modeling of carbon nanotube field-eﬁect transistorsnano/pub/castro_thesis.pdf · 2006. 6. 23. ·...

Modeling of Carbon NanotubeField-Effect Transistors

by

Leonardo de Camargo e Castro

B.A.Sc. (Electrical Engineering), The University of British Columbia, 2001

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Electrical and Computer Engineering)

The University Of British Columbia

July 2006

c© Leonardo de Camargo e Castro 2006

Abstract

In this thesis, models are presented for the design and analysis of carbon nanotube field-effecttransistors (CNFETs). Such transistors are being seriously considered for applications in theemerging 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-massSchrödinger Equation (SE) to compute the electron and hole charges in the nanotube, and byusing the Landauer Equation to compute the drain current. A Schrödinger-Poisson (SP) solveris 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 thecurrent predicted by the Landauer Equation. The scattering matrix solution is employed tocompute the wavefunction, and an adaptive integration scheme to obtain the charge. Overallconvergence is sought via the Picard or Gummel iterative schemes. An AC small-signal circuitmodel, employing the DC results from the SP solver, is also constructed to obtain estimates ofthe 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 fordifferent device applications. The model qualitatively agrees with some experimental results inthe literature, and gives confidence that the performance of coaxial devices, when they becomeavailable, will be well predicted by the models. In the AC regime, it was found that undersomewhat ideal operating conditions the operating limit of these devices might just reach intothe 1-10 THz regime.

In addition to the development of rigorous modeling procedures for CNFETs, a preliminarycompact model is developed, in which some of the essence of the detailed model is distilled intoa set of simpler equations, which may prove useful in guiding device design towards CNFETsfor 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 11.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Electronic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Carbon Nanotube Field-Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1 Planar Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Coaxial Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Modeling Coaxial CNFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5 Specific Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs 362.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 3. Electrostatics of Coaxial Schottky-Barrier CNFETs 443.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Coaxial Nanotube Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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Chapter 4. An Evaluation of CNFET DC Performance 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4 DC Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Ambipolarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5.2 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5.3 Subthreshold Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5.4 ON Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5.5 Transconductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 5. A Schrödinger-Poisson Solver for Modeling CNFETs 755.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 6. An Improved Evaluation of the DC Performance of CNFETs 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Correspondence of the Compact and Quantum Models . . . . . . . . . . . . . . . . . . . . . . . . . 866.3 Quantum-Mechanical Reflection for the Thermionic Case . . . . . . . . . . . . . . . . . . . . . . . 896.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 7. Quantum Capacitance in Nanoscale Device Modeling 967.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.2 Equilibrium Quantum Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2.1 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2.2 One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.3 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.4 Application: CNFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Chapter 8. Method for Predicting ft for CNFETs 1108.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.2 The Small-Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2.1 Equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.2.2 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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8.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter 9. High-frequency Capability of Schottky-Barrier CNFETs 1239.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Chapter 10. Extrapolated fmax for CNFETs 12910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.2 Modeling Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13010.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Chapter 11. Conclusion and Recommendations for Future Work 140

Appendix A. Complete Schrödinger-Poisson Model 145A.1 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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List of Figures

1.1 Hybridized carbon atom and graphene lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Carbon nanotube lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Graphene energy dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Nanotube energy dispersion relation, density of states, and carrier velocity . . . . . . . . . . 81.5 Nanotube properties for various tube radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Examples of planar CNFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Examples of electrolyte-gated CNFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Coaxial CNFET structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9 Solution to Laplace’s equation in 2D cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . 211.10 Electric field on nanotube surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.11 CNFET band diagrams for Laplace solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1 Model geometry: closed cylindrical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Conduction energy band diagram for various bias conditions . . . . . . . . . . . . . . . . . . . . . . . 402.3 Electron distribution in the mid-length region of nanotube . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Drain current-voltage characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Comparison of drain I-V characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Equilibrium energy band diagram: (16,0) tube, tins = 5.6 nm and ²ins = 3.9 . . . . . . . . 483.2 Equilibrium energy band diagram: (16,0) tube, tins = 30.9 nm and ²ins = 3.9 . . . . . . . 493.3 Equilibrium energy band diagram: (16,0) tube, tins = 5.6 nm and ²ins = 19.5 . . . . . . . 503.4 Sub-threshold current: (16,0) tube, tins = 5.6 nm and ²ins = 3.9 . . . . . . . . . . . . . . . . . . . . 523.5 Non-equilibrium energy band diagram: (16,0) tube, tins = 5.6 nm and ²ins = 3.9 . . . . 533.6 Drain I-V characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 Coaxial CNFET model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Band diagram illustrating CNFET ambipolarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 ID-VGS for various contact work functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Band diagrams for various contact work functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Ratio of equilibrium quantum capacitance to insulator capacitance . . . . . . . . . . . . . . . . . 674.6 ID and (b) gm as a function of gate-source voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.7 ID-VDS for various contact work functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.8 Band diagrams for various gate work functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 CNFET net carrier density as a function of position and VDS . . . . . . . . . . . . . . . . . . . . . . 805.2 Conduction band edges for VDS = 0 and 0.4V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Conduction band edges and transmission probabilities for electrons . . . . . . . . . . . . . . . . 815.4 Energy distribution of mid-channel, source-originated electron concentration . . . . . . . 82

6.1 Drain current versus gate-source voltage for various models . . . . . . . . . . . . . . . . . . . . . . . . 916.2 Drain current and transconductance for various models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Transmission probabilities of source-injected electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.4 Drain current versus drain-source voltage for various models . . . . . . . . . . . . . . . . . . . . . . . 93

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List of Figures

7.1 Equilibrium 1D quantum capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Non-equilibrium 1D quantum capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.3 Energy band diagram comparison for Laplace and Schrödinger-Poisson solutions . . 1037.4 CNFET quantum capacitance as a function of gate- and drain-source voltages . . . . 1057.5 CNFET transconductance and its constituent elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.1 Small-signal equivalent circuit for ft calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.2 Charge supply to and through the source and drain electrodes . . . . . . . . . . . . . . . . . . . . 1138.3 Charge density in CNFET as a function of position and energy . . . . . . . . . . . . . . . . . . . 1158.4 Components of the source and drain capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.5 ft and its components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.6 Bias and temperature dependences of capacitance and transconductance . . . . . . . . . . 118

9.1 Structure of the modeled CNFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.2 Small-signal equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.3 Capacitances and transconductance for the model device . . . . . . . . . . . . . . . . . . . . . . . . . 1269.4 Charge density versus energy and position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.5 Extrapolated figures of merit for various contact resistances . . . . . . . . . . . . . . . . . . . . . . 127

10.1 Unilateral power-gain for Device 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.2 τeff estimates for Device 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13510.3 fmax estimates for Device 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.4 Error in fmax prediction for Device 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.5 fmax for improved Device 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.1 Simulation space for CNFET with metal contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147A.2 Complex bands in semiconducting nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.3 Dispersion relation and density of states in 1D metal contact . . . . . . . . . . . . . . . . . . . . . 149A.4 Agreement with experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154A.5 Agreement with atomistic simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

vii

List of Symbols

k WavevectorE EnergyV Electric potentialm Carrier effective massT Temperaturez Position along nanotube axis of symmetryρ Radial distance from nanotube axis of symmetryχt Electron affinity of semiconducting nanotubeφ WorkfunctionEg Energy bandgapEc Conduction band edgeEv Valence band edgeEF , µ Fermi energy levelE0 Charge neutrality level∆ Distance between sub-band bottom and charge neutrality levelEvac Vacuum energy levelLt Nanotube lengthLu Gate underlapLg Gate lengthLc Source/drain contact length (when assumed equal)Rt Nanotube radiusRg Cylindrical gate radiustins Insulator thicknesstg Gate thicknesstc Source/drain contact thickness (when assumed equal)²t Nanotube relative permittivity²ins Insulator relative permittivityQ Charge, charge densityn, p Electron and hole carrier densitiesCins Insulator capacitanceCQ Quantum capacitance (also known as semiconductor capacitance)ψ Wavefunctionf Fermi functionT Transmission functiong Density of statesG Local density of statesL Linear operator in Poisson’s equationH Hamiltonian operator in Schrödinger’s equationq Magnitude of electronic chargekB Boltzmann’s constanth Planck’s constant (~ = h2π is sometimes labeled Dirac’s constant)²0 Permittivity of free space

viii

Acknowledgments

I would like to acknowledge my colleagues Dylan McGuire and Sasa Ristic, and the membersof 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 collaborationwas 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 manyopportunities 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, Erika, Fernando and Olavo, as it could not havebeen 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 membersof the UBC Nanoelectronics Group during the period 2002-2005. This statement acknowledgesthe role of this thesis’ author in their creation. Most of the derivations in Chap. 3 and Chap. 7were the work of David L. John. Chap. 5 and Chap. 6 had equal contributions from this authorand David L. John. The author of this thesis was responsible for all the remaining researchpresented herein, with the exception of a few equations present in Chap. 4 and Chap. 8, whichare due to co-authors. As far as manuscript preparation is concerned, all the publicationsincluded here were collaborative efforts by their co-authors, and the author of this thesis hada major role in the production of all but Chap. 3 and Chap. 7, wherein he was involved mostlyin editing. Nearly all the data and figures in this thesis were prepared by its author.

x

Chapter 1Introduction

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 (ε-k) relation, density of states (DOS)

calculations, and scattering mechanisms.

2


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 1s22s22p2 (6 electrons in 3 orbitals). Covalent bonding occurs via hybridization of the two

outermost shells (4 electrons), and in graphene this takes shape via sp2 orbitals, as illustrated

in Fig. 1.1A.

(A) (B)

Figure 1.1: Pictorial representation of (A) sp2-hybridized carbon and (B) graphene latticestructure. Note that the orbitals are for illustration only, and are neither rigorously-derivedprobability densities nor drawn to scale.

A carbon atom in graphene will assemble in a single-sheet hexagonal lattice resembling the

surface of a honeycomb, as illustrated in Fig. 1.1B. This is also known as a trigonal-planar

σ-bond framework, with an inter-atomic spacing, acc, of approximately 1.42 Å along the bonds

that are separated by 120 degrees. The 2p electrons from all the atoms on the sheet constitute

a “cloud” of delocalized π-orbitals surrounding the carbon cores, and these valence electrons,

once excited, are responsible for conduction in graphene. Note that in Fig. 1.1B, 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


weakly by dispersion forces1 and have an inter-layer spacing of about 3.35 Å. While the strength

of the σ-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.

(A)

â1

ζ

â2

zig−zag

chiral

armcha

ir

Ch

→

(n1,n

2)=(4,1)

(B)

Figure 1.2: Pictorial representation of (A) unrolled and (B) rolled carbon nanotube latticestructures. The latter shows a (16,0) tube.

Single-wall nanotubes are characterized by a wrapping (or chiral) vector Ch = n1â1 + n2â2,

where [â1, â2] are lattice unit vectors separated by 60 degrees and the indices (n1, n2) are

positive integers (0 ≤ n2 ≤ n1) 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 (n1, n2)

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|/(2π) = acc√

3(n21 + n22 + n1n2)/(2π), where, for example, Rt ≈ 0.63 nm

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 Å [7], whereas the upper limit on radius is in theseveral nanometre range. Depending on their (n1, n2) indices, nanotubes are placed in one of

three groups, which are named according to the shape of the cross-section established by the

1Dispersion 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 Waalsforces.

4


chiral vector slicing across the hexagonal pattern: armchair (n1 = n2 and ζ = 30◦), zig-zag

(n2 = 0 and ζ = 0◦), and chiral (all other cases), where ζ is the angle between Ch and â1 [6].

The indices also serve to quickly identify the conduction properties of a nanotube—when (n1−n2) 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

Natoms ≈ 2 AcylAhex

1Lt

=8πRtLt3√

3a2cc

1Lt≈ 240Rt ,

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 nm−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 ε-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


of radius smaller than 0.5 nm [7, 9, 10]. These effects are a manifestation of re-hybridization

(mixing) of the σ and π 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 π-orbitals, and following the lattice vector conventions given in Ref. [6] is:

Egraphene(kx, ky) = ±t[1 + 4 cos

(3kxacc

2

)cos

(√3kyacc

2

)+ 4 cos2

(√3kyacc

2

)]1/2, (1.1)

where the positive and negative terms correspond to the symmetrical bonding and anti-bonding

energy bands, respectively, the k’s are wavevectors, and t is the transfer integral (or nearest-

neighbour parameter), the value of which is typically taken to have magnitude 2.8 eV [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 Γ). 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 dispersionrelation is approximately cone-shaped and from the constant slope the Fermi velocity is given

by [6]

vF =32~

acc|t| .

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 integerrepresenting 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(1 + 4 cos γ1 cos γ2 + 4 cos2 γ2

)1/2,

6


Figure 1.3: Energy dispersion relation for graphene, from nearest-neighbour tight-binding cal-culation, using Slater-Koster approximation. Γ, M, and K are high-symmetry points, wherethe K-points lie on the plane of E = 0 (the Fermi level).

where the cosine arguments are given by

γ1 =3acc4

n2 − n1η

kz +π

2n1 + 3n2

η2p

γ2 =3acc4

n2 + n1η

kz +π

23n1 − n2

η2p ,

in which η2 = n21 + n22 + n1n2, −π < (3accη/dg)kz < +π, p = 0...(2η/dg − 1), and dg =

gcd(2n1 − n2, 2n2 − n1).

The one-dimensional density of states, g(E), is obtained via 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) =∑

p

∑

kz

(2) =∑

p

(2 · 1

2π

∫dkz

)=

∑p

(2π

∫∂kz∂Et

dE)

, (1.2)

where the factors of 2 in the numerator account for spin degeneracy and include ±k-states(∂Et/∂k is an even function). Fig. 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

7


bandgap for most cases, but a progressively worse fit to the DOS as one moves up in energy

and includes more bands.

0

0.5

1

1.5

Wavevector

Ene

rgy

(eV

)

Density of States0 5 10

x 105Velocity (m/s)

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 aboutthe x-axis. The carrier velocity is computed from v(E) = 1/~(∂E/∂k).

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 tosuppress 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]:

Eg =|t|acc2Rt

.

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, ni,

typically employed in describing bulk semiconductors; although it may not be as useful of a

8


measure in quantum devices involving quasi-1D carbon nanotubes, it is worthwhile mentioning.

0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Nanotube radius (nm)

Ene

rgy

gap

(eV

)

0

2

4

6

8

log 1

0(n i

) (

m−

1 )

0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

Nanotube radius (nm)

Effe

ctiv

e m

ass

(m

0)

(a) (b)

(16,0)

Figure 1.5: Nanotube properties for various tube radii (larger chiral numbers correspond tolarger radii): (a) intrinsic electron concentration (at T =300K) and energy gap; (b) carriereffective mass of lowest subband, normalized to the free electron mass m0 = 9.11 · 10−31 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 σ-π hybridization due to curvature in small-radius

tubes causes a charge transfer from the portion of the π-orbitals lying inside the tube to the

σ-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


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 4.8 eV, which is about 0.1-0.2 eV 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


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 ∼ 104 cm2/V·s [25, 26]. Theoreticalpredictions, 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 semiconductingtubes 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-Büttiker 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

2A Fermi liquid is a population of electrons whose interactions do not significantly alter their energydistribution near the Fermi level, and thus they are governed by the Fermi function. A Luttinger Liquidis one in which electron-electron interactions give rise to exotic properties.

11


been demonstrated to be more than 109 A/cm2, 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 10µA, which normalized to its circular cross-section

corresponds to a current density of order 108 A/cm2.

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


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 Planar Devices

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


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 SiO2, 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 Al2O3, 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 metallicnanotubes, studying both end- and side-contacted tubes, concluded that Ti contacts yield superiorconductance over their Au and Al counterparts [38,39].

14


contact resistance.

Fig. 1.6C shows a further improvement in CNFETs through the placement of the gate electrode

over the semiconducting nanotube, thus improving the channel electrostatics via the thin gate

oxide [36,40]. Moreover, the Ti source/drain metalizations in this device form titanium carbide

abrupt junctions with the nanotube, yielding increased conductance [41]. Another attempt

to obtain better gate electrostatics involved materials with high dielectric constants, such as

zirconia (ZnO2) and hafnia (HfO2), being used as gate insulators [42]. Fig. 1.6D illustrates a

device built with Pd 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 Fig. 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 DC characteristics was fabricated

with Pd end contacts, Al gate, and hafnia for the insulator [43].

1.2.2 Coaxial Devices

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. Capitalizing 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 CNFETs: (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 Fig. 1.7A, using a multi-wall

15


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 LiClO4 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 CNFETs

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


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 1D 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 tins = Rg−Rt; theend-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


Figure 1.8: Coaxial CNFET structure. The insulator fills the entire simulation space notoccupied 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

∂2V

∂ρ2+

(1ρ

+1²

∂²

∂ρ

)∂V

∂ρ+

∂2V

∂z2= −Qv

², (1.3)

where V (ρ, z) is the potential within the device cylinder, Qv is the volumetric charge density, ²

is the permittivity, and the ∂²/∂ρ 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 = −φS/q

VD = VDS − φD/q

VG = VGS − φG/q ,

18


where the φ terms represent the work function of each electrode, and VGS and 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

VGS 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, φ, and on

the electron affinity, χ, 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 thiswork, the contact series resistances are included in the small-signal model of the CNFET.

5In a semiconductor, the work function φ = χ + (Ec − EF ).

19


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 SiO2 (²r ≈ 3.9) [56], ZrO2 (²r ≈ 25) [42] or HfO2 (²r ≈ 16) [57]. High-permittivity dielectrics are preferable, since they allow for better electrostatic coupling of the

gate to the channel; employment of non-SiO2 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

6This is under the assumption of symmetrical density of states, as presented in Section 1.1.2.

20


qualitative band diagrams that serve to illustrate some important behaviours of the devices.

We begin by taking an arbitrary pair of voltages, and Fig. 1.9A illustrates the solution for

the bias conditions of VGS = 0.3 V and VDS = 0.5V, 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 isentirely controlled by the gate electrode, tracking its voltage identically.

Figure 1.9: Complete solution (A) and cross-section of solution (B) to Laplace’s equation intwo-dimensional cylindrical coordinates for a cylinder 50 nm in length, 1 nm in diameter, andbiased at VGS = 0.3 V and VDS = 0.5V.

Fig. 1.9B, a cross-section of the solution in Fig. 1.9A, taken at ρ = Rt = 0.5 nm, shows the

nanotube surface potential, VCS(z), i.e., 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 limit these regions will 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 Fig. 1.10, for the same conditions given

in Fig. 1.9.

Alternate contact geometries will undoubtedly affect these profiles. For instance, decreasing the

21


0 20 40−4

−2

0

2

x 106

z (nm)

Ele

ctric

Fie

ld (

V/m

)

Radial Component

0 20 40−8

−6

−4

−2

0

x 107

z (nm)

Ele

ctric

Fie

ld (

V/m

)

Longitudinal Component

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 ε-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× 108 V/m and tubes with radii Rt ≤ 1.5 nm, worsening with larger radii and strongerfields [61].

Moreover, in the absence of mid-channel Ec discontinuities, the solution to Poisson’s equation,

22


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 anequal 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)− χ

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 VGS and then VDS . As

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


0 10 20 30 40 50−0.5

0

0.5

z (nm)

Ene

rgy

(eV

)

VGS

= 0, VDS

= 0(A)

0 10 20 30 40 50−1

−0.5

0

0.5

z (nm)

Ene

rgy

(eV

)

VGS

= 0.5 V, VDS

= 0(B)

0 10 20 30 40 50−1

−0.5

0

0.5

z (nm)

Ene

rgy

(eV

)

VGS

= 0.5 V, VDS

= 0.4 V(C)

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

z (nm)E

nerg

y (e

V)

VGS

= 0.5 V, VDS

= 0.9 V(D)

Figure 1.11: CNFET band diagrams for Laplace solutions. There is no work function differencebetween 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, Schrödinger’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 Schrödinger’s

equation to subsequent chapters.

Under equilibrium conditions, i.e., when VDS = 0 and there is no net charge flow, the carrier

24


distributions are 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. 1.11B,

with Ec dropping relative to the fixed EF for VGS > 0, or rising for VGS < 0. In this situation, a

self-consistent solution for the system is obtained by solving Poisson’s equation in conjunction

with the 1D carrier density equation, which accounting for both carrier types is

Qv(z) = q(p− n) = q∫ ∞

0g(E) {f [E + qVCS(z)]− f [E − qVCS(z)]} dE , (1.4)

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 1D electron concentration, with a dependence on ∆E = Ec−EF . Under a positivegate-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 = 0, and 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


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

EF 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, +k-states, are

primarily occupied by source-injected electrons, while drain-injected electrons primarily fill the

−k-states. This method therefore amounts to a tunneling-barrier modulation of the Fermifunctions 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


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-Büttiker

formalism, we treat the device as being composed of two carrier reservoirs, separated by a

1D 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 1D nature of the system) toaccount 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)=(1/π)(∂k/∂E), and carrier group velocity

v(E)=(1/~)(∂E/∂k), we find the net current between the source and drain to be

I = −2qh

∫[f(E − µS)− f(E − µD)]T (E) dE ,

where µS and µD 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

and ±k.

7Due to the bandgap of ∼ 0.7 eV being employed in typical devices, photoemission produces light ofwavelength λ ≈ hc/Eg ≈ 1.8 µm, lying in the infrared range of the spectrum.

8Recall that, under the elastic conditions here, each energy “level” of transport is independent of allothers.

27


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 Schrödinger-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 µD 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 Schrödinger-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 1D CNFET devices. Finally, a small-

signal model, in conjunction with an improved Schrödinger-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 fT and fmax 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 Schrödinger-Poisson (SP) model gave results that were faithful

28


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 Iij

modeling of carbon nanotube field-eﬁect transistorsnano/pub/castro_thesis.pdf · 2006. 6. 23. ·...

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