," Phys. Rev. Lett, 92(10), 106804-1-106804-4 (2004). [10] Francois Leonard and J. Tersoff, "Dielectric response of semiconducting carbon nan-otubes," Appl. Phys. Lett, 81(25), 4835-4837 (2002). [11] Jing Guo, Supriyo Datta, and Mark Lundstrom, "A numerical study of scaling issues for Schottky-barrier carbon nanotube transistors," IEEE Trans. Electron Devices, 51(2), 172-177 (2004). [12] Anisur Rahman, Jing Guo, Supriyo Datta, and Mark S. Lundstrom, "Theory of ballistic nanotransistors," IEEE Trans. Electron Devices, 50(9), 1853-1864 (2003). [13] D. L. John, L. C. Castro, and D. L. Pulfrey, "Quantum capacitance in nanoscale device modeling," J. Appl. Phys., 96(9), 5180-5184 (2004). 95 Chapter 7 Quantum Capacitance in Nanoscale Device Modeling 7.1 Introduction The concept of "quantum capacitance" was used by Luryi [1] in order to develop an equivalent circuit model for devices that incorporate a highly conducting two-dimensional (2D) electron gas. Recently, this term has also been used in the modeling of one-dimensional (ID) systems, such as carbon nanotube (CN) devices [2,3]. Here, we derive expressions for this capacitance in one- and two-dimensions, showing the degree to which it is quantized in each case. Our discussion focuses primarily on the ID case, for which we use the carbon nanotube field-effect transistor (CNFET) as the model device, although the results apply equally well to other types of ID semiconductors. The 2D case has been discussed in Ref. [1], and is included here only to illustrate key differences. Equilibrium expressions are derived, and these are extended to cover two extremes in the non-equilibrium characteristic, namely: phase-coherent and phase-incoherent transport. In the former case, the wavefunction is allowed to interfere with itself, and may produce resonances depending on the structure of the device. This results in the charge, and the quantum ca-© 2004 American Institute of Physics. Reprinted, with permission, from D. L . John, L . C. Castro, and D. L. Pulfrey, "Quantum Capacitance in Nanoscale Device Modeling," J. Appl. Phys., 96(9), 5180-5184 (2004). 96 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling pacitance, becoming strong functions of the length of the semiconductor. In the latter case, this type of resonance is not allowed, and the quantum capacitance is more uniform. Finally, we show how the quantum capacitance affects the transconductance of a CNFET, where the Landauer expression can be used to compute the current [4]. 7.2 Equilibrium Quantum Capacitance In order to derive analytical expressions, it is assumed that our device is in quasi-equilibrium, and that the carrier distribution functions are rigidly shifted by the local electrostatic potential. If the density of states (DOS) is symmetric with respect to the Fermi level, EF, as in graphene, then we can write the charge density, Q, due to electrons and holes in the semiconductor, as roo Q = q g(E) [f (E + Eg/2 + qVa) - f (E + Eg/2 - qVa)\ dE , (7.1) Jo where q is the magnitude of the electronic charge, E is the energy, g(E) is the ID or 2D DOS, f(E) is the Fermi-Dirac distribution function, Va is the local electrostatic potential, Eg is the bandgap, and EF is taken to be mid-gap when Va = 0. The quantum capacitance, CQ, is defined as and has units of F /m 2 and F/m in the 2D and ID cases, respectively. 7.2.1 Two Dimensions In the two-dimensional case, if we employ the effective-mass approximation with parabolic bands, the DOS is given by 9(E) = £ « B ) , where v(E) is the number of contributing bands at a given energy, m is the effective mass, and h is Dirac's constant. If we combine this with Eqs. (7.1) and (7.2), and exchange the order of differentiation and integration, we get ^ (E + E/2-qvA + s e c h 2 f* + y + «r. dE, 97 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling where ks is Boltzmann's constant, and T is temperature. If v is a constant, we can perform the integration to get umq 2nh 2 sinh (Eg/2kBT) c o s h ( * £ M W ^ / 2 + ^ 2kBT 2 k B T which reduces to CQ = vmq (7.2) when Eg = 0 in agreement with Ref. [1], where metallic properties were assumed. Note that this function is quantized in the metallic case, but continuous for a semiconductor. For Eg greater than about 15kBT, however, the function makes a rapid transition from a small value to that given by Eq. (7.2) when Va crosses Eg/2, and is thus effectively quantized. 7.2.2 O n e D i m e n s i o n In the one-dimensional, effective-mass case, we have The explicit energy dependence of this DOS complicates the evaluation of our integral for CQ. The approach suggested in Ref. [2], i.e., using the fact that the derivative of f(E) is peaked about Ep in order to approximate this integral using a Sommerfeld expansion [5], cannot be done in general, due to the presence of singularities in the ID DOS. The capacitance is given by CQ = m 2kBTh V ~2 f Jo v{E) E ^ ( E + E g / 2 - q V a ] gech2 ( E + E g / 2 ^ 2kBT 2kBT dE, (7.3) where h is Planck's constant. For sufficiently large \Va\, we can completely neglect one of the sech2(-) terms. As a simple example, if Va = 0.1 V for a material with Eg ~ leV, the contribution to the integral from the first term is roughly four orders of magnitude greater than the second. This approximation is equivalent to neglecting hole charge for positive Va, and electron charge for negative Va. The solid line in Fig. 7.1 shows the equilibrium CQ as a 98 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling function of Va for a semiconductor with two valence and conduction bands: at 0.2 and 0.6 eV away from the Fermi level. An effective mass of 0.06mo is assumed, where mo is the free-electron mass. The van Hove singularities, at each band edge, result in corresponding peaks in CQ. For a linear energy-wavevector relation, such as that near the Fermi level in graphene or a metallic CN, the DOS is constant. This is the case considered by Burke [3], and is valid when Va is such that f(E) is approximately zero before the first van Hove singularity is encountered in the integral. Since the higher energy bands are not relevant to the integration under such a condition, v is constant, and the DOS is given by where VF is the Fermi velocity. The result is C hvF which agrees with the expression quoted in Ref. [3]. Note that in Eq. (7.3) CQ does not manifest itself as a multiple of some discrete amount, so "quantum capacitance" is not an appropriate description for a ID semiconductor, unlike in the-metallic 2D and metallic ID CN cases, where the capacitance is truly quantized. 7.3 General Considerations We can now extend our discussion to include the non-equilibrium behaviour for a general, ID, intrinsic semiconductor. All of the numerical results are based on the methods described in Refs. [6] and [7], which consider the cases when transport in the ID semiconductor is either coherent or incoherent, respectively. While these methods were developed in order to describe CNFETs, their use of the effective-mass approximation allows them to be used for any device, and bias, where the semiconductor is described well by this approximation. For phase-incoherent transport, we utilize a flux-balancing approach [7,8] to describe the charge in an end-contacted semiconductor. If we consider only the electrons that are far away from 99 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling the contacts, i.e., in the mid-length region, Eq. (7.1) becomes [8] (7.4) where Vbias is the potential difference between the end contacts, TL(E) and TR(E) are the transmission probabilities at the left and right contacts respectively, T*(E) = TL,TR/(TI, + TR — TLTR) is the composite transmission probability for the entire system, and VA is evaluated in the mid-length region. A similar expression holds for holes. The first term in Eq. (7.4) resembles the equilibrium case, so we expect a similar form for that contribution to CQ. The peak for each contributing band will occur at the same VA, but the overall magnitude will be smaller due to the multiplication by the transmission function. The second term is also similar except that these peaks will now be shifted by Vbias- This is depicted by the dashed curve in Fig. 7.1, where the case illustrated by the solid curve has been driven from equilibrium by Vbias = 0.2 V. Note the splitting of each large peak into two smaller peaks: one at the same point, and the other shifted by Vbias- Of course, the numerical value of the non-equilibrium capacitance depends on the exact geometry considered, as it will influence both VA and the transmission probabilities in Eq. (7.4), but the trends shown here are general and geometry-independent. For the coherent, non-equilibrium case, it is instructive to consider a metal-contacted device, in which the band discontinuities at the metal-semiconductor interfaces are sufficient to allow significant quantum-mechanical reflection of carriers even above the barrier. Further, we restrict our attention to short devices since the importance of coherence effects is diminshed as the device length is increased. Due to the phase-coherence, then, we have a structure very much like a quantum well, even for devices where tunneling through the contact barriers is not important. For our device, we expect quasi-bound states to emerge at the approximate energies „ n2ir 2h 2 ^ ~ 2mL 2 ' where L is the semiconductor length. For m ~ 0.06mn, such as in a (16,0) CN, E, 100 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling 0.45 0.4 0.35 0.3 1" 0.25 LL C o° 0 2 0.15 0.1 0.05 0 -1 -0.5 0 0.5 1 V a ( V ) Figure 7.1: ID quantum capacitance as a function of the local electrostatic potential at equi-librium (solid), and for the mid-length region of an end-contacted semiconductor with a bias voltage of 0.2 V (dashed) between the end contacts. The effective mass is taken to be 0.06mo, and energy bands are situated at 0.2 and 0.6 eV on either side of the Fermi level. 6.3(n/L)2eV, where L is in nanometres. This may be compared with the result for metallic CNs, where the linear energy-wavevector relationship yields a 1/L dependence [3,9]. Fig. 7.2(a) displays CQ as a function of position and VA (in the mid-length region) for this choice of m. The maxima, indicated as brighter patches, show a dependence on VA that reveals the population of quasi-bound states. Moreover, the maxima in position clearly show the characteristic modes expected from our simple square-well analogy. Note that the peak-splitting occurs for coherent transport as well, as shown in Fig. 7.2(b), where the peaks have been split by Vbias — 0.1 V. The main difference, between the coherent and incoherent cases, is the presence of the quasi-bound states. These serve to increase the number of CQ peaks, since each quasi-bound state behaves like an energy band, and they also give rise to a strong spatial dependence. While Fig. 7.2 shows only a single-band, coherent result, inclusion of multiple bands would cause CQ to exhibit peaks corresponding to each band, and to each quasi-bound state. 101 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling ( b) 0.4 > s ra -0.4 -0.3 -0.1 -0.2 0.1 0.2 0.3 0 0 10 20 Distance from source (nm) 0 10 20 Distance from source (nm) Figure 7.2: ID quantum capacitance, in arbitrary units, for a short-channel, phase-coherent semiconductor as a function of position and the local electostatic potential in the mid-length region for applied bias voltages of (a) 0 and (b) 0.1 V between the end contacts. The bright areas indicate higher capacitance. We now elaborate on the above in the context of CNFET modeling. In particular, for the purpose of developing compact models, it would be useful to ascertain the conditions under which the quantum capacitance is small in comparison with that due to the insulator geometry, a regime previously described as the "quantum capacitance limit" [2,10]. To this end, we examine a coaxial CNFET, and treat Va as the potential, with respect to the source contact, on the surface of the CN in the mid-length region. CQ can be considered to be in series with the insulator capacitance C i n s [11], however, the ratio of these capacitances is related not to Va and the gate-source voltage, VGS, but to dVcs/dVa. If the charge accumulation were linear over some bias range, as might be deemed appropriate at the local extrema of CQ, we could relate this ratio directly to the potentials. Knowledge of the " C Q limit" is beneficial since a relatively low CQ implies that changes in Va will closely track changes in VGS, obviating the need to calculate CQ when computing the energy band diagram. Note, however, that CQ cannot be neglected when considering performance 7.4 Application: C N F E T s 102 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling metrics that depend on the total capacitance, i.e., the propagation delay may be dominated by CQ in this limit. We find that CQ C i n s only when Q is small enough as to allow for the employment of a Laplace solution [12] for the position-dependent potential: eliminating the need for a cumbersome self-consistent Schrodinger-Poisson solution. The difference between these solutions is illustrated in Fig. 7.3 for a coaxial CNFET with an insulator thickness and CN radius of 2.5 and 0.6 nm, respectively, and an end contact work function that is 0.6 eV less than that of the CN. Figs. 7.3(a) and (b) correspond to the off and turn-on states, respectively. While equilibrium band diagrams are shown for simplicity, similar trends prevail with the application of a drain-source voltage. For a device dominated by thermionic emission, such as the one depicted here, the discrepancy shown in (a), close to the contacts, will not significantly affect the current calculation, while in (b), the error would clearly be much greater. For a device dominated by tunneling, i.e., if the energy bands had the opposite curvature, a similar discrepancy would result in a large error in the current calculations due to the exponential dependence of the tunneling probability on the barrier shape. (a) 0.2 0.1 > & 0 >* D) O "0.1 c LU -0.2 -0.3, (b) 0.1 > 0 iS _ 0 2 -0.3 0 5 10 15 20 Distance from source (nm) Figure 7.3: Comparison of the equilibrium energy band diagrams, for a model CNFET, at gate-source voltages of (a) 0.2 and (b) 0.32 V, computed via the solutions to a self-consistent Schrodinger-Poisson system (solid), and to Laplace's equation (dashed). The Fermi energy is at OeV. Now, we seek to theoretically quantify the condition under which CQ <C C; n s . From Fig. 7.1, we see that the first local maxima is on the order of 0.3nF/m. For this peak to be insignificant, 103 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling we would require Q n s to be orders of magnitude higher than this. For a 2 nm-thick dielectric in a coaxial device, we would require a relative permittivity of ~ 530 in order to give two orders of magnitude difference between C i n s and CQ. Reports of solid, high-permittivity dielectrics for CN devices [13-16] have quoted values only as high as 175 for the relative permittivity [16], so we conclude that, for realistic dielectrics, we can expect to only marginally enter the CQ limit, and that the first CQ peak will be significant. If we consider an electrolytically-gated CNFET [17], we could perhaps achieve a relative permittivity of 80, and an effective thickness of 1 nm, as considered in Ref. [2], but this would yield C i n s ~ 2 5 C Q , and would, again, only marginally be entering this limit. For a short-channel, phase-coherent device, the requirement for negligible CQ is that the Fermi levels for the injecting contacts should be far away from E\ ~ 6 . 3 / 1 ? eV. If we consider positive applied voltages to the gate and drain, this would imply that qVa should.be more than about SksT below Eg/2+E\. For the long-channel or phase-incoherent cases, this condition is given by Ei = 0, corresponding to the conduction band edge. The relative importance of CQ, computed in the mid-length region of the device, is depicted in Fig. 7.4 for a phase-incoherent device as a function of VGS and the drain-source voltage, VDS, where we note that VDS corresponds to Vbias, and that Va is influenced by both VDS and VGS- Here, the aforementioned peak-splitting for non-zero VDS is clearly evident in the diverging bright lines. Only for low bias voltages can CQ be neglected, as shown by the black region in the centre of the figure. However, this figure also reveals the regions where it becomes approximately constant, i.e., the bias ranges where the series capacitance relationship can be used to estimate Va from V G S [H]- Note, though, that this is a single-band calculation, and these regions may not be as prevalent when higher transverse modes are considered. Finally, we consider the influence of CQ on the transconductance for our model device, which has a doubly-degenerate lowest band. If we employ the Landauer equation [4] for transport in 104 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling 0.41 0.21 -0.21 -0.41 : -0.4 -0.2 0 0.2 0.4 -.0.7 0.6 0.5 0.4 , C y" o 0.3 " 0.2 0.1 0 Figure 7.4: Quantum capacitance for a long-channel CNFET as a function of the gate- and drain-source voltages. Numerical values are displayed as a fraction of the insulator capacitance. two conducting channels, the current is / = J Tn(E) [f(E) - f ( E + qVDS)\ dE - J TP{E) [f(E) - f ( E - qVDS)] dE^ , where Ec = Eg/2 — qVa is the spatially constant conduction band edge in the mid-length region of a long-channel device, Ev = Ec — Eg is the valence band edge, and Tn(E) and TP(E) are the transmission probabilities for electrons and holes, respectively, from one end contact to the other. The transconductance is defined as di 9m = dVcs' which yields 4<7 2 r CQ + Cms J {Tn(Ec) [f(Ec)-f(Ec + qVDS)} ~Tp (Ey) [f (Ey) ~f(Ey- qVDS)}} . (7.5) Note that, if we assume only electron transport with CQ Cjng, low temperature, and high VDS, this expression reduces to the classic Landauer result [4] for two conducting channels - - 4 q 2 T 9m — fi 105 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling which is the ultimate transconductance in this case [7,10]. Fig. 7.5(a) shows the theoretical transconductance, from Eq. (7.5), for our model device, while (b) and (c) show the energy-distribution term (in curly braces), and the capacitance ratio term (in square brackets), re-spectively. The decrease in <7m, at high VQS, is due primarily to the decreasing difference in the contact distribution functions as, for example, Ec becomes closer to qVns- However, the exact magnitude of g m is dependent on the capacitance ratio. Further, CQ will be responsible for additional oscillations in QM, if higher bands or quasibound states are considered in the calculation. Such transconductance features have been observed experimentally in Ref. [18], and have also been described in Ref. [19]. (a) 0.8 Figure 7.5: (a) Electron transconductance for a model CNFET as a function of the gate-source voltage for drain-source voltages of 0.2 (solid) and 0.4 V (dashed). Constituent elements of the theoretical transconductance from Eq. (7.5) are (b) the energy-distribution term (in curly braces), and (c) the capacitance-ratio term (in square brackets). 106 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling 7.5 Conclusions From this theoretical study on the charge-voltage relationship in one- and two-dimensional systems, it can be concluded that: 1. the "quantum capacitance" occurs in discrete quanta for 2D and ID metals if Va is such that the Fermi level falls in a linear portion of the energy-wavevector relationship; 2. for 2D semiconductors, this capacitance is approximately quantized if the bandgap is greater than about lbksT, and varies continuously otherwise; 3. for long, ID systems with parabolic bands, and with Va such that these bands contribute to the charge density, the equilibrium capacitance exhibits maxima that are related to the number of contributing bands at a given energy; 4. application of a bias to a ID semiconductor causes each equilibrium capacitance peak to split into two smaller peaks, with one remaining at the equilibrium position, and the other shifting by the applied bias; 5. the potential in the mid-length region of a ID semiconductor cannot be computed, in general, from potential division due to two capacitors in series due to the nonlinearity of C Q ; 6. for short, phase-coherent structures, the quasi-bound states cause the capacitance peaks to occur at higher local electrostatic potentials, with additional maxima corresponding to the occupation of these states; 7. for a CNFET, it is unlikely that the insulator capacitance can become high enough to allow the quantum capacitance to be neglected in energy band calculations, except in cases where the accumulated charge is low enough that the solution to Laplace's equation is sufficient for the calculation, or if extremely high permittivity dielectrics are used; 8. the quantum capacitance has a significant effect on the transconductance, and should be considered when modeling CNFETs. 107 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling References [1] Serge Luryi, "Quantum capacitance devices," Appl. Phys. Lett., 52(6), 501-503 (1988). [2] Anisur Rahman, Jing Guo, Supriyo Datta, and Mark S. Lundstrom, "Theory of ballistic nanotransistors," IEEE Trans. Electron Devices, 50(9), 1853-1864 (2003). [3] P. J. Burke, "An RF circuit model for carbon nanotubes," IEEE Trans. Nanotechnol., 2(1), 55-58 (2003). [4] David K. Ferry and Stephen M. Goodnick, Transport in Nanostructures (Cambridge Uni-versity Press, New York, 1997). [5] Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976), 1st ed. [6] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson solver for modeling carbon nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Conf. and Trade Show, vol. 3, 65-68 (Boston, U.S.A., 2004). [7] L. C. Castro, D. L. John, and D. L. Pulfrey, "An improved evaluation of the DC perfor-mance of carbon nanotube field-effect transistors," Smart Mater. Struct., 15(1), S9-S13 (2006). [8] L. C. Castro, D. L. John, and D. L. Pulfrey, "Towards a compact model for Schottky-barrier nanotube FETs," Proc. IEEE Conf. on Optoelectronic and Microelectronic Materials and Devices, 303-306 (Sydney, Australia, 2002). [9] Zhen Yao, Cees Dekker, and Phaedon Avouris, "Electrical transport through single-wall carbon nanotubes," Mildred S. Dresselhaus, Gene Dresselhaus, and Phaedon Avouris, eds., Carbon Nanotubes, vol. 80 of Topics Appl. Phys., 147-171 (Springer-Verlag, Berlin, 2001). [10] L. C. Castro, D. L. John, and D. L. Pulfrey, "Carbon nanotube transistors: An evalu-ation," Proc. SPIE Conf. Device and Process Technologies for MEMS, Microelectronics, and Photonics III, vol. 5276, 1-10 (Perth, Australia, 2004). 108 Chapter 7. Quantum Capacitance in Nanoscale Device Modeling [11] Jing Guo, Sebastien Goasguen, Mark Lundstrom, and Supriyo Datta, "Metal-insulator-semiconductor electrostatics of carbon nanotubes," Appl. Phys. Lett., 81(8), 1486-1488 (2002). [12] S. Heinze, M. Radosavljevic, J. Tersoff, and Ph. Avouris, "Unexpected scaling of the performance of carbon nanotube Schottky-barrier transistors," Phys. Rev. B, 68, 235418-1-235418-5 (2003). [13] Adrian Bachtold, Peter Hadley, Takeshi Nakanishi, and Cees Dekker, "Logic circuits with carbon nanotube transistors," Science, 294, 1317-1320 (2001). [14] J. Appenzeller, J. Knoch, V. Derycke, R. Martel, S. Wind, and Ph. Avouris, "Field-modulated carrier transport in carbon nanotube transistors," Phys. Rev. Lett, 89(12), 126801-1-126801-4 (2002). [15] Ali Javey, Hyoungsub Kim, Markus Brink, Qian Wang, Ant Ural, Jing Guo, Paul Mclntyre, Paul McEuen, Mark Lundstrom, and Hongjie Dai, "High-K dielectrics for advanced carbon-nanotube transistors and logic gates," Nature Mater., 1, 241-246 (2002). [16] B. M. Kim, T. Brintlinger, E. Cobas, M. S. Fuhrer, Haimei Zheng, Z. Yu, R. Droopad, J. Ramdani, and K. Eisenbeiser, "High-performance carbon nanotube transistors on SrTi0 3 /Si substrates," Appl. Phys. Lett, 84(11), 1946-1948 (2004). [17] Sami Rosenblatt, Yuval Yaish, Jiwoong Park, Jeff Gore, Vera Sazonova, and Paul L. McEuen, "High performance electrolyte gated carbon nanotube transistors," Nano Lett, 2(8), 869-872 (2002). [18] Minkyu Je, Sangyeon Han, Ilgweon Kim, and Hyungcheol Shin, "A silicon quantum wire transistor with one-dimensional subband effects," Solid-State Electron., 44, 2207-2212 (2000). [19] D. Jimenez, J. J. Saenz, B. Ihiguez, J. Suhe, L. F. Marsal, and J. Pallares, "Modeling of nanoscale gate-all-around MOSFETs," IEEE Electron Device Lett, 25(5), 314-316 (2004). 109 Chapter 8 Method for Predicting ft for C N F E T s 8.1 Introduction Carbon nanotube field-effect transistors ( C N F E T s ) are being seriously considered for meeting the requirements of the 11-nm technology node [1]. Their D C performance is predicted to be superior to that of ultimately scaled silicon M O S F E T s [2,3], and impressive values for drain current and transconductance have already been reported i n prototype devices [4]. The A C capabilities of C N F E T s are not yet so obvious. Thus far, measurements on laboratory devices have been l imited by experimental difficulties and parasitics [5-7]. Thus, the highest reported frequency of 580 M H z , for operation without signal degradation, cannot be viewed as a representative value for an intrinsic device 1 . One way to investigate the A C capabilities of C N F E T s would be to perform A C simulations with the same rigour that has characterized earlier D C simulations [9,10]. This means using a self-consistent Schrodinger-Poisson solver to compute values for the parameters appearing in, for example, a small-signal equivalent circuit, from which a useful metric, such as ft, could be obtained. Through the use of this self-consistent procedure, we expect a more accurate result than that predicted in Ref. [11], where capacitances in the equivalent circuit model were computed assuming a metallic nanotube and electrostatics for an infinitely long coaxial system. © [2005] IEEE. Reprinted, with permission, from L. C. Castro, D. L. John, D. L. Pulfrey, M . Pourfath, A. Gehring, and H . Kosina, "Method for Predicting fo for Carbon Nanotube FETs," IEEE Trans. Nanotechnol., 4(6), 699-704 (2005). Very recently, operation up to 10 GHz has been reported [8], albeit with considerable signal attenuation. 110 Chapter 8. Method for Predicting ft for CNFETs In this paper, we perform a rigorous calculation of the gate-voltage dependencies of both the transconductance [12], and the capacitances, including the so-called "quantum capaci-tance" [13], in order to compute the small-signal, equivalent-circuit parameters, from which our improved estimates of ft for CNFETs are obtained. This analysis reveals a bias depen-dence that is quite unusual, and which may prove useful in voltage-controlled, high-frequency circuitry. 8.2 The Small-Signal Model 8.2.1 Equivalent circuit Starting from Maxwell's first two equations, and considering a system with three electrodes, through which charge can enter or leave the system, it follows that dQ dQs dQd 8Q9 where Q is the total charge within the system and Qs, Qd> and Qg refer to charges associ-ated with each of the device's three terminals, namely: source, drain, and gate, respectively. Labelling displacement currents with a superscript d, it follows from Eq. (8.1) that i ds+ij + i dg = 0. The standard notations are used for the voltages, e.g., the total gate voltage vG comprises a DC voltage VG and an AC small-signal vg. If we suppose that Qs, Qd, and Qg are functions of time through the application of time-dependent voltages vs(t), vo(t), and vG(t), then each displacement current will comprise three terms, e.g., h - Qt U 9 d gt + ^ 9 9 m > where the capacitances come from the set Cjj = T - 7 j ^ , i,j = s,d,g, (8.2) where the minus sign is taken when i ^ j, and the plus sign when i = j [14]. Not all the capacitances in this set are independent, and it can be easily shown that, for example, Cgg = {Cgs + Cgd) = {Csg + Cdg) • 111 Chapter 8. Method for Predicting ft for CNFETs Thus, if we now consider vs and V£> to be held constant, as is appropriate for an estimation of ft in the absence of parasitic source and drain resistances, then it follows that id _ ( R , R \DVGS lg - V^ag + ^dg) ^ • This displacement current, which is also the total gate current ig, can be computed from an equivalent circuit, such as is shown in Fig. 8.1. 'gs G 'gd 's 9mV, gs id Figure 8.1: Small-signal, equivalent circuit for the total AC currents, under the condition that only the potential on the gate is time-dependent. Turning now to the conduction currents, we employ the standard quasi-static approach for a current that depends on two potential differences, i.e., VQS and VDS in this case. In its linear implementation, this leads to a drain conduction current of i°d = 9mVgs + 9dsVds • Here, as we are keeping VDS constant, the term involving the output conductance gds need not be considered. This completes the specification of the small-signal equivalent circuit. It is a well-founded circuit, with the only approximation being the use of quasi-statics to obtain linear expres-sions for the conduction currents. Note that the interfacial conductance of Aq 2T/h, due to transverse-mode reduction on passing from a large, many-mode electrode to a two-mode, quasi-1-D nanotube with a transmission probability T [15], is not shown explicitly in Fig. 8.1, as it is implicit in the transconductance g m . On the basis of the circuit shown in Fig. 8.1, the common-source, short-circuit, unity-current-gain frequency, as extrapolated from a frequency at which the square of the gain rolls off at 112 Chapter 8. Method for Predicting ft for CNFETs —20 dB/decade, is given by ft = 2ir(Csg + Cdg) 8.2.2 Model parameters In order to relate Csg and Cdg to meaningful physical quantities, firstly we split the charges Qs and into two components: Qs = Qse ~T" Qst Qd = Qde + Qdt where Qse and Qde are charges on the actual source and drain electrodes, respectively; and Qst and Qdt are charges on the nanotube that enter via the source and drain electrodes, respectively. Each of these charges is supplied by the appropriate displacement current, as illustrated in Fig. 8.2. A capacitance can now be related to each of the charge components. For the source capacitance, for example, we have C =®Q± = dQse | dQst s g dvc dvG dvc = • CSe + Cst • These capacitive components are readily calculated because their related charges can be com-Source se Lu Wrap around gate to > Q s t lins Semiconducting nanotube Q^i < Insulator Drain Qde : i d d L, Figure 8.2: Charge supply to and through the the source and drain electrodes in a coaxial CNFET. puted from a recently described self-consistent DC Schrodinger-Poisson solver [9]. This solver 113 Chapter 8. Method for Predicting ft for CNFETs has been adapted to employ Neumann boundary conditions at the non-metallic bounding sur-faces in the structure depicted in Fig. 8.2. In our solver, the source- and drain-related nanotube charges, Qst and Q^t, are computed from integrations of the line charges that are related to the wavefunctions associated with carriers communicating with the system via the source and drain, respectively. Wavefunctions are computed via the effective-mass Schrodinger equation with plane-wave solutions assumed in the metal contacts. A phenomenological band discon-tinuity is used to model the electrode-nanotube heterointerfaces as simple Schottky barriers. Normalization of the wavefunctions is achieved by setting the probability density current equal to the current expected from the Landauer equation for ballistic transport. The small-signal parameters Cst and Cdt are computed for a given drain bias via numerical derivatives with a perturbation in the gate voltage on the order of 0.1 mV. Similarly, g m is computed from Landauer's equation. Cse is associated with the change in the charge that resides on the actual source electrode Qse. Similarly, Cde is related to a change in Qde- These charges are computed from appropriate applications of Gauss' Law in integral form. 8.3 Results and Discussion Results are presented for a coaxial transistor structure, as shown in Fig. 8.2, with Schottky-barrier contacts at the source/tube and drain/tube interfaces. This embodiment, which avoids the need to dope the nanotube, and which employs the ultimate "multi-gate" to combat short-channel effects, is being actively pursued experimentally [16]. Here, by way of an example, we consider a (16,0) carbon nanotube with a radius of 0.63 nm, a length Lt = 20 nm, and a relative permittivity of 1 [17]. The insulator has a thickness £;ns = 2.5 nm, and its relative permittivity is taken to be 25, as is appropriate for zirconia, which is used in some high-performance CNFETs [18]. The gate electrode is separated from the source and drain electrodes by L u = 4nm, and has a thickness tg — 3nm. The work function of the gate is taken to be the same as that of the intrinsic nanotube (4.5 eV), whereas the source and drain metallizations have a work function of 3.9 eV. Although this yields an n-type device, inasmuch as the dominant 114 Chapter 8. Method for Predicting ft for CNFETs carriers are electrons, p-type operation is directly analogous, and, owing to the symmetry of the nanotube's band-structure about the midgap energy, can be obtained through the use of a higher work function metal. Thus, the CNFET considered here can be classed as a negative-Schottky-barrier device, such as has been predicted to give DC characteristics that are superior to those of devices with either zero- or positive-Schottky barriers at the end contacts [2]. In negative-barrier CNFETs the conduction current is due to thermionic emission at the source-tube and drain-tube interfaces. Quantum-mechanical reflection at these interfaces, due to the band discontinuities mentioned earlier, leads to resonances, and the appearance of quasi-bound states, at least in nanotubes of the short length considered here. This plays a significant role in determining the values for the model parameters discussed below. An illustrative example of the charged quasi-bound states, and the conduction-band profile in the device, is presented in Fig. 8.3. The example shows charge in the first and second quasi-bound states, and the appearance of charge in an additional quasi-bound state in the potential well at the drain end of the device. z (nm) Figure 8.3: Grey-scale representation of the energy- and position-dependence of the electronic charge in the nanotube at VGS = 0.4 V and VDS = 0.5 V. The uniform columns to the left and right represent the energy range below the Fermi level in the source and drain, respectively. The conduction-band edge is superimposed (solid line). The results presented here are intended to illustrate the ability of the proposed method to provide meaningful estimates of ft for CNFETs. An optimization of the CNFET structure to 115 Chapter 8. Method for Predicting ft for CNFETs suggest an upper bound for ft is not attempted at this stage, but some comments are offered after the discussion of the present results as to the factors that might be important in this regard. All the results presented below are for operation at VDS = 0.5 V. The various components of the capacitance are shown in Fig. 8.4. Considering, firstly, the tube 6 5 LL «> 4 CD O CD 3 S- 2 CO O 1 c Of 0 0.2 0.4 0.6 0.8 1 Gate voltage (V) Figure 8.4: Components of the source and drain capacitances. capacitance Cst due to charge injected from the source, the peak at around VQS = 0.35 V corresponds to alignment in energy of the source Fermi level and the first quasi-bound state for electrons in the nanotube [13]. The peak in Cdt is displaced from the peak in Cst by approximately VDS [13], and corresponds to alignment in energy of the drain Fermi level and the first quasi-bound state. Considering now the capacitances associated with changes in charge on the actual end contacts, it can be seen that these are relatively bias-independent. In fact, this is due to Cse and Cde being dominated by the regions of overlap of the end contacts with the edges of the gate electrode. Obviously, this capacitance could be reduced by increasing the separation between the end contacts and the edges of the gate, or by making the gate electrode thinner, or by making the end contacts more "needle-like" [19]. The latter could be achieved by utilizing metallic 116 Chapter 8. Method for Predicting ft for CNFETs nanotubes for the source and drain. The total capacitances associated with each electrode are shown in Fig. 8.5(a). 12 Gate voltage (V) Figure 8.5: ft and its components, (a) The total capacitances associated with the source, Csg, and with the drain, Cdg; (b) transconductance; (c) ft-We now discuss the transconductance, as shown in Fig. 8.5(b). Firstly, note that the choice of end-contact work function renders the device unipolar, except at very low bias. Thus, the hole contribution to the transconductance at moderate and high VQS is negligible. Secondly, it can be seen that g m reaches a maximum, and then decreases as VQS increases. This phenomenon has been reported elsewhere [12,13]. The overall reduction in g m at high VGS relates to the increasing electron injection from the drain as the potential energy in the mid-length region of the tube is reduced. The considerable structure in the transconductance plot is due to the presence of the 117 Chapter 8. Method for Predicting ft for CNFETs quasi-bound states referred to earlier. As VGS increases, the conduction band edge is pushed below the source Fermi level, and as the quasi-bound states cross this level, g m increases. Thus, the situation is not dissimilar to that which gives rise to the peaks in capacitance. Indeed, at low temperatures, our simulations reveal that the peaks in transconductance and capacitance do occur at the same biases (see Fig. 8.6). Evidently, in going from T=4K to T=300K, thermal broadening causes peaks that are close together to merge, with the taller one dominating. Thus, the second peak in transconductance dominates the first, while the opposite is true in the capacitance case. 16 i i i i i i 0 0.2 0.4 0.6 0.8 1 Gate voltage (V) Figure 8.6: Bias dependence, at two temperatures, for (a) capacitance, and (b) transconduc-tance. Note that the peaks in transconductance coincide with peaks in capacitance at the lower temperature. The changes in capacitance and transconductance discussed above lead to a very interesting and unusual bias dependence in the cut-off frequency ft, as illustrated in Fig. 8.5(c). For the 118 Chapter 8. Method for Predicting ft for CNFETs example of a 20 nm tube, as used here, ft peaks at about 600 GHz. This is a long way from the value of 4THz, which can be inferred from a recent model that ignored the bias dependence of the transconductance and capacitances, and attributed the device capacitance to that of an infinite coaxial system, with the quantum capacitance given by that of a metallic, rather than a semiconducting, nanotube [11]. In the finite coaxial system considered here, the mid-tube quantum capacitance is not explicitly identified, as it does not relate to the terminals on which the equivalent circuit is based. It is contained within Cst and Cdt, the peak values of which turn out to have comparable magnitudes to the electrostatic electrode capacitances, Cse and C<ie, in this particular example; thus, the overall capacitance shows significant bias dependence. In future studies, we will attempt to optimize the Schottky-barrier CNFET as regards high-frequency performance. However, in concluding this paper, we can make a few comments regarding the parameters that are likely to be of importance. Clearly, the magnitude of the band discontinuity at the end contacts is significant. We have used a value of -5.5 eV for the depth of the metal conduction band below the Fermi level [9]. Higher values may be appropriate for noble metals of the type that appear suited to end contacts for CNFETs, in which case one can expect more quantum mechanical reflection and a lower transconductance, leading to a reduced ft. Increasing the barrier height by increasing the work function of the end-contact metal (in the case of n-type devices) will significantly degrade performance because of the appearance of a thick tunneling, barrier in the ungated portion of the nanotube. Changing the nanotube to one of larger bandgap, yet maintaining the barrier height at about —Eg/2, may also degrade transconductance, at a given bias, because the ON current can be expected to be smaller, at least at low gate bias. Further, the peaks in transconductance and capacitance will be displaced to higher VGS as more depressing of the conduction-band edge under the gate will be required to align the quasi-bound states with the source Fermi energy. Increasing the ungated regions L u should be advantageous in a negative-barrier device because the electrostatic electrode capacitance will be reduced without a degradation in g m . We have been quite aggressive in the vertical scaling of our device as we have used a high permittivity and a small thickness for the gate insulator. Relaxing these values will not change the resonances, but will shift the 119 Chapter 8. Method for Predicting ft for CNFETs peaks in capacitance and transconductance to higher biases, due to the poorer electrostatic coupling between gate and nanotube. Finally, we should mention that in non-Schottky barrier CNFETs, in which the source and drain regions are formed by doping the ungated portions of the nanotube [20,21], potential wells will form between the end contacts and the intrinsic, gated part of the nanotube, and could lead to resonances somewhat similar to those described here if the doped regions are short enough. 8.4 Conclusions From this work on AC small-signal simulations of Schottky-barrier carbon nanotube field-effect transistors, it can be concluded that: 1. the generic small-signal, equivalent-circuit model for FETs is appropriate for studying the quasi-static AC performance of CNFETs, provided the model parameters are rigorously derived; 2. in the case of short nanotubes with Schottky-barrier end contacts, a resonant structure is formed, leading to the appearance of quasi-bound states; 3. the quasi-bound states lead to gate-bias dependencies of the capacitances and transcon-ductance, which, in turn, give rise to a short-circuit, unity-current-gain frequency ft, which displays a dependence on VQS that is unusual in its oscillatory nature. References [1] Ken David, "Silicon research at Intel," (2004). [Online.] Available: ftp://download.intel.com/research/silicon/Ken_David_GSF_030604.pdf. [2] L. C. Castro, D. L. John, and D. L. Pulfrey, "Carbon nanotube transistors: An evalu-ation," Proc. SPIE Conf. Device and Process Technologies for MEMS, Microelectronics, and Photonics III, vol. 5276, 1-10 (Perth, Australia, 2004). [3] Jing Guo, Mark Lundstrom, and Supriyo Datta, "Performance projections for ballistic carbon nanotube field-effect transistors," Appl. Phys. Lett, 80(17), 3192-3194 (2002). 120 Chapter 8. Method for Predicting ft for CNFETs [4] Ali Javey, Jing Guo, Qian Wang, Mark Lundstrom, and Hongjie Dai, "Ballistic carbon nanotube field-effect transistors," Nature, 424, 654-657 (2003). [5] David J. Frank and Joerg Appenzeller, "High-frequency response in carbon nanotube field-effect transistors," IEEE Electron Device Lett, 25(1), 34-36 (2004). [6] J. Appenzeller and D. J. Frank, "Frequency dependent characterization of transport prop-erties in carbon nanotube transistors," Appl. Phys. Lett, 84(10), 1771-1773 (2004). [7] Dinkar V. Singh, Keith A. Jenkins, J. Appenzeller, D. Neumayer, Alfred Gill, and H.-S. Philip Wong, "Frequency response of top-gated carbon nanotube field-effect transistors," IEEE Trans. NanotechnoL, 3(3), 383-387 (2004). [8] X. Huo, M. Zhang, Philip C. H. Chan, Q. Liang, and Z. K. Tang, "High frequency S parameters characterization of back-gate carbon nanotube field-effect transistors," IEDM Tech. Digest, 691-694 (2004). [9] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson solver for modeling carbon nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Conf. and Trade Show, vol. 3, 65-68 (Boston, U.S.A., 2004). [10] Jing Guo, Supriyo Datta, and Mark Lundstrom, "A numerical study of scaling issues for Schottky-barrier carbon nanotube transistors," IEEE Trans. Electron Devices, 51(2), 172-177 (2004). [11] Peter J. Burke, "AC performance of nanoelectronics: Towards a ballistic THz nanotube transistor," Solid-State Electron., 48, 1981-1986 (2004). [12] L. C. Castro, D. L. John, and D. L. Pulfrey, "An improved evaluation of the DC perfor-mance of carbon nanotube field-effect transistors," Smart Mater. Struct., 15(1), S9-S13 (2006). [13] D. L. John, L. C. Castro, and D. L. Pulfrey, "Quantum capacitance in nanoscale device modeling," J. Appl. Phys., 96(9), 5180-5184 (2004). 121 Chapter 8. Method for Predicting ft for CNFETs [14] Yannis P. Tsividis, Operation and Modeling of the MOS Transistor (McGraw-Hill, Toronto, 1987). [15] Supriyo Datta, Electronic Transport in Mesoscopic Systems, vol. 3 of Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Cambridge University Press, New York, 1995). [16] Wolfgang Hoenlein, Franz Kreupl, Georg Stefan Duesberg, Andrew Peter Graham, Maik Liebau, Robert Viktor Seidel, and Eugen Unger, "Carbon nanotube applications in micro-electronics," IEEE Trans. Compon. Pack. T., 27(4), 629-634 (2004). [17] Frangois Leonard and J. Tersoff, "Dielectric response of semiconducting carbon nan-otubes," Appl. Phys. Lett, 81(25), 4835-4837 (2002). [18] Ali Javey, Hyoungsub Kim, Markus Brink, Qian Wang, Ant Ural, Jing Guo, Paul Mclntyre, Paul McEuen, Mark Lundstrom, and Hongjie Dai, "High-«; dielectrics for advanced carbon-nanotube transistors and logic gates," Nature Mater., 1, 241-246 (2002). [19] Enzo Ungersboeck, Mahdi Pourfath, Hans Kosina, Andreas Gehring, Byoung-Ho Cheong, Wan-Jun Park, and Siegfried Selberherr, "Optimization of single-gate carbon-nanotube field-effect transistors," IEEE Trans. Nanotechnol, 4(5), 533-538 (2005). [20] Ali Javey, Ryan Tu, Damon B. Farmer, Jing Guo, Roy G. Gordon, and Hongjie Dai, "High performance n-type carbon nanotube field-effect transistors with chemically doped contacts," Nano Lett, 5(2), 345-348 (2005). [21] Yu-Ming Lin, Joerg Appenzeller, and Phaedon Avouris, "Novel carbon nanotube FET design with tunable polarity," IEDM Tech. Digest, 687-690 (2004). 122 Chapter 9 High-frequency Capability of Schottky-Barrier CNFETs 9.1 Introduction Carbon nanotube field-effect transistors (CNFETs) are predicted to have superior DC char-acteristics to those of foreseeable silicon MOSFETs [1,2]. Experimentally, impressive values for drain current and transconductance have already been reported in prototype devices [3]. However, the AC performance of CNFETs has not yet received much attention [4], and the likelihood of high-frequency operation needs to be established. Here, we take a step in this direction by predicting two useful figures of merit for high-frequency transistors, namely: the unity current-gain and unity power-gain frequencies, fx and / m a x , respectively. We employ the standard small-signal method [5], abetted by a self-consistent Schrodinger-Poisson solver [6], in order to obtain preliminary estimates of these valuable figures of merit. In particular, we consider the cylindrically gated device depicted in Fig. 9.1. The CNFET geometry consists of a nanotube with radius Rt, coated by an insulator of thickness t i n s , wrapped by a cylindrical gate of thickness tg, and capped at the ends by planar source and drain contacts of radius t\ns+tg+Rt- These end contacts are separated longitudinally from the gate by a distance L u . A version of this chapter has been accepted for publication. L. C. Castro, D. L. Pulfrey, and D. L. John, "High-Frequency Capability of Schottky-Barrier Carbon Nanotube FETs," Solid-State Phenomena (2005). Accepted December 22, 2005. 123 Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs •-9 ^ Suurco t i n e V Wrap-around gato < • Drain 1 y Semiconducting nanotube R . t Insulator ^ ', , w L, Figure 9.1: Structure of the modeled CNFET. 9.2 Model Since the CNFET is much like a traditional field-effect transistor, we may employ the standard equivalent circuit model for this device [5], where the small-signal parameters themselves, such as the transconductance and the various transcapacitances, are computed by taking numerical derivatives based on the results of self-consistent DC charge-voltage calculations [6,7]. Through our use of a Schrodinger-Poisson solver for these DC results, we are able to include the effects of geometry, quantum capacitance, and spatial non-uniformity of the charge in our calcula-tions. While we have previously presented results for the intrinsic ft with only three circuit elements [7], here we consider the additional effects of series resistances associated with each of the terminals, and we include the remaining capacitance components. This allows us to com-pute the extrinsic fr, and also to consider / m a x - For clarity, we include the standard equivalent circuit in Fig. 9.2, where we note that the subscripts s, d and g refer to the source, drain and gate, respectively. In the usual way, we consider the fx and / m a x expressions, which may be arrived at by extrapolating the characteristic decay in gain, from its value at an appropriate low frequency, to the 0 dB point. 9.3 Results and Discussion The model device in this study employs a (16,0) nanotube of radius 0.63 nm, length 30 nm, bandgap 0.62eV, unity relative permittivity, and effective mass 0.06mo, where mo is the free 124 Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs Figure 9.2: Full small-signal equivalent circuit for the CNFET. The transcapacitance Cm = electron rest mass. Moreover, tins is 2.5 nm, tg is 3nm, the insulator permittivity is 25 as is appropriate for zirconia [8], and L u is 5nm for this initial investigation. The work function of the nanotube is taken to be 4.5 eV, while that of the end-contacts is 3.9 eV, yielding negative-Schottky barrier, n-type transistor operation. All results are for a drain-source voltage of 0.5 V. Fig. 9.3 illustrates the intrinsic parameters for our device. The oscillatory peaks in the capacitances and transconductance are related to the formation of quasi-bound states in the short channel [7,9]. With the modulation of the applied voltage, the states, indicated by the bright patches in Fig. 9.4, are shifted in energy by band bending in the channel. As they cross the source or drain Fermi level, they become populated, and we obtain peaks in the charge accumulation and, consequently, in the capacitances. Since the charge accumulation affects the amount of band bending in the channel through our self-consistent DC calculations, we also see peaks in the transconductance gm. The g m behaviour is complicated due to its dependence both on the charge and on the Fermi distributions at the injecting contacts. In Fig. 9.5, we present our main results, the predictions of fx and / r a a x for our model CNFET. Fig. 9.5(a) shows fx for Rs = Rd set to lkft, lOkfi, and 100 kfi, and we recall that Rg has no influence on. this figure of merit. Note that the values of the parasitic resistors Rs and Rd are chosen to be comparable to the contact resistance that results from mode constriction when carriers pass from a many-moded material to a material with only a few modes [10]. In the carbon nanotube case, we consider the lowest two degenerate modes in energy for an equivalent contact resistance of around 6-7 kQ. Note that the contact resistance is automatically included in our self-consistent 125 Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gate voltage (V) Gate voltage (V) Figure 9.3: Capacitances and transconductance for the model device. z (nm) Figure 9.4: Charge density for the model device subject to a gate-source voltage of 0.38 V. Brighter patches indicate higher charge density, while a portion of the conduction band edge is shown by the white line. The energy values are referenced to the source Fermi level. DC calculations, so an explicit resistor is not needed to represent it in the equivalent circuit.. Thus, the resistors shown in Fig. 9.2 are solely parasitic. It is evident that the maximum value of fx occurs at the first peak in g m . Turning now to / m _ x , shown in Fig. 9.5(b), we focus on the effect of Rg, and show results for R g set to 100Q, lkfl, and 10kf2, with R s = Rd = 10kf.. Again, a pronounced peak coincides with the first peak in g m . 126 Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gate voltage (V) Gate voltage (V) Figure 9.5: Extrapolated figures of merit: (a) fx with R s = set to IkO (solid), 10kf2 (dashed), and 100 kQ (dotted), and (b) / m a x with R s = R d = lOkft for R 9 set to 100ft (solid), lkft (dashed), and lOkft (dotted). 9.4 Conclusions From this simulation of the high-frequency performance of CNFETs, it can be concluded that, in short-channel devices, with negative-barrier Schottky contacts for the source and drain, and with very short ungated regions, and where coherent transport is possible, resonance effects can lead to a strong bias dependence of the high-frequency figures of merit, and / m a x - At the resonance peaks, these frequencies may reach the THz level. References [1] L. C. Castro, D. L. John, and D. L. Pulfrey, "An improved evaluation of the DC perfor-mance of carbon nanotube field-effect transistors," Smart Mater. Struct., 15(1), S9-S13 (2006). [2] Jing Guo, Mark Lundstrom, and Supriyo Datta, "Performance projections for ballistic carbon nanotube field-effect transistors," Appl. Phys. Lett, 80(17), 3192-3194 (2002). [3] Ali Javey, Jing Guo, Qian Wang, Mark Lundstrom, and Hongjie Dai, "Ballistic carbon nanotube field-effect transistors," Nature, 424, 654-657 (2003). 127 Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs [4] Dinkax V. Singh, Keith A. Jenkins, J. Appenzeller, D. Neumayer, Alfred Gill, and H.-S. Philip Wong, "Frequency response of top-gated carbon nanotube field-effect transistors," IEEE Trans. Nanotechnol, 3(3), 383-387 (2004). [5] Yannis P. Tsividis, Operation and Modeling of the MOS Transistor (McGraw-Hill, Toronto, 1987). [6] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson solver for modeling carbon nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Conf. and Trade Show, vol. 3, 65-68 (Boston, U.S.A., 2004). [7] L. C. Castro, D. L. John, D. L. Pulfrey, M. Pourfath, A. Gehring, and H. Kosina, "Method for predicting fT for carbon nanotube FETs," IEEE Trans. Nanotechnol, 4(6), 699-704 (2005). [8] Ali Javey, Ryan Tu, Damon B. Farmer, Jing Guo, Roy G. Gordon, and Hongjie Dai, "High performance n-type carbon nanotube field-effect transistors with chemically doped contacts," Nano Lett, 5(2), 345-348 (2005). [9] D. L. John, L. C. Castro, and D. L. Pulfrey, "Quantum capacitance in nanoscale device modeling," J. Appl. Phys., 96(9), 5180-5184 (2004). [10] Supriyo Datta, Electronic Transport in Mesoscopic Systems, vol. 3 of Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Cambridge University Press, New York, 1995). 128 Chapter 10 Extrapolated / m a x for CNFETs 10.1 Introduction The frequency / m a x , at which the extrapolated power gain becomes unity, is a well-established figure-of-merit for characterizing the high-frequency performance of transistors. A useful, com-pact expression for / m a x is available for heterojunction bipolar transistors [1], but, in Si metal-oxide-semiconductor field-effect transistors, the need to consider the electrical properties of the substrate makes for a more complicated situation. However, in carbon nanotube field-effect transistors ( C N F E T s ) , the substrate is not an active part of the device, so the traditional, small-signal equivalent circuit, in which there are no elements representing the substrate [2, p.441], can be used as a basis for deriving a useful expression for / m a x - Moreover, because of the small size of C N F E T s , the quasi-static approximation should be valid up to very high frequencies. Here, starting from the small-signal parameters of the equivalent circuit, we systematically make a series of approximations that lead to compact expressions for the extrapolated / m a x -These expressions are shown to be applicable over a wide range of conditions, and to be useful in guiding the design of high-frequency devices. A version of this chapter has been published. L. C. Castro and D. L . Pulfrey, "Extrapolated / m a x for Carbon Nanotube Field-Effect Transistors," Nanotechnology, 17(1), 300-304 (2006). Online at ht tp: / /www .iop.org/journals/nano. 129 Chapter 10. Extrapolated fmax for CNFETs 10.2 Modeling Procedures The small-signal, extrinsic ^-parameters for the equivalent circuit previously shown in Fig. 9.2 are given by the standard expressions [2, p.440]: Zlle = V22/Y + Rsg Z\2e = -Vu/Y + Rs Z2U = -V2i/Y + Rs (10.1) Z22e = yil/Y + Rsd Y = y\ 12/22 - 2/122/21 , where R s g = R s + Rg, Rsd = R s + Rj, and the intrinsic 2/-parameters are [3, p.378]: 2/n = j u ( C g s + C g d ) V l 2 = ~ 3 U j C a d (10.2) J/21 = 9m ~ j u { C m + C g d ) 922 = 9ds + j v ( C s d + Cgd) • The transcapacitance Cm relates non-reciprocal capacitance pairs, and is given by, for exam-ple, Cdg — Cgd- From the components in Eq. (10.1), an expression for the radian frequency LOT at which the short-circuit, common-source, current gain reaches unity, when extrapolated from some lower frequency at which the square of the gain rolls-off at -20dB/decade, follows, namely [2, p.441]: — = — [1 + g d s R s d ] + RsdCgd , (10.3) where the intrinsic "cut-off" frequency is given by {Cgs + Cgd) The assumptions made in arriving at the expressions for the extrapolated unity-current-gain frequencies are that the frequency at which the extrapolation can properly begin is subject to 130 where Chapter 10. Extrapolated / m a x for CNFETs the following restrictions: u 2 « g 2 J ( C m + C g d ) 2 (10.5) J 2 <C gm/(ARs) (10.6) u, 2 « g 2 J { C d g + B R s ) 2 (10.7) « (CO T + B R s d ) 2 / { A R s d f , (10.8) -<4 = CdgCgd — CggCdd B ~ Cgg9ds + C g d g m Cgg  = Cgs + Cgd Cdd = Csd + Cgd . The above limitations should easily be satisfied by CNFETs intended for operation at frequen-cies of several hundreds of GHz . For the power gain, we use Mason's unilateral gain [4]: r / = k21e-*12e|2 q ) 4 mzneMz22e) ~ K ( « 1 2 e ) K ( « 2 1 e ) ] ' ^ ' One further restriction on the extrapolation frequency is required to obtain an expression for U with the desired dependence on UJ~2: u>2 < B 2/A 2 . (10.10) In fact, Eq. (10.10) is a more stringent restriction than the fourth assumption of Eq. (10.8), which is, therefore, rendered redundant. It follows, after lengthy algebraic manipulation, that U can be written in the form that is familiar for bipolar transistors [1]: " - c f e - < 1 0 n ) where (i?C)eff is an effective time constant. Because the assumptions we wish to make in order to simplify the expression for / m a x affect both U>T and (RC)es we elect not to isolate the 131 Chapter 10. Extrapolated /max for CNFETs expression for the latter [1], but, instead, to work on the expression for the reciprocal cyclic frequency reff, where 2 _ (-RC)eff and <ff = , (10-12) /max = -T^— • (10.13) With the assumptions made so far, reff is given by Teff,l = {Rg{Cgg + 2RCB) + Rc [Cgg - C„ +2Csd + R c B + £ {gm + 2gdsy] } , (10.14) where, for convenience, we have assumed similar source and drain contacts, and set R s = R d = Rc-To make progress in simplifying Eq. (10.14), one has to compare component values, which, because of their bias- and device-dependence, cannot be expected to result in relations that are as generally applicable as the frequency limitations stated earlier in Eqs. (10.5)-(10.8) and (10.10). We start by asserting Cgs = Cgd . (10.15) The motivations for doing this are the small size and longitudinal symmetry of CNFETs: the electrodes are inevitably very close together, so the extrinsic contributions to Cgs and Cgd will be significant; and the symmetry would make them equal. We can anticipate this equality breaking down at low- and high-gate bias, when the electrode-dependent quantum-capacitance contribution to Cgs and Cgd, respectively, is particularly significant [5]. Using Eq. (10.15) in Eq. (10.14) leads to a considerable simplification: 7eff,2 = — (1 + —) VR9 + Rc)[l + Rc(9m + 2gds)\ - (10.16) 9m \ 9m ) Finally, in the interests of further simplification, we suggest: 9m » 2gds . (10.17) This inequality may break down in CNFETs with small-diameter (large-bandgap) nanotubes, for which the transconductance is generally less than in those with large-diameter tubes. The 132 Chapter 10. Extrapolated /max for CNFETs result of this additional assumption is a very compact expression: C 2 7e2ff,3 = — &Rg + Rc){l + 9mRc) • (10.18) 9m In the next section we evaluate the validity of Eqs. (10.14), (10.16) and (10.18) for several Schottky-barrier CNFETs. The component values are evaluated as described previously [5], using a Schrodinger-Poisson solver [6], with the inclusion of the complex band structure of the nanotube [7]. We found that it was not necessary to consider more than the lowest, doubly-degenerate band for the tubes and bias ranges considered in this work. 10.3 Results and Discussion In seeking ultimate performance limits we examine devices of the coaxial structure shown previously in Fig. 1.8, but we base values for the physical properties on those of presently realizable planar structures, such as a recent, high-DC-performance device [8]. All the devices considered here have a gate of length L g = 50 nm and of thickness tg = 20 nm, an insulator relative permittivity of 16 (HfO^), and Pd end-contacts of radius tc = 4nm. Unless otherwise stated, the contact length is L c = 100 nm, the gate underlaps are L u s = Lud = 5nm, and the contact resistances are computed from a Pd resistivity of 0.48 kfi-nm, which can be inferred from Ref. [8]. The data of Ref. [9] was used for the tube-dependent, end-contact barrier heights, while the work function of the gate was set equal to that of the nanotube [10]. The latter assignment is arbitrary in view of the lack of information on other factors, such as oxide charge, that will affect the threshold voltage in practice, and serves only to change the effective gate potential. The gate resistance can be expected to have a large effect on fmax [11], but, in the present absence of knowledge about practical gate connection configurations, we take, unless otherwise stated, R g = lkS.. With the dimensions listed above, for example, R c « 0.9 kQ. The trends in the capacitances illustrated in Fig. 9.2 have been discussed previously [5,12], and for the particular devices described here Cgs and Cgd are on the order of 10 aF. Firstly, we consider Device 1, which has a nanotube diameter dt = 1.7 nm (taken to corre-spond to a tube of chirality (22,0)), for which the Pd contacts produce a negative barrier for 133 Chapter 10. Extrapolated /max for CNFETs ~100 % 50 o Q. Figure 10.1: Unilateral power-gain for Device 1 using Eq. (10.9) (solid), and Eq. (10.11) with Eq. (10.14) (dots), Eq. (10.16) (circles), and Eq. (10.18) (crosses). The inset magnifies the curves near OdB. VGs = VDS = -0.5 V. holes of — 0.04 eV [9]. The combination of low barrier height and an insulator thickness of i i n s = 2.5 nm should produce a device of high transconductance. Mason's power gain U is shown in Fig. 10.1, from which it is clear that, for this particular device at the given biases of VQS = VDS — —0.5 V, all the assumptions leading to Eqs. (10.14), (10.16) and (10.18) are rea-sonable. The effects of the assumptions do appear, however, at different VQS, as illustrated in Fig. 10.2. It can be seen that the lowest Teg is « 0.16 ps, which corresponds to fmax ss 500 GHz. At high, negative, gate bias, injection of holes from the drain is facilitated [11], leading to an increase in the quantum-capacitance contribution to Cgd- Thus, assumption Eq. (10.15) overestimates Cgs, leading to Eq. (10.16) overestimating the true reff at the most negative bias considered. The effect of assumption Eq. (10.17) is more severe at high bias because g m falls off considerably. Again, this is due to holes being injected into the nanotube from the drain: the resulting hole flow bucks that issuing from the source, reducing g m . Moreover, g d s rises in that bias range, ultimately yielding a ratio 2gds/gm « 1 near VQS — —0.8V and invalidating assumption Eq. (10.17). We turn now to Device 2, which has a nanotube diameter dt = 0.8 nm (taken to correspond to a tube of chirality (10,0)), a positive hole-barrier of 0.3 eV at the Pd end-contacts [9], an increased insulator thickness i ; n s of 8 nm and shorter contacts L c of 30 nm. The higher barriers and thicker insulator will reduce g m below that of Device 1. These features should lead to a lower / m a x than predicted for Device 1. However, this should be mitigated somewhat by lower 134 Chapter 10. Extrapolated / m a x for CNFETs 0.6 0.5 w~ 0.4 Q . 0.3 0.2 0.1 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 Gate voltage (V) Figure 10.2: reg estimates for Device 1 using Eq. (10.9) (solid), and Eq. (10.13) with Eq. (10.14) (dots), Eq. (10.16) (circles), and Eq. (10.18) (crosses). VDS = -0.5 V. capacitances Cgs and Cgd, due to the larger £;ns and smaller Lc. The results shown in Fig. 10.3 show that / m £ t x is, indeed, significantly lower than for Device 1. Interestingly, res,2 is a better approximation to the true reff in this case, which is perhaps unexpected, given that the shorter Lc and thicker i j n s should reduce the inter-electrode capacitances that would otherwise help to equalize Cgs and C a_. The reason lies in the positive barrier heights and larger bandgap, which restrain charge injection into the nanotube (see inset to Fig. 10.3), thereby reducing the quantum capacitance contributions to Cgs and Cgd even more than the above physical changes reduce the inter-electrode contributions. The higher barrier at the source generally reduces the drain current, so both gm and gds are affected, and regt3 is no worse an approximation, relatively speaking, than it was for Device 1. One of the reasons for the low / m a x shown for Device 2 in Fig. 10.3 is that the effective gate bias is lower than for Device 1 because of the higher threshold voltage due to the thicker gate insulator. While this could be ameliorated by application of a higher negative bias to the gate, or by using a higher work function for the gate metal, Fig. 10.3 is useful because it illustrates that our equations are reasonable over an effectively different bias range than applies to Device 1. So far, we have used resistances of R c « 0.9 kJ. and « 0.3 kCl for Devices 1 and 2, respectively, and R g = 1 kO. To examine the effect of parametrically changing these values, results are 135 Chapter 10. Extrapolated /max for CNFETs Gate voltage (V) Figure 10.3: / m a x estimates for Device 2, at VDS — —0.5 V, using Eq. (10.9) (solid) and Eq. (10.13) with Eq. (10.14) (dots), Eq. (10.16) (circles), and Eq. (10.18) (crosses). The in-set illustrates the valence band edge profiles near the source contact for Devices 1 (dotted) and 2 (solid) at VGS = VDS = —0.5 V. Energies are referenced to the source Fermi level. presented in Fig. 10.4 for Device 1. The error in the estimation of the prediction of / m a x was examined, after making each of the assumptions leading to the three expressions for reff. For the first two, the error in / m a x is less than 1 % over the range of resistances shown in Fig. 10.4(a). Fig. 10.4(b) depicts the case after, additonally, making assumption Eq. (10.17), and shows that the error is greatest at large R c . This is because the approximated term, (gm + 2gas) —> g m in simplifying Eq. (10.16), is multiplied by the square of Rc, whereas R g appears without exponentiation. Fig. 10.4 indicates that the compact expressions are useful over a wide range of resistance values. Finally, we demonstrate the utility of the compact expressions in guiding design towards CN-FETs that should lead to improved / m a x - Obviously, reducing Cgd would be helpful because of its domination of the output admittance. Eqs. (10.16) and (10.18) highlight this by elucidating the direct dependence of reff on Cgd- By contrast, reff has a lesser dependence on transcon-ductance. One way to trade-off g m against Cgd would be to increase t-ms. Ways to reduce Cgd directly would be to shorten the drain contact L c , and to increase the gate-drain underlap Lud-Although the functional dependencies of g m and Cgd on tms and Lud are not readily attainable, the beneficial effect to Device 1 of making these changes is illustrated in Fig. 10.5, where the 136 Chapter 10. Extrapolated /max for CNFETs 10 10" 10" 10" 10 J R c (Ohms) 1 0 1 1 0 1 R g(Ohms) R c (Ohms) 1 0 1 1 0 1 R g (Ohms) Figure 10.4: Error in / m a x prediction for Device 1, incurred by the use of: (a) E q . (10.16); (b) Eq. (10.18). VDS = - 0 . 5 V . 100 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 Gate voltage (V) Figure 10.5: / m a x for Device 1 and / m a x improvement for Device 1 computed from E q . (10.9) without assumptions, both at VDS — -0 .5 V . Solid line: Device 1 as originally specified; dotted line: Device 1 wi th t-ms = 8 n m , Lc = 30 nm, Lus = 5nm, and Lud = 15 nm. peak value of / m a x is raised by about 15 % to 580 G H z . 10.4 Conclusions From this study of the extrapolated / m a x in Schottky-barrier C N F E T s it can be concluded that: 1. compact expressions for / m a x can be derived that are useful over wide ranges of physical properties, parasitic resistances and gate biases; 2. the compact expressions provide a useful guide to the design of high-frequency devices; 3. /max values in excess of 0.5 T H z should be realizable. 137 Chapter 10. Extrapolated /max for CNFETs References [1] M. Vaidyanathan and D. L. Pulfrey, "Extrapolated fmax of heterojunction bipolar transis-tors," IEEE Trans. Electron Devices, 46, 301-309 (1999). [2] W. Liu, Fundamentals of III-V Devices (John Wiley, Toronto, 1999). [3] Yannis P. Tsividis, Operation and Modeling of the MOS Transistor (McGraw-Hill, Toronto, 1987). [4] S. J. Mason, "Power gain in feedback amplifier," IRE Trans, on Circuit Theory, 1, 20-25 (1954). [5] L. C. Castro, D. L. John, D. L. Pulfrey, M. Pourfath, A. Gehring, and H. Kosina, "Method for predicting fx for carbon nanotube FETs," IEEE Trans. NanotechnoL, 4(6), 699-704 (2005). [6] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, "A Schrodinger-Poisson solver for modeling carbon nanotube FETs," Tech. Proc. of the 2004 NSTI Nanotechnology Conf. and Trade Show, vol. 3, 65-68 (Boston, U.S.A., 2004). [7] H. Flietner, "The E(k) relation for a two-band scheme of semiconductors and the applica-tion to the metal-semiconductor contact," Phys. Stat. Sol. (b), 54, 201-208 (1972). [8] Ali Javey, Jing Guo, Damon B. Farmer, Qian Wang, Erhan Yenilmez, Roy G. Gordon, Mark Lundstrom, and Hongjie Dai, "Self-aligned ballistic molecular transistors and elec-trically parallel nanotube arrays," Nano Lett, 4(7), 1319-1322 (2004). [9] Zhihong Chen, Joerg Appenzeller, Joachim Knoch, Yu-Ming Lin, and Phaedon Avouris, "The role of metal-nanotube contact in the performance of carbon nanotube field-effect transistors," Nano Lett, 5(7), 1497-1502 (2005). [10] Jijun Zhao, Jie Han, and Jian Ping Lu, "Work functions of pristine and alkali-metal inter-calated carbon nanotubes and bundles," Phys. Rev. B, 65, 193401-1-193401-4 (2002). 138 Chapter 10. Extrapolated /max for CNFETs [11] L. C. Castro, D. L. Pulfrey, and D. L. John, "High-frequency capability of Schottky-barrier carbon nanotube FETs," Solid-State Phenomena (2005). Accepted December 22, 2005. [12] D. L. John, L. C. Castro, and D. L. Pulfrey, "Quantum capacitance in nanoscale device modeling," J. Appl. Phys., 96(9), 5180-5184 (2004). 139 Chapter 11 Conclusion and Recommendations for Future Work This thesis describes a body of work that yielded insightful results regarding the DC and AC performance of carbon nanotube field-effect transistors (CNFETs). Various models were devised and applied to several types of devices, with the aim of explaining device behaviour and related physical phenomena, as well as ultimately obtaining predictive performance metrics for guiding device design. Given that various papers were published from this work, the thesis was composed by their incorporation as chapters herein, arranged in a roughly chronological order. The preliminary version of a CNFET compact model (Chap. 2) introduced a self-consistent, non-equilibrium compact model for Schottky-barrier CNFETs, incorporating a crude JWKB tunneling expression for positive barriers at the contacts. It represented an improvement on concurrent models in the literature in that it included the effects of the Schottky barriers— assuming transparent contacts overestimates both the channel charge and current. However, the model was still based on compact electrostatics, i.e., it implicitly assumed the behaviour of an infinite-length tube, and relied on other equilibrium models to estimate the barrier shapes out of equilibrium. As discussed in Chap. 3, an equilibrium electrostatics model served as a guide to the behaviour of the potential profile in the channel under non-equilibrium conditions, and was the basis for the fitting parameters in the aforementioned compact model. Finally, the non-equilibrium charge did not account for metal-induced gap states nor include any short-channel effects, both of which require a Schrodinger-Poisson (SP) solver. The compact model was then employed in a preliminary evaluation of CNFET DC performance, 140 Chapter 11. Conclusion and Recommendations for Future Work as outlined in Chap. 4. With the incorporation of hole transport, the ambipolar feature of CNFETs was revealed, whereby under certain bias- and work-function conditions it is possible to have injection of electrons and holes in different ranges of the device's /-^-characteristics. In particular, at high positive VDS applied to an n-type transistor, hole injection from the drain increases the CNFET current even though it has already reached the saturation regime as typically defined for silicon devices. A logarithmic I-VGS plot for a given device and non-zero VDS reveals a v-shaped characteristic that indicates the electron and hole branches of the current. From these, it was possible to examine the expected performance of various metal contacts, given that their particular work function would yield a different-polarity Schottky barrier for electrons or holes. Such "work function engineering" allows for negative barriers to facilitate majority carrier transport and stave off injection of the supposedly minority carrier. The suppression of ambipolar behaviour is particularly important for digital applications, as it severely affects the ON/OFF ratio and subthreshold slope of the CNFET. The evaluation of CNFETs further studied their limits for ON current, conductance and transconductance. With the aim of improving the non-equilibrium charge computation, a SP solver was elaborated, as illustrated in Chap. 5. This model was based on the effective mass approximation, the pa-rameters for which were drawn from a tight-binding nearest-neighbour calculation. The results were self-consistent solutions of the phase-coherent charge with the local electrostatic potential, allowing for a more accurate study of short-channel ballistic devices. The inclusion of evanes-cent modes injected from the metal contacts into the forbidden energy gap of the nanotube, also known as metal-induced gap states, revealed the importance of these for positive-barrier devices, since the exponential dependence of the tunneling currents on the barrier shape makes these CNFETs more susceptible to slight variations in the near-contact carrier density. The model also served to demonstrate the importance of including phase-coherent charge in short devices, because the carrier distributions in energy were highly modified from their equilibrium form, owing to the formation of quasi-bound states in the channel. Subsequent to the implementation of the SP solver, its equivalence to the compact model under certain conditions was proven and described in Chap. 6. An improvement for the trans-141 Chapter 11. Conclusion and Recommendations for Future Work mission function was sought, with the aim of obtaining a more accurate estimate of the I- V-characteristics in negative barrier devices, i.e., those that had yielded the best DC performance. It was also found that this kind of CNFET would not, as was previously thought possible, come as close to the ultimate DC current limits of a ballistic carbon nanotube channel. This is due to the fact that the quantum mechanical transmission of above-barrier carriers is less than unity in much of the current-carrying energy range. The formation of quasi-bound states, between two large potential steps in close spatial proximity, produces a weak localization of the current in energy, i.e., restricts which energy ranges transmit or not. Another study, presented in Chap. 7, delved further into the concept of "quantum capacitance" (CQ) in nanoscale transistors, and illustrated its application to CNFET devices. In particular, the "quantum capacitance limit" being mentioned in the literature was methodically exam-ined and its range of validity specified. This limit is represented by the insulator capacitance overwhelming that of the semiconductor, and is thus desirable because it allows the gate to exercise a stronger control of the channel over a wider range of biases. This limit was claimed to be reachable with CNFETs, but was found here to require unviable material properties. In CNFETs, unlike in a transistor with a 2DEG channel, the ID density of states was found to produce a CQ that was neither quantized nor readily computed from a simple formula. Moreover, with the employment of the aforementioned compact model, the bias dependence of CQ was also studied, and found to reveal highly nonlinear features, owing particularly to the non-equilibrium nature of the carrier distributions. Having studied the DC performance of the CNFETs, a small-signal model, in conjunction with an improved Schrodinger-Poisson solver, was employed to compute their AC figures-of-merit fx and / m a x , a first in the CNFET literature. In its final version, the SP solver included the effects of interband tunneling and an energy-dependent correction to the effective mass. The small-signal model also proved useful in studying the effects of parasitic capacitances and resis-tances brought about by the employment of various device dimensions and contact geometries. Several trends were studied, particularly the oscillations of the capacitances and transconduc-tance with gate bias, and their effect on fx and / m a x - These oscillations were correlated to 142 Chapter 11. Conclusion and Recommendations for Future Work the interaction between the Fermi energy levels of the injecting contacts and the quasi-bound states in the channel, the position in energy of the latter being controlled by the gate. This behaviour was demonstrated clearly by analysing the low-temperature operation of CNFETs. Further illustrated by the small-signal analyses was the dependence of the figures-of-merit on the nanotube bandgap. It was found that larger tubes, with smaller bandgaps, benefited from higher transconductances but these gains were often offset by increased capacitances, brought about by the larger charge densities present in the devices. Finally, compact expressions for /max were derived, from which the importance of the gate-drain capacitance and the transcon-ductance were ascertained. Extrinsic fx and / m a x values in the order of tenths of THz were computed for representative devices. However, the high-frequency models employed herein have been quasi-static in nature and the validity of this must be established if they are to be employed at the frequencies that are be-ing expected of these devices as potential replacements or supplements of current technologies. Various works examining this issue in other nanoscale devices can be found in the recent liter-ature, and many could be employed in modeling CNFETs. Another avenue to be pursued is to study the role of various scattering mechanisms at the frequency and bias ranges of interest here. At these dimensions, and under certain conditions, scattering could play an important role in nanoscale devices. Moreover, as the device length approaches the nanotube diameter, the effective-mass, mode-space approach employed here should give way to a more detailed quantum model in order to account for variations at the atomistic scale of quantities such as the density-of-states. Finally, the relevance and applicability to carbon nanotube transistors of the concept of kinetic inductance, which has emerged in the recent literature, merits further investigation. Based on the current efforts of the semiconductor industry, and barring improvements in man-ufacturability and prototype performance, research on carbon nanotube transistors may be supplanted by that based on other nanowires, given they are easier to manufacture repeatably. Moreover, existing technologies that are already well-developed, such as those based on silicon, may resolve their impending scalability problems and prove to be more viable solutions. Still, 143 Chapter 11. Conclusion and Recommendations for Future Work carbon nanotubes remain an attractive material for employment in state of the art semicon-ducting devices, owing to their high current capability, near-ballistic transport, and minute scale. 144 Appendix A Complete Schrodinger-Poisson Model Here we describe a numerical model for the CNFET, an improvement on that of Chap. 5 and presented in more detail. The objective is to describe the final version of the Schrodinger-Poisson solver developed in this work. The key changes are the inclusion of interband tunneling and the handling of both metallic and doped-semiconductor contacts. Throughout the ensuing description of the model it is assumed that the contacts are metallic (as treated in the preceding chapters), unless otherwise stated. We seek a self-consistent solution of the potential V, charge Q, and current I for the CNFET. This is accomplished via the employment of the Schrodinger, Poisson and Landauer equations, which are useful inasmuch as they yield an accurate portrait of the electronic charge in a ballistic device where quantum-mechanical effects (e.g. tunneling, resonance) are present. The problem being modeled here is specified by the system CV = Q + B (A.l) Q = q}2Db / Gs,b [1 - u - fs] + GD,b [1 - u - fD] dE (A.2) b J Emm I = T^2Db (fs-fo)TdE (A.3) Hip = Eip, (A.4) where C and B are, respectively, a linear differential operator and a boundary condition vector for Poisson's equation, Db is the degeneracy factor of sub-band b, Qc>b = Gc,b(z,E) = (ipty)c,b is the local density of states associated with injection from contact c, ip is the wavefunction of a carrier with energy E, fc = f(E — fic) is the contact Fermi function with Fermi level fxc, 145 Appendix A. Complete Schrodinger-Poisson Model T = T(E) is the transmission function, and Ti is the effective-mass Hamiltonian in Schrodinger's equation. The unit step function, referenced to the midgap energy Eo (also known as the charge neutrality level), is specified by 1 , E > E 0 u = u ( E - E0) = ' 0 , E < E 0 . To address Eq. (A.l), we deal specifically with the coaxial geometry of the CNFET as shown in Fig. 1.8 and recall the discussion in Sect. 1.3.1, noting that care must be taken with discretization and derivative discontinuities. Given the charge vector Q, which is non-zero only for elements on the tube surface, and boundary condition vector B, we compute the potential vector V by inversion, i.e., V = £~X(Q + B). In building the operator C and vector B, we employ three equations for the interior points of the simulation space (all points excluding the contact interfaces and open boundaries): (1) since we only consider charge on the tube surface, a jump discontinuity for the normal-component of the electric flux is enforced there (all points in set St, solid line at p « 0.6nm in Fig. A.l) ; (2) a homogeneous Neumann condition is applied on the axis of revolution by symmetry arguments (all points in set >Sa, dash-dot line in Fig. A.l); (3) a Laplace solution is carried out for all other interior points (in set Si). Thus, dV £ins n dp dV - e t — Q y(p,z)eSt R- 2-KRtto dv - Q - = 0 V(p,z)eSa bAf l d V d2V n w , , _ ^ + -pJp- + W = ° *(P>*)e*' where Rt is the tube radius, ei n s and et are, respectively, the relative permittivities of the in-sulator and nanotube, eo is the permittivity of free-space, and the ID charge density Q has been uniformly smeared around the tube circumference. Open boundaries (dotted lines in Fig. A.l) are treated with a homogeneous Neumann condition, equivalent to assuming that the normal displacement is zero, and this is typically valid with appropriate choices of contact dimensions [1]. Points lying on a contact surface are assigned fixed (Dirichlet) boundary condi-tions, specified by the bias voltages. Note that in the case of doped-semiconductor contacts, no Dirichlet conditions are specified; homogeneous Neumann conditions are enforced in the entire 146 Appendix A. Complete Schrodinger-Poisson Model 20 15 | 10 10 20 30 z (nm) 40 50 Figure A . l : Simulation space for CNFET with metal contacts. Neumann boundary conditions are specified on dotted perimeter, and Dirichlet conditions on edges of contacts. Linear charge specified on nanotube surface (solid line at p « 0.6nm). perimeter of the simulation space. The solution of Eq. (A.l) is effected via the finite-difference scheme, chosen primarily for sim-ulation speed. This involves discretizing the two-dimensional solution space of Fig. A . l into a grid and solving difference equations for every grid point [2]. We now turn to the computation of the linear charge density Q from the ID time-independent Schrodinger's equation (Eq. (A.4)). The general ID effective-mass Hamiltonian is given by where mb is the carrier effective mass in sub-band b (equal for electrons and holes, and constant across the device), h is Dirac's constant, and EPOT is the potential energy of the electron or hole. Employing a plane wave solution, Schrodinger's equation reduces to where ip(z, E) is the electron wavefunction. The wavevector k = k(z, E) in the nanotube is given by h 2 \E - EQ\ - A 6 \ E - E 0 \ 2 - A 2 2Ab \E - EQ\ > A 6 \E-E0\ < AB, (A.5) where A;, is the energy distance between the bottom of sub-band b and EQ, and the two energy ranges distinguish, respectively, between carriers outside and inside the bandgap (also defined 147 Appendix A. Complete Schrodinger-Poisson Model as propagating and evanescent modes). The intra-gap wavevector employs a correction for the non-parabolicity of the e-k relation there, via an energy-dependent effective mass [3]. This is consistent with the so-called "complex band picture" [4] that provides a smooth transition of the e-k relation between the conduction and valence bands, as illustrated in Fig. A.2. Note that for the lowest sub-band (and omitting b subscript) the condition \E — EQ\ > A = Eg/2 may be rewritten as a double requirement E — EQ > A and E — EQ < —A, or E > EQ + A = EC and E < E0 - A = EV. > o LU O *—< 0) > TO S >> O) i CD c LU 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 y i „ „ j? m „T ... / / T . electron . ,.,„„.,„„.,~.^,^< N. \ <- imaginary k hole i ' real k -> \ . Wavevector Figure A.2: Complex bands in semiconducting nanotube: comparison between nearest-neighbour tight binding (solid), effective mass (dashed) and energy-dependent effective mass within bandgap, as per Ref. [3] (circles). The agreement between the energy-dependent effective mass and the tight-binding case is clear. The distinction between electrons and holes is of concern insofar as the carriers react differently to an applied field. This is reflected in the wavefunction computation by labeling a carrier according to its energy-position in the e-k relation, i.e., at a given position in the device channel, the solution ip^ip is counted as an electron if E > Eo or as a hole otherwise. This book-keeping does not keep track of the actual electron and hole concentrations, but rather models deviations from charge neutrality. Under this scheme, it is possible that a carrier traversing a region of strong band-bending, while maintaining a constant energy under the elastic conditions assumed here, may be labeled differently at different points of its path. This allows for the possibility of interband tunneling, as in the case, for example, of a reverse-biased 148 Appendix A. Complete Schrodinger-Poisson Model p-n junction, where one may describe the situation by two equivalent pictures: that a valence-band hole on the p-side tunnels through the bandgap into the conduction-band on the n-side, or that a conduction-band electron on the n-side tunnels into the p-side valence-band. Note that the implementation herein only allows for uncoupled interband tunneling, i.e., it only accounts for transitions within a corresponding pair of valence and conduction sub-bands. With regards to the current computation, the existence of open states on the opposite side of the device is ensured in the derivation of the Landauer equation, which takes the Fermi level of both contacts into account and applies identically to an electron or hole picture. In the metal contacts the e-k relation is given by fc2 = ^{EIEF - \E - fxc\) , \E - Mc| < EREI, (A.6) where nib is the effective mass of the nanotube sub-band into which the carriers are being injected, and ERE{ is a fitting parameter related to the type of metal being used. A half-full metal band is assumed, such that EIE{ = E t 0 p - Mc = /•*_ — -^bottom and fic = EQ>C (the contact charge neutrality level). A typical value used in the simulations was 5.5eV, as estimated for various metals commonly used in CNFETs from the data in Ref. [5]. The metal e-k relation and density of states are illustrated in Fig. A.3. 0 1 2 3 0 1 2 3 Wavevector (nm-1) Density of states (nm-1) Figure A.3: Dispersion relation and density of states in ID metal contact. 149 Appendix A. Complete Schrodinger-Poisson Model Given that the wavefunction amplitude is arbitrary to any multiplying factor in Eq. (A.4), we must normalize it to compute the correct carrier density in the.device. In many quantum physics problems, such as the near-infinite quantum well, the electron is confined in space, and normalization may be effected by a spatial restriction on the electron's location, via the requirement that the integral of the positional probability density in the confinement region is unity. In our problem, however, this is not viable, as we have open boundary conditions at the contacts (infinite-length contact regions) and we cannot assume the entire carrier-wavefunction is confined within the device or its near vicinity. We thus resort to a flux normalization, whereby the amplitude of the wavefunction is related to the injecting carrier flux, emanating from the metallic contact regions. In essence, this is achieved by equating, at each energy level being considered, the particle current to that in Landauer's equation (Eq. (A.3)). We illustrate the normalization for carriers injected from the source contact; injection from the drain contact may be derived in analogous fashion. With a wavefunction of the form i>{z, E) = A e i k ^ z , where A = A(z, E) is a constant to be determined from the normalization, we may write the particle current with effective mass m at some energy E as and the Landauer current as (A.7) m IL(E) = ^ - f ( E - u . s ) T ( E ) , (A.8) TTtl where the transmission function, defined as a ratio of outbound to inbound particle currents, is k ( L u E ) \ A ( L t , E ) \ 2 [ ] k(0,E) \A(0,E)\* • Equating Eq. (A.8) with Eq. (A.7) in the drain region (z = Lt), i.e., at the output for a source-injected wave, we obtain the requirement for normalization 150 Appendix A. Complete Schrodinger-Poisson Model Note that this normalization condition is related to the ID density of states in the metal contact (assuming parabolic bands): , 2 dk 2m ,, ^ E) = -,dE = ^  (A' 10) where the apparent discrepancy by a factor of 2 is due to the fact that g(E) includes both ±fc-states, while |^4|2 is related to a single-direction flux. Substitution of either Eq. (A.5) or Eq. (A.6) into Eq. (A.9) yields the normalization condition for, respectively, semiconductor or metal contacts. Note that the effective-mass Hamiltonian is the simplest, yet quickest to simulate, implemen-tation of Eq. (A.4). More detailed models were not pursued here, as we did not expect large discrepancies, for the devices simulated in this work, between the current and more elaborate models. A major reason for this is that the ID density of states of CNs (see Fig. 1.4) should lend itself well to the employment of the parabolic approximation, given that most carriers reside near the sub-band edge, where the approximation is at its best. An agreement in charge profiles would yield commensurate accuracy in the computation of band profiles and currents. Very recently, some works have begun to address the validity of the effective mass approxi-mation in CNFET performance computations [6,7], with Ref. [7] supporting the use of this approximation, for the sake of computational savings, under most practical bias conditions. Regarding the computation of the wavefunction (i.e., solving Eq. (A.4) for tp), a scattering ma-trix method was employed in lieu of the simpler transfer matrix method so as to circumvent nu-merical issues related to rounding errors [8]. The implementation of the algorithm for obtaining the wavefunction profile along the channel, for a particle at a given energy, is rather straight-forward and will not be detailed here. However, it should be noted that in solving Eq. (A.2) for a given Qc,b (computed from the appropriate ip's), an adaptive integration scheme should be used, given that the integrand can be highly peaked, particularly in short-channel devices where quasi-bound states are prominent. Here we employ the Adaptive Simpson Quadrature method, modified to process various energy ranges in parallel, a procedure that speeds up computation by reducing the number of function calls to the system's computational bottleneck, Eq. (A.4). 151 Appendix A. Complete Schrodinger-Poisson Model We normally solve the non-linear Schrodinger-Poisson system for the potential and charge density using the Picard iterative scheme. The (A;+l)-th iteration is given by Vlb+i = V)b - a L ~ l r k (A.ll) rk = CVK-[Q(VK) + B], where rk is the residual of the A>th iteration, and 0 < a < 1 is a damping parameter. Note that if a = 1, the scheme reduces to back-substitution. The complete algorithm for achieving a self-consistent solution under a given set of simulation parameters is as follows: 1. build matrix L and vector B, and initialize A; to 1; 2. compute VQ by solving Laplace's equation (Q = 0 in Eq. (A.l)); 3. solve Schrodinger's equation for all relevant E to get corresponding >^'s; 4. integrate V»'s to obtain carrier density QK(Vk-i); 5. compute new VK by solving Poisson's equation with new QK in Eq. (A.l l ) ; 6. check that ||rfc||oo has reached tolerance and go to step 3 if not. Typically, a solution is deemed to have converged if the absolute potential reached a tolerance of 1 /LiV. In the actual implementation, a further check is performed on the current, as computed with Eq. (A.3), such that the error was at most InA. The choice of a has a strong influence on convergence and unfortunately, the appropriate value depends on too many simulation pa-rameters to be ascribed a simple formula. In general, a « 0.2 virtually guarantees convergence, but often at prohibitive simulation times; in most cases, a value of a sa 0.7 gave a better performance trade-off. An alternative solution method, the Gummel iterative scheme, is employed for simulations that have difficulty converging with the above method. Under this scheme, we employ a quasi-Fermi level (EqF(z)) formulation that effectively localizes the charge-this is important because the charge density obtained with Schrodinger's equation is highly non-local. The delocalized charge 152 Appendix A. Complete Schrodinger-Poisson Model Qapprox is specified by an equation identical to Eq. (A.2), except the functions Qcf, are replaced by the position-independent ID density of states (akin to that in Eq. (A.10)). The outer loop is solved by undamped fixed-point iterations, while an inner loop to solve for the potential as a function of Qapprox employs a Newton's method, where the derivative <9<3approx/<9V required for the Jacobian operator is analytic. The algorithm is given by the following steps: 1. build matrix C and vector B; make initial guess for Eqp; 2. inner loop to find Vr(Qapprox(-SgF)), employing Newton's method: compute Qapprox^gF) and analytic dQapprox /^Vcs , update potential with Kpdate = - ( £ - dQa.ppTox./dV)~ 1(CV - Q a p P r o x - B), and loop until tolerance reached; 3. compute new Q{V) using potential from step 2 and Eq. (A.2); 4. check if V = C~ 1(Q + B) has reached desired tolerance relative to last iteration of outer loop; if not, find new EqF by inverting Q(Eqp) and go to step 2. In simulations with doped-semiconductor contacts, this method produces much better conver-gence than Picard. It is also found that the initial guess for the quasi-Fermi level plays a significant role in whether convergence is reached. A typical initial guess is to make the source and drain Fermi levels reach into a quarter of the (intrinsic) channel region and make the centre of the channel region take an average of the contact Fermi levels. The best results in using an initial guess were obtained by employing previous solutions at close values of bias for the same device geometry. Once a converged solution has been attained, we have a self-consistent description of the charge and potential profiles in the device. The drain current may then be computed from Eq. (A.3). A . l Sample Results The model described above and variations of it were used to obtain much of the data in this thesis. Here, in the interest of establishing its validity, we directly compare this model with experimental data from the literature and with a more detailed, atomistic model. We present 153 Appendix A. Complete Schrodinger-Poisson Model results for two distinct devices. Firstly, we show an approximate fit to the I- V characteristics of the high-performance SB-CNFET device of Ref. [9], based on the data of Figure l(c,d) of that work. Secondly, we compare the results of this model to that of the more detailed, atomistic approach in Ref. [10], as applied to a p-i-n tunneling device. In the comparisons, the e-k relation of the nanotube is used in three permutations, defined by the simulation flag "F", which takes on the value F=0 when the e-k relation is parabolic (top component of Eq. (A.5)) for all energies, F=l when the e-k relation is described exactly by Eq. (A.5), and F=2 when the e-k relation has an energy-dependent effective mass (bottom component of Eq. (A.5)) for all energies, i.e., for both \E — EQ\ < A;, and \E — EQ\ > A(,. Two sub-bands were included for all simulations in this section, although omitting the second sub-band would likely have a minor effect in the comparison given the lowest sub-band accounts for nearly all charge and current. Gate voltage (V) Drain voltage (V) Figure A.4: Agreement between Schrodinger-Poisson model and data from experimental device of Ref. [9]: (a) I-VGs plot for VDS = -0.3 V, and (b) I-VDS plot for (from top to bottom of figure) VQS — —0.43, —0.73 and —1.03 V (these values correspond to VQS = —0.1, —0.4 and —0.7V in Ref. [9]; see text for explanation). Solid lines are experimental results, while symbols are simulated values. Simulation parameters are given in text. In fitting the curves of Ref. [9], as illustrated in Fig.A.4, we employ a (20,0) nanotube (£'9=0.52eV), a self-aligned gate with L g = 50nm (underlap L u = 0) and thickness tg = 4nm, an insulator of thickness i i n s = 8 nm and relative permittivity of 15 (HfOg), and Pd end-contacts (4>=5 eV) of radius tc = 6 nm and length L c = 4 nm. We set ETef — 1 eV for the metal contacts, 154 Appendix A. Complete Schrodinger-Poisson Model and F=2. Note also that all the simulation curves in Fig. A.4 are offset in the gate bias by 0.33 V in order to better match the experimental curves—this can be attributed to the poten-tial presence of trapped oxide charges in the actual device, causing a shift in the transistor threshold voltage Vt. No other scaling or manipulation was employed. 1 0 " s 10" O c ' 2 Q 10" f + + + F=0 x F=1 o F=2 • NEGF -0.2 0 0.2 0.4 0.6 Gate Voltage (V) 0.8 Figure A.5: Comparison of the solver described in this section to the atomistic simulations of Ref. [10], in modeling the I-V characteristics of a p-i-n device. Refer to text for parameters employed in simulation and an explanation of figure caption. In another comparison, we illustrate in Fig. A. 5 the simulated /- V characteristics for a p-i-n device, as given by this model and one of more detail from Ref. [10]. The device parameters are: (16,0) tube, L a = 7nm, L u = 0, L c = 10 nm, tg — 0.1 nm, £ m s = 2nm, em s = 3.9, and VDS = 0.4 V. The p-type source and n-type drain nanotube contacts are doped, respectively with +0.39 and —0.39 carriers per nanometre. The solid circles in Fig. A.5 are the results of the model in Ref. [10], and the remaining points are obtained by this work, in the three permutations of the e-k relation as described above. Note that because the p-i-n transistor is a tunneling device, and correctly estimating the interband tunneling current is more difficult than estimating currents dominated by propagating modes, the discrepancy between this model and the atomistic simulation of Ref. [10] likely indicates an upper bound on the error of this model. 155 Appendix A. Complete Schrodinger-Poisson Model References [1] D. L. McGuire and D. L. Pulfrey, "Error analysis of boundary condition approximations in the modeling of coaxially-gated carbon nanotube field-effect transistors," Phys. Stat. Sol. (a) (2006). Accepted February 21, 2006. [2] G. D. Smith, Numerical Solutions of Partial Differential Equations: Finite Difference Methods (Clarendon Press, Oxford, 1985), 3rd ed. [3] H. Flietner, "The E(k) relation for a two-band scheme of semiconductors and the applica-tion to the metal-semiconductor contact," Phys. Stat. Sol. (b), 54, 201-208 (1972). [4] T.-S. Xia, L. F. Register, and S. K. Banerjee, "Quantum transport in carbon nanotube transistors: Complex band structure effects," J. Appl. Phys., 95(3), 1597-1599 (2004). [5] Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976), 1st ed. [6] Andres Godoy, Zhicheng Yang, Umberto Ravaioli, and Francisco Gamiz, "Effects of non-parabolic bands in quantum wires," J. Appl. Phys., 98, 013702-1-013702-5 (2005). [7] S. O. Koswatta, N. Neophytou, D. Kienle, and M. S. Lundstrom, "Dependence of DC characteristics of CNT MOSFETs on bandstructure models," IEEE Trans. NanotechnoL (2006). To be published. [8] David Yuk Kei Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B, 38(14), 9945-9951 (1988). [9] Ali Javey, Jing Guo, Damon B. Farmer, Qian Wang, Erhan Yenilmez, Roy G. Gordon, Mark Lundstrom, and Hongjie Dai, "Self-aligned ballistic molecular transistors and elec-trically parallel nanotube arrays," Nano Lett, 4(7), 1319-1322 (2004). [10] David Llewellyn John, Simulation studies of carbon nanotube field-effect transistors, Ph.D. thesis, University of British Columbia, Vancouver, Canada (2006). 156 

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