Castro Thesis

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-massSchrodinger Equation (SE) to compute the electron and hole charges in the nanotube, and byusing the Landauer Equation to compute the drain current. A Schrodinger-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 Schrodinger-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 Schrodinger-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 εins = 3.9 . . . . . . . . 483.2 Equilibrium energy band diagram: (16,0) tube, tins = 30.9 nm and εins = 3.9 . . . . . . . 493.3 Equilibrium energy band diagram: (16,0) tube, tins = 5.6 nm and εins = 19.5 . . . . . . . 503.4 Sub-threshold current: (16,0) tube, tins = 5.6 nm and εins = 3.9 . . . . . . . . . . . . . . . . . . . . 523.5 Non-equilibrium energy band diagram: (16,0) tube, tins = 5.6 nm and εins = 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 Schrodinger-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)εt Nanotube relative permittivityεins 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 Schrodinger’s equationq Magnitude of electronic chargekB Boltzmann’s constanth Planck’s constant (~ = h

2π is sometimes labeled Dirac’s constant)ε0 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 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 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 A. 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

armchair

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 = n1a1 + n2a2,

where [a1, a2] 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(n2

1 + 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 A [7], whereas the upper limit on radius is in the

several 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 a1 [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 ≈ 2Acyl

Ahex

1Lt

=8πRtLt

3√

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 dispersion

relation 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 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(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 =3acc

4n2 − n1

ηkz +

π

2n1 + 3n2

η2p

γ2 =3acc

4n2 + n1

ηkz +

π

23n1 − n2

η2p ,

in which η2 = n21 + n2

2 + 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

∂EtdE

), (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 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]:

Eg =|t|acc

2Rt.

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 10

(ni)

(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]. 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-Buttiker 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; 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


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

+1ε

∂ε

∂ρ

)∂V

∂ρ+

∂2V

∂z2= −Qv

ε, (1.3)

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

is the permittivity, and the ∂ε/∂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 (εr ≈ 3.9) [56], ZrO2 (εr ≈ 25) [42] or HfO2 (εr ≈ 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 is

entirely 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 stronger

fields [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 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)− 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, 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


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 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 = 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 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


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-Buttiker

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) 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)=(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 = −2q

h

∫[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 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 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 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 1D 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 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 Schrodinger-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.

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35

Chapter 2Towards a Compact Model forSchottky-Barrier CNFETs

2.1 Introduction

Nanoscale transistors fashioned from carbon nanotubes are an exciting possibility [1,2]. Trans-

port is essentially one-dimensional, and there is very little carrier-phonon interaction [3]. 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 via capacitative coupling between the gate and the contacts [6, 7]. In the present

work we concentrate on coaxial, Schottky-barrier devices formed with intrinsic nanotubes, and

seek to develop a compact model for the prediction of the drain I-V characteristics. The basic

transistor structure is shown in Fig. 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 VGS 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

c© [2002] IEEE. Reprinted, with permission, from L. C. Castro, D. L. John, and D. L. Pulfrey, “Towardsa Compact Model For Schottky-Barrier Nanotube FETs,” Proc. IEEE Conf. on Optoelectronic andMicroelectronic Materials and Devices, 303–306 (Sydney, Australia, 2002).

36

Chapter 2. Towards a Compact Model for Schottky-Barrier CNFETs

Source Drain

Gate

Oxide

Gate

Oxide

CNTCNT

Lt

Rg

Figure 2.1: Schottky-barrier carbon nanotube FET model geometry. The gate forms the curvedsurface of the outer cylinder, and the source and drain form the two ends. The semiconductingnanotube 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 will deviate

considerably from a Maxwellian 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


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 Cins is

the insulator capacitance for an infinitely long coaxial system. Out of equilibrium, i.e., when

VDS 6= 0, VCS is influenced by VDS :

VCS = VCS + αVDS

QC = −Cins(VGS − VCS) , (2.2)

where α is a parameter that needs to be determined in order to specify VCS . For 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, fS/2 denotes the positive- or

forward-directed part of the Fermi-Dirac distribution outside the tube at the source, whereas

fD/2 represents the negative- or backward-directed part of the distribution outside the tube at

the drain. Thus:

QC = −q

∫ ∞

Ec

g(E)fS(E)

2

(2

TD− 1

)T ? dE

+∫ ∞

Ec

g(E)fD(E, VDS)

2

(2TS

− 1)

T ? dE

, (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 αVDS , and

the tunneling probabilities at the source and drain, TS and TD respectively, are computed using

38


the JWKB approximation and are dependent on E, VGS and αVDS . The overall transmission

probability T ? = TSTD/(TS + TD − TSTD). In our method, we equate Eq. (2.2) and Eq. (2.3)

and solve for α, thereby determining VCS . An iterative procedure is necessary because of the

dependence on VCS of TS and TD, via the potential profile, which are represented by

V (z) =

VCS − VCS

eβ − 1

[e−

βa(z−a) − 1

]0 ≤ z ≤ a

VCS a ≤ z ≤ L− a

VCS − VCS − VDS

eβ − 1

[e−

βa(L−a−z) − 1

]L− a ≤ z ≤ L ,

(2.4)

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 ≈ 2RG, where RG 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 β ≈ 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:

ID =∑

i

2q

π~

∫ ∞

Ec,i

[fS(E)− fD(E, VDS)]T ∗ dE ,

where the sum is over i doubly-degenerate conduction bands, the edges Ec,i of which are func-

tions of α, VGS 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


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 Fig. 2.2. Note the VDS-dependence of VCS and of

the barrier shapes.

0 20 40 60 80 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Distance from source contact (nm)

Ene

rgy

(eV

)q V

CS

EF E

F

VGS

= 0V

DS = 0

Sou

rce

C

onta

ct

q VCS

~

q VDS

Dra

in

Con

tact

VGS

> 0V

DS = 0

VGS

> 0 0 < V

DS < V

GSVGS

> 0 V

DS > V

GS

Figure 2.2: Conduction energy band diagram for various bias conditions.

−0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

Kin

etic

ene

rgy

at z

=0.

5Lt (

eV)

Electron distribution ±k−states Drain−

source voltage

Electrons travelling fromSource to Drain

Forward Channel

Backward Channel Electrons travelling fromDrain to Source

VGS

= 0.5 V

Figure 2.3: Electron distribution in the “mid-length” of the tube as a function of VDS forVGS = 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 within 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 Fig. 2.3. As VDS increases, there is less injection from the drain, and less reflection from the

40


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.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

Drain−source voltage, VDS

Dra

in c

urre

nt, I

D (

µA)

VGS

= 0.4 V

VGS

= 0.5 V

VGS

= 0.3 V

Figure 2.4: Drain current-voltage characteristics.

0 0.1 0.2 0.3 0.4 0.50

2

4

6

This workReference 9

Dra

in c

urre

nt, I

D (

µA)

Drain−source voltage, VDS

VGS

= 0.5 V

Figure 2.5: Comparison of drain current-voltage characteristics at VGS = 0.5 V. The solid lineis 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 VGS = 0.5V is

around 1µA, 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 VDS,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 ID,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


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.

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43

Chapter 3Electrostatics of Coaxial Schottky-BarrierCNFETs

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 (FET) [5, 6]. Here we concentrate on the coaxial

Schottky-barrier carbon nanotube FET (SB-CNFET), in which the cylindrical gate surrounds

the tube and is insulated from it by a dielectric. The basic structure was shown previously

in Fig. 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 via 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

c© [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 firstauthor.

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 are

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, VDS , 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 6= 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 VDS 6= 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


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 (ρ, θ, z) are:

V (Rg, θ, z) = VGS − φG/q

V (ρ, θ, 0) = −φS/q

V (ρ, θ, L) = VDS − φD/q

V (ρ, θ, z) = V (ρ, θ + 2π, z) ,

where φG, φS and φD 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, θ, 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) =1

2πρδ(ρ−Rt)Qz(z) ,

where Qz(z) is the net 1-D charge concentration, Rt is the CN radius, and δ(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

1FEMLAB, see http://www.comsol.com

46


the CN DOS [14, 15]. Using the nearest-neighbour tight-binding approximation [9], the DOS

is symmetrical about EF , so the net 1-D carrier density at some point along the intrinsic tube

may be computed as

Qz(z) = q

∫ ∞

0g(E) [f(E + qV (z) + φt)− f(E − qV (z)− φt)] dE ,

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 φt 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.


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, εins, and source- and drain-work functions, are considered. The electron affinity

for the carbon nanotube is taken to be 4.18 eV, based on a work function of 4.5 eV [17], and

an intrinsic-tube band gap of 0.64 eV. Unless otherwise stated, the relative permittivity of the

nanotube, εt, 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 φS = φD, 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, εins = 3.9 and for various φS = φD with

VDS = 0 V and VGS = 0.2 and 0.5 V. In Fig. 3.1(a), φS = 4.5 eV and corresponds to the case of

equal work functions for the metal and the nanotube, whereas φS = 4.33 eV [Fig. 3.1(b)] and

φS = 4.63 eV [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 φS,D and

47


0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (c)

0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (a)

Ene

rgy

(eV

)

0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (b)

Distance from source (nm)

Figure 3.1: Self-consistent equilibrium energy band diagram near the source for a (16,0) tubewith a 5.6 nm gate dielectric thickness (Rg/Rt = 10), and φS = φD set to (a) 4.5 eV, (b) 4.33 eV,and (c) 4.63 eV. 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).

VGS . Only in the low-φS case at low VGS 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 VGS 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 2 nm [12]. Regarding the permittivity of the dielectric, recent work has reported the use of

48


0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Ene

rgy

(eV

)

(a)

0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (b)

Distance from source (nm)0 20 40

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (c)

Figure 3.2: Self-consistent equilibrium energy band diagram near the source for a (16,0) tubewith a 30.9 nm gate dielectric thickness (Rg/Rt = 50), and the same sequence of work functionsas in Fig. 3.1, namely: φS = φD set to (a) 4.5 eV, (b) 4.33 eV, and (c) 4.63 eV. Data are forVDS = 0 V and VGS = 0.2 (dashed line) and 0.5 V (solid line). Energies are with respect to theFermi level (dotted line).

zirconia [6], for which εins is around five times higher than that used in obtaining the above

figures. The effect of such a change in εins can be seen by comparing Figs. 3.1 and 3.3. At

VGS = 0.5V, 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 εins. At lower VGS , e.g., 0.2V, the increased εins 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 Rg for the contact geometry considered here,

as has been remarked upon elsewhere [18], and here we indicate that it also depends barely at

49


0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (a)

Ene

rgy

(eV

)

0 20 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (b)

Distance from source (nm)0 20 40

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4 (c)

Figure 3.3: Self-consistent equilibrium energy band diagram near the source for a (16,0) tubewith a 5.6 nm gate dielectric thickness (Rg/Rt = 10), a gate dielectric with permittivity fivetimes higher than used in Fig. 3.1, and the same sequence of work functions as in Fig. 3.1,namely: φS = φD set to (a) 4.5 eV, (b) 4.33 eV, and (c) 4.63 eV. Data are for VDS = 0 V andVGS = 0.2 (dashed line) and 0.5 V (solid line). Energies are with respect to the Fermi level(dotted line).

all on VGS .

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 φG 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


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,

I =4q

h

∫T (E)[f(E)− f(E + qVDS)] dE .

Some results for various gate/tube dimensions are shown in Fig. 3.4: note that the dielectric

thickness tins = Rg − Rt. The sub-threshold current can be due to either mainly electrons or

mainly holes, depending on the bias conditions. For VDS = 0.1V, 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 VGS , 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 = VDS/2, when the tunneling barriers for

source electrons and drain holes are of equal thickness. The magnitudes of the sub-threshold

slopes, |S| = |(d log10 I/d VGS)−1|, for both mainly electron conduction (VGS > VDS/2), and

mainly hole conduction (VGS < VDS/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√

tins. This suggests

that we seek a scaling relationship for S in our coaxial devices by examining how the VGS-

dependent part of the potential in the vicinity of the source contact varies with tins. This can

be accomplished by expanding the first term of the first mode of an analytical Laplace solution

(Ref. [19], Appendix), i.e.,

V ' 4(qVGS − φG)I0(πRtL

−1t

)πzL−1

t

qπI0(πRgL

−1t

) ,

51


−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

10−10

10−9

10−8

10−7

10−6

10−5

Gate−source voltage (V)

Dra

in c

urre

nt (

A)

Hole Majority

Electron Majority

(16,0) tube

VDS

= 0.1 V ε

t = 1

εins

= 3.9

RG

/RT = 2 (S ≈ 76 mV/dec)

RG

/RT = 5 (S ≈ 80 mV/dec

RG

/RT = 10 (S ≈ 99 mV/dec)

RG

/RT = 20 (S ≈ 150 mV/dec)

Figure 3.4: Sub-threshold current for the same tube properties as in Fig. 3.1(a), but with variousratios of gate radius to tube radius, for VDS = 0.1 V. Note: the dielectric thickness is Rg −Rt.

where I0 is the zeroth-order Modified Bessel Function of the First Kind. Thus

∂V

∂VGS∝ 1

I0(πRgL

−1t

)

and, therefore, we can expect

S ' αI0(βtins) ,

where α and β are fitting parameters. It is found that a reasonable fit to S, from the data

included in Fig. 3.4, results with α ≈ 79mV/decade and β ≈ 0.15 nm−1. The fact that a fit

can be obtained confirms that the dependence of S on tins is related to the specific geometry

of the transistor, with a Bessel function being involved in this case because of the cylindrical

structure. Electrically, tins 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 Rg. This fact has been used, in

52


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 VGS and VDS [10].

0 10 90 100

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4


Ene

rgy

(eV

)

/ /

Figure 3.5: Energy band diagrams for the same tube properties as in Fig. 3.1(a) for VGS = 0.3Vand various VDS : 0 V (dotted), 0.2V (dashed), 0.5 V (solid). For the equilibrium case the profileis the exact solution of Poisson’s equation; for VDS 6= 0 the solutions are from Castro’s compactmodel [10], with the base widths of the potential profiles at the source and drain being takenas 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 VDS 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

53


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1.0

Drain−source voltage (V)

Dra

in c

urre

nt (

µA)

VGS

= 0 V V

GS = 0.2 V

VGS

= 0.4 V

VGS

= 0.5 V

Figure 3.6: Drain I-V characteristics, as calculated by the method of Ref. [10], for a (16,0)tube, Rg/Rt = 10, φS = φD = 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 VGS < VDS [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].

54


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 I-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).





(2002).

55




126801–1–126801–4 (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).





1–235418–5 (2003).

56




[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).



[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).



2(3), 175–180 (2003).



57

Chapter 4An Evaluation of CNFET DC 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, “CarbonNanotube Transistors: An 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 (ε-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


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

Ie =4q

h

∫ ∞

Emid

Te(E)[f(E)− f(E + qVDS)] dE , (4.1)

where q is the magnitude of the electronic charge, h is Planck’s constant, Emid 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 Th. Thus, once the T ’s are known, the problem of

the DC drain current is solved.

4.4 DC Modeling

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-

60


ically gated devices [24], more closely resemble the coaxial structure with wrap-around gate

that would be ideal, given the cylindrical form of the nanotube. As we are interested in

ultimate-performance assessment, we take a coaxial geometry for modeling purposes, as illus-

trated in Fig. 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 CNFET model geometry.

The transmission probabilities T are functions of energy and local potential V (ρ, θ, 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 within 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(r) =1

2πρδ(ρ−Rt)Qz(z) ,

where Qz(z) is the net 1-D carrier concentration, Rt is the nanotube radius, and δ(x) is the

Dirac delta function.

If we allow for a spatially varying permittivity in the radial direction, Gauss’ Law implies that:

∂2V

∂ρ2+

(1ρ

+1ε

∂ε

∂ρ

)∂V

∂ρ+

∂2V

∂z2= −Q(r)

ε,

61


where the potential within the device reduces to a function of just two cylindrical coordinates

V (ρ, 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 (ρ, 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 (ρ, 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, Emid 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


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 Emid, 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) = VGS − φG/q

V (ρ, 0) = −φS/q

V (ρ, Lt) = VDS − φD/q ,

where φG, φS and φD are the work functions of the gate, source and drain metallizations, re-

spectively, and VGS 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



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

0.63 nm, 0.62 eV and 4.2 eV, respectively [4]. The tube length is taken to be 20 nm, 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., φBn = φM−χt. All simulations are performed for a temperature

T = 300 K.

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 φBn = Eg/2, where Eg is the nanotube bandgap. The

figure shows that, for a constant VDS , the conduction can be due to either electrons or holes,

depending on the gate bias VGS . In this case, at VGS +∆φ = VDS/2, where ∆φ 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


IOFF . This is because at zero gate bias Emid is determined only by φG, 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 Emid 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 5 10 15 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

z (nm)

Ene

rgy

(eV

)

Figure 4.2: Band diagram illustrating ambipolarity in a device with φS,D,G = 4.5 eV andVDS = 0.4 V. Hole injection at VGS = 0.05V (dotted line and arrow); electron and hole injectionat VGS = 0.2V (solid line and arrows); electron injection at VGS = 0.35 V (dashed line andarrow).

4.5.2 Conductance

The quantized interfacial conductance, as mentioned earlier, has a maximum value Gmax =

4q2/h for a nanotube with two transverse modes [20]. Measurements of conductance, G =

ID/VDS , can only be expected to approach Gmax 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 ≈ 0.4Gmax have been reported already [28].

65


−0.5 −0.25 0 0.25 0.5

10−8

10−7

10−6

10−5

10−4


Dra

in C

urre

nt (

A)

(a)

−0.5 −0.25 0 0.25 0.5

10−8

10−7

10−6

10−5

10−4

Gate−source voltage (eV)

Dra

in C

urre

nt (

A)

(b)

Figure 4.3: ID-VGS at VDS = 0.4V. (a) φG = 4.2 eV and various φS,D: 3.9 (solid line); 4.2(dotted line); 4.5 eV (dashed line). (b) φS,D = 3.9 eV and various φG: 3.9 (solid line); and4.37 eV (dashed line).

0 5 10 15 20

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

z (nm)

Ene

rgy

(eV

)

Figure 4.4: Band diagrams at VGS = 0 and VDS = 0.4V for the three devices used in Fig. 4.3a.φG = 4.2 eV and various φS,D: 3.9 (dotted line); 4.2 (solid line); and 4.5 eV (dashed line).

4.5.3 Subthreshold Slope

The limiting value for the sub-threshold slope S, in situations where the sub-threshold cur-

rent is thermionically determined, is about 60mQ mV/decade, where mQ = 1 + CQ/Cins is the

“quantum capacitance coefficient”, with CQ being the quantum capacitance [5], and Cins the

insulator capacitance. The use in CNFETs of high-permittivity dielectrics, such as zirconia [22]

66


or aqueous solutions [24], opens up the possibility of attaining values of mQ 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 tins = 2 nm, for ex-

ample, S = 110mV/decade has been measured for planar CNFETs [23], and S ≈ 80 mV/decade

has been predicted for coaxial devices, in which the capacitive coupling is superior. For a SiO2

gate oxide with tins = 67nm, 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

10−4

10−2

100


CQ

/Cin

s

Figure 4.5: Ratio of equilibrium quantum capacitance to insulator capacitance for an insulatorrelative permittivity and thickness of 25 and 2.5 nm, respectively, and for all work functionsequal to 4.5 eV.

4.5.4 ON 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 in

Eq. (4.1) and setting T = 1, which also implies neglecting quantum-mechanical reflection at the

67


0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

Dra

in c

urre

nt (

µA

)

Gate−source voltage (V)0 0.1 0.2 0.3 0.4

0

20

40

60

80

100

120

140

160

Tra

nsco

nduc

tanc

e ( µ

S)


(a) (b)

Figure 4.6: (a) ID, and (b) gm, as a function of gate-source voltage. The solid lines are forthe “quantum-capacitance limit” from Eqs. (4.2), (4.3) and (4.4). The dashed lines are for adevice with φS,D = 3.9 eV and φG = 4.37 eV, i.e., the device that gave the lowest OFF currentin Fig. 4.3.

metal/nanotube interfaces. Integrating Eq. (4.1) then leads to

Ie, max =4q

hkBT ln(1 + e−Emid/kBT ) , (4.2)

where kB 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

Emid =Eg

2− qVGS + ∆φ

mQ. (4.3)

Note that the relationship would be essentially linear if CQ ¿ Cins, i.e., in the “quantum-

capacitance limit” of mQ → 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 ¿ Cins 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


far in a CNFET is about 25µA [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 φG is examined in the band diagrams of Fig. 4.8, which are for the case of VGS =

VDS = 0.4V. At φG = 3.9 eV, Emid is depressed to the extent that a tunneling barrier forms

for the low energy electrons in the source. With low φG, clearly a larger VDS is required to

suppress the drain-injected electron current, leading to a higher saturation voltage VDS,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 10 nm-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 0.60

10

20

30

40

50

60

70

80

90


(a)

0 0.2 0.4 0.60

10

20

30

40

50

60

70

80

90


(b)

Dra

in c

urre

nt (

µA)

Dra

in c

urre

nt (

µA)

Figure 4.7: Drain characteristics at VGS = 0.4V. (a) φG = 4.2 eV and various φS,D: 3.9 (dashedline); 4.2 (solid line); and 4.5 eV (dotted line). (b) φS,D = 3.9 eV and various φG: 3.9 (dashedline); 4.2 eV (solid line); and 4.5 eV (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 φG = 4.37 eV and φS,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 = 0.4V,

69


0 5 10 15 20

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

z (nm)

Ene

rgy

(eV

)

Figure 4.8: Band diagrams for φS,D = 3.9 eV, VGS = VDS = 0.4V and various φG: 3.9 (dashedline); 4.2 (solid line); and 4.5 eV (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 (ID,max/2Rt) of about 70 mA/µm!

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

gm,max =4q2

h

[1 + exp

(Eg/2− qVGS −∆φ

kBT)]−1

. (4.4)

As Fig. 4.6b reveals, at high VGS , gm, max attains its limiting value of 4q2/h, which, interestingly,

is the same value attainable by Gmax, as noted elsewhere [33]. Taking once more the device

with φG = 4.37 eV and φS,D = 3.9 eV as an example, at VDS = 0.4V, 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


These features, allied to the excellent thermal and mechanical properties of carbon nanotubes,

make these molecules strong contenders for implementation in nanoscale integrated circuits.

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[9] Jason Clifford, D. L. John, and David L. Pulfrey, “Bipolar conduction and drain-induced

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[10] Jing Guo, Jing Wang, Eric Polizzi, Supriyo Datta, and Mark Lundstrom, “Electrostatics

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[18] Won Bong Choi, Jae Uk Chu, Kwang Seok Jeong, Eun Ju Bae, Jo-Won Lee, Ju-Jin Kim,

and Jeong-O Lee, “Ultrahigh-density nanotransistors by using selectively grown vertical

carbon nanotubes,” Appl. Phys. Lett., 79(22), 3696–3698 (2001).

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nanotube transistors,” Appl. Phys. Lett., 83(12), 2435–2437 (2003).

[27] Anisur Rahman, Jing Guo, Supriyo Datta, and Mark S. Lundstrom, “Theory of ballistic

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[29] Ali Javey, Jing Guo, Magnus Paulsson, Qian Wang, David Mann, Mark Lundstrom, and

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74

Chapter 5A Schrodinger-Poisson Solver for ModelingCNFETs

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 εins) 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,

c©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 Conf. and Trade Show , vol. 3, pp. 65–68, March 7–11,Boston, U.S.A.

75

Chapter 5. A Schrodinger-Poisson Solver for Modeling CNFETs

Rg, the nanotube radius, Rt, the insulator thickness tins = Rg −Rt, and the device length, Lt.

In this closed, metallic cylinder system, Poisson’s equation, restricted to just two dimensions

by azimuthal symmetry, is∂2V

∂ρ2+

1ρ

∂V

∂ρ+

∂2V

∂z2= −Q

ε, (5.1)

where V (ρ, z) is the potential within the outer cylinder, Q is the charge density, and ε 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 VCS(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

∂2ψ

∂z2= −2m

~2(E − Epot) ψ , (5.2)

where ψ(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 k + 1 is given by

Vk+1 = Vk − αL−1rk ,

rk = LVk −Q(Vk) ,

where rk is the residual of the k-th iteration, 0 < α ≤ 1 is a damping parameter, and Lrepresents the linear, differential operator allowing Eq. (5.1) to be written as LV = Q.

76


5.2.1 Potential

The boundary conditions for V are given by

V (Rg, z) = VGS − φG/q ,

V (ρ, 0) = −φS/q ,

V (ρ, Lt) = VDS − φD/q ,

∂V

∂ρ(0, z) = 0 ,

(5.3)

where φG,S,D represent the work functions of the gate, source, and drain metallizations, respec-

tively, and VGS and VDS are the gate- and drain-source voltages. Due to the discontinuity in ε

across the nanotube surface, we must also apply the usual matching condition

εins∂V

∂ρ

∣∣∣∣R+

t

− εt∂V

∂ρ

∣∣∣∣R−t

= −q(p− n)2πRtε0

,

where p and n are the one-dimensional hole and electron carrier densities, εt is the nanotube

relative permittivity, and ε0 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 ρ = 0 was addressed by applying

l’Hopital’s Rule to the offending term, yielding

1ρ

∂V

∂ρ' ∂2V

∂ρ2.

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 areEpot,e(z) = Ec(z) = Evac(z)− χt ,

Epot,h(z) = −Ev(z) = −(Ec(z)−Eg) ,(5.4)

where Eg and χt are, respectively, the nanotube band-gap and electron affinity.

77


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

Q = −q(p− n)δ(ρ−Rt)

2πρ,

where δ 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 VCS(z).

We solve Eq. (5.2) using the scattering-matrix method in which a numerical solution is prop-

agated by cascading 2×2 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

ψn = ψn+1 ,

∂ψn

∂z=

∂ψn+1

∂z.

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

ψ =

ASeikSz + BSe−ikSz , z < 0 ,

ADeikDz + BDe−ikDz , z > Lt ,

where kS and kD 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


for the flux, and must be equal to the probability current. For the transmitted wave, this yields

2q

π~fST =

q~m

kD|AD|2 , (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 =kD|AD|2kS |AS |2 .

Simple manipulation yields the normalization condition

|AS |2 =2m

π~2

fS

kS. (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,

n(z) =∫ ∞

Eref,e

( |ψe,S |2 + |ψe,D|2)

dE ,

p(z) =∫ ∞

Eref,h

( |ψh,S |2 + |ψh,D|2)

dE ,

where the Eref 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

ψ, 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


5.3 Results

We now present results for a CNFET with a (16, 0) nanotube (Rt ≈ 0.63 nm; Eg ≈ 0.62 eV),

Lt = 20 nm, tins = 2.5 nm, and εins = 25. All work functions are taken to be 4.5 eV unless

otherwise noted, and χt = 4.2 eV. The nanotube is presumed to have a free-space relative

permittivity εt = 1 [9], and Eref 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 andVDS .

0 5 10 15 20

−0.2

−0.1

0

0.1

0.2

0.3


Con

duct

ion

band

edg

e (e

V)

Figure 5.2: Conduction band edges for the model device with VGS = 0.5V, and VDS = 0(dashed) and 0.4V (solid). Energies are with respect to the source Fermi level.

80


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 VGS = 0.5V 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.

−0.6

−0.4

−0.2

0

0.2

Ene

rgy

(eV

)

(a)

0 10 20

−0.2

−0.1

0

0.1

0.2

0.3

z (nm)

Ene

rgy

(eV

)

(b)

0 0.5 1Probability

Figure 5.3: Conduction band edges and transmission probabilities for electrons at VGS = 0.5Vand VDS = 0.4V: (a) φS = φD = 3.9 eV and (b) φS = φD = 4.5 eV. Energies are with respectto 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


as Fig. 5.3 shows, wherein the significant reflection is due to Eref 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

−0.2

−0.1

0

0.1

Normalized n(E)

Ene

rgy

(eV

)

Figure 5.4: Source-originated electron concentration at Lt/2, normalized to its maximum value.VGS = 0.5V, VDS = 0.4V, and φS = φD = 3.9 eV. Energies are with respect to the source Fermilevel.

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


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(3), 175–180 (2003).



1–235418–5 (2003).




[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).



[8] Lee W. Johnson and R. Dean Riess, Numerical Analysis (Addision-Wesley, Don Mills,

Ontario, 1977).

83






84

Chapter 6An Improved Evaluation of the DCPerformance 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 ImprovedEvaluation of the DC 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, Emid, 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 3 nm 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


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 ? =TSTD

TS + TD − TSTD, (6.1)

where TS,D are, respectively, the source and drain transmission probabilities. The mid-length

electron density is given by

nmid =∫ ∞

Emid

g1DT ?

[fS

2

(2

TD− 1

)+

fD

2

(2TS

− 1)]

dE, (6.2)

where g1D is the 1-D nanotube density-of-states, numerically computed from a tight-binding

method, f 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 nmid 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

g1D(E) =4π

dk

dE≈ 2

π~

√2m

E − Emid,

and consider, for brevity, just the component originating at the source, we obtain

nmid,S =√

2m

π~

∫ ∞

Emid

1√E − Emid

fS

[TS(2− TD)

TS + TD − 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


the source and drain barriers, the total probability density in the mid-length region of the chan-

nel due to source injection, |ψmid,S |2, can be found from the infinite geometric series relation,

and is given by

|ψmid,S |2 =|P |2 + |P |2(1− TD)TS + TD − TSTD

. (6.4)

However, the source transmission probability can also be written as

TS ≡ jtrans

jinc=

kmid

kS|P |2 =

√E −Emid

E − Eref,S|P |2,

where the j’s are probability density currents, the k’s are wavevectors, and Eref,S is the energy

of the conduction band edge of the source metal. Therefore, from Eq. (6.4),

TS(2− TD)TS + TD − TSTD

=√

E − Emid√E − Eref,S

|ψmid,S |2 .

Substituting into Eq. (6.3), and including the analogous term for injection from the drain, we

get

nmid =√

2m

π~

∫ ∞

Emid

(fS |ψmid,S |2√E −Eref,S

+fD|ψmid,D|2√

E − Eref,D

)dE . (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

88


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:

ψ(z) =

Aeikconz + Be−ikconz , z < 0 ,

1√kbar(z)

(Cei

∫ z0 kbar(z)dz + De−i

∫ z0 kbar(z)dz

), 0 < z < w ,

F eikmid(z−w) + Ge−ikmid(z−w) , z > w ,

where A to G are constants, w is the barrier width, and kcon, kbar(z), and kmid 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 kbar 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 ψ 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


note that kbar(w) = kmid and k′bar(w) = 0, where the primed notation denotes a derivative with

respect to z. This provides a compact expression for the source transmission probability,

TS =16kconk

3bar0

(k′bar0)2 + 4(k2

bar0 + kconkbar0)2, (6.6)

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 ?=1.


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

εins, 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, εins = 25, and εt,

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 [11]. This is because, with a saturating

ID-VDS characteristic, selection of VDS 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


bias of VGS = 0 [1,11].

−0.2 0 0.2 0.4 0.6

10−2

10−1

100

101

102

VGS

(V)

Dra

in c

urre

nt (

µA)

Figure 6.1: Drain current versus gate-source voltage at VDS = 0.4V 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 Emid [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


0 0.3 0.6

0

10

20

30

40

50

60

VGS

(V)

Dra

in c

urre

nt (

µA)

0 0.3 0.6

0

0.2

0.4

0.6

0.8

1

VGS

(V)

Tra

nsco

nduc

tanc

e (4

q2 /h)

(a) (b)

Figure 6.2: (a) Drain current and (b) transconductance, as a function of gate-source voltage atVDS = 0.4V. The dotted lines are for the ultimate limit (see text). Other curves illustrate SP(circles), CM1 (dashed), and CM2 (solid).

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

Probability

Ene

rgy

(eV

)

Figure 6.3: Transmission probabilities, above Emid, of source-injected electrons for the SP(solid) and CM2 (dashed) models, at VGS = 0.4V and VDS = 0.4V.

shows a peak close to unity, then CM1’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


0 0.1 0.2 0.3 0.4 0.5 0.6 0

2

4

6

8

10

VDS

(V)D

rain

cur

rent

(µA

)

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 Lt > 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


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).









[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).



94


[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).



[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).



[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 7Quantum Capacitance in Nanoscale DeviceModeling

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 (1D) 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 1D 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 1D 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-

c© 2004 American Institute of Physics. Reprinted, with permission, from D. L. John, L. C. Castro, andD. 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

Q = q

∫ ∞

0g(E) [f (E + Eg/2 + qVa)− f (E + Eg/2− qVa)] dE , (7.1)

where q is the magnitude of the electronic charge, E is the energy, g(E) is the 1D 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

CQ =∂Q

∂Va,

and has units of F/m2 and F/m in the 2D and 1D 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

g(E) =m

π~2ν(E),

where ν(E) is the number of contributing bands at a given energy, m is the effective mass, and

~ 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

CQ =mq2

4π~2kBT∫ ∞

0ν(E)

[sech2

(E + Eg/2− qVa

2kBT)

+ sech2

(E + Eg/2 + qVa

2kBT)]

dE ,

97


where kB is Boltzmann’s constant, and T is temperature. If ν is a constant, we can perform

the integration to get

CQ =νmq2

2π~2

2− sinh (Eg/2kBT )

cosh(

Eg/2− qVa

2kBT)

cosh(

Eg/2 + qVa

2kBT)

,

which reduces to

CQ =νmq2

π~2, (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 One Dimension

In the one-dimensional, effective-mass case, we have

g(E) =ν(E)π~

√2m

E.

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 EF in order to approximate this integral using a Sommerfeld expansion [5], cannot be

done in general, due to the presence of singularities in the 1D DOS.

The capacitance is given by

CQ =q2

2kBT h

√m

2

∫ ∞

0

ν(E)√E

[sech2

(E + Eg/2− qVa

2kBT)

+ sech2

(E + Eg/2 + qVa

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 = 0.1V for a material with Eg ' 1 eV, 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


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 0.06m0 is assumed, where m0 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, ν is constant, and the DOS is given by

g(E) =2ν

hvF,

where vF is the Fermi velocity. The result is

CQ =2νq2

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 1D semiconductor, unlike in the

metallic 2D and metallic 1D 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, 1D,

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 1D 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


the contacts, i.e., in the mid-length region, Eq. (7.1) becomes [8]

Q = −q

2

∫ ∞

0g(E)T ?(E)

[f (E + Eg/2− qVa)

(2

TR(E)− 1

)

+f (E + Eg/2 + q (Vbias − Va))(

2TL(E)

− 1)]

dE , (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) = TLTR/(TL +

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 = 0.2V. 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

En ' n2π2~2

2mL2,

where L is the semiconductor length. For m ' 0.06m0, such as in a (16, 0) CN, En '

100


−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Va (V)

CQ

(nF

/m)

Figure 7.1: 1D 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 biasvoltage of 0.2V (dashed) between the end contacts. The effective mass is taken to be 0.06m0,and energy bands are situated at 0.2 and 0.6 eV on either side of the Fermi level.

6.3(n/L)2 eV, 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 = 0.1V.

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


Figure 7.2: 1D quantum capacitance, in arbitrary units, for a short-channel, phase-coherentsemiconductor as a function of position and the local electostatic potential in the mid-lengthregion for applied bias voltages of (a) 0 and (b) 0.1 V between the end contacts. The brightareas indicate higher capacitance.

7.4 Application: CNFETs

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 Cins [11], however, the ratio of these capacitances is related not to

Va and the gate-source voltage, VGS , but to ∂VGS/∂Va. 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 “CQ 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

102


metrics that depend on the total capacitance, i.e., the propagation delay may be dominated

by CQ in this limit. We find that CQ ¿ Cins 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.

0 5 10 15 20−0.3

−0.2

−0.1

0

0.1

Ene

rgy

(eV

)


(b)

0 5 10 15 20−0.3

−0.2

−0.1

0

0.1

0.2


Ene

rgy

(eV

)

(a)

Figure 7.3: Comparison of the equilibrium energy band diagrams, for a model CNFET, atgate-source voltages of (a) 0.2 and (b) 0.32 V, computed via the solutions to a self-consistentSchrodinger-Poisson system (solid), and to Laplace’s equation (dashed). The Fermi energy isat 0 eV.

Now, we seek to theoretically quantify the condition under which CQ ¿ Cins. From Fig. 7.1, we

see that the first local maxima is on the order of 0.3 nF/m. For this peak to be insignificant,

103


we would require Cins 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 Cins 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 Cins ' 25CQ, 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 E1 ' 6.3/L2 eV. If we consider positive

applied voltages to the gate and drain, this would imply that qVa should be more than about

5kBT below Eg/2+E1. For the long-channel or phase-incoherent cases, this condition is given by

E1 = 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 VGS [11]. 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


Figure 7.4: Quantum capacitance for a long-channel CNFET as a function of the gate- anddrain-source voltages. Numerical values are displayed as a fraction of the insulator capacitance.

two conducting channels, the current is

I =4q

h

∫ ∞

Ec

Tn(E) [f(E)− f (E + qVDS)] dE

−∫ Ev

−∞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

gm =∂I

∂VGS,

which yields

gm =4q2

h

[Cins

CQ + Cins

]Tn (Ec) [f (Ec)− f (Ec + qVDS)]

−Tp (Ev) [f (Ev)− f (Ev − qVDS)] . (7.5)

Note that, if we assume only electron transport with CQ ¿ Cins, low temperature, and high

VDS , this expression reduces to the classic Landauer result [4] for two conducting channels

gm =4q2

hTn ,

105


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 gm, at high VGS , is due primarily to the decreasing difference in

the contact distribution functions as, for example, Ec becomes closer to qVDS . However, the

exact magnitude of gm is dependent on the capacitance ratio. Further, CQ will be responsible

for additional oscillations in gm, 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].

0 0.1 0.2 0.3 0.4 0.5 0.60

0.4

0.8

g m (

4q2 /h

)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60.5

0.8

1.1

VGS

(V)

Cap

acita

nces

(c)0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.4

0.8

Dis

trib

utio

ns

(b)

Figure 7.5: (a) Electron transconductance for a model CNFET as a function of the gate-sourcevoltage for drain-source voltages of 0.2 (solid) and 0.4V (dashed). Constituent elements ofthe theoretical transconductance from Eq. (7.5) are (b) the energy-distribution term (in curlybraces), and (c) the capacitance-ratio term (in square brackets).

106


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 1D 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 15kBT , and varies continuously otherwise;

3. for long, 1D 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 1D 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 1D semiconductor cannot be computed, in

general, from potential division due to two capacitors in series due to the nonlinearity of

CQ;

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


References

[1] Serge Luryi, “Quantum capacitance devices,” Appl. Phys. Lett., 52(6), 501–503 (1988).



[3] P. J. Burke, “An RF circuit model for carbon nanotubes,” IEEE Trans. Nanotechnol.,

2(1), 55–58 (2003).








[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).




[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).




108




(2002).



1–235418–5 (2003).





126801–1–126801–4 (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

SrTiO3/Si substrates,” Appl. Phys. Lett., 84(11), 1946–1948 (2004).



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. Inıguez, J. Sune, L. F. Marsal, and J. Pallares, “Modeling of

nanoscale gate-all-around MOSFETs,” IEEE Electron Device Lett., 25(5), 314–316 (2004).

109

Chapter 8Method for Predicting ft for CNFETs

8.1 Introduction

Carbon nanotube field-effect transistors (CNFETs) are being seriously considered for meeting

the requirements of the 11-nm technology node [1]. Their DC performance is predicted to

be superior to that of ultimately scaled silicon MOSFETs [2, 3], and impressive values for

drain current and transconductance have already been reported in prototype devices [4]. The

AC capabilities of CNFETs are not yet so obvious. Thus far, measurements on laboratory

devices have been limited by experimental difficulties and parasitics [5–7]. Thus, the highest

reported frequency of 580 MHz, for operation without signal degradation, cannot be viewed

as a representative value for an intrinsic device1. One way to investigate the AC capabilities

of CNFETs would be to perform AC simulations with the same rigour that has characterized

earlier DC 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.

c© [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 fT for Carbon Nanotube FETs,” IEEE Trans.Nanotechnol., 4(6), 699–704 (2005).

1Very 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

∂Q

∂t=

∂Qs

∂t+

∂Qd

∂t+

∂Qg

∂t= 0 , (8.1)

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

ids + idd + idg = 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), vD(t), and vG(t), then each

displacement current will comprise three terms, e.g.,

idg = −Cgs∂vS

∂t− Cgd

∂vD

∂t+ Cgg

∂vG

∂t,

where the capacitances come from the set

Cij = ∓∂Qi

∂vj, i, j = s, d, g , (8.2)

where the minus sign is taken when i 6= 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


Thus, if we now consider vS and vD 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

idg = (Csg + Cdg)∂vGS

∂t.

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.

Cgs Cgd

S D

G

gmvgs

ig

idis

Figure 8.1: Small-signal, equivalent circuit for the total AC currents, under the condition thatonly 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., vGS and vDS in this case. In its linear

implementation, this leads to a drain conduction current of

icd = gmvgs + gdsvds .

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 4q2T/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 gm.

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


−20 dB/decade, is given by

ft =gm

2π(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 Qd into two components:

Qs = Qse + 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

Csg =∂Qs

∂vG=

∂Qse

∂vG+

∂Qst

∂vG

= Cse + Cst .

These capacitive components are readily calculated because their related charges can be com-

Insulator

Wrap-around gateSource Drain

Semiconducting nanotube

Lt

Lg

Lu

tins

tg

Qst Qdt

QdeQseisd

idd

Figure 8.2: Charge supply to and through the the source and drain electrodes in a coaxialCNFET.

puted from a recently described self-consistent DC Schrodinger-Poisson solver [9]. This solver

113


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 Qdt, 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, gm 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.


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 tins = 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 Lu = 4 nm, 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


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.

−5 0 5 10 15 20 25

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

z (nm)

Ene

rgy

(eV

)

Figure 8.3: Grey-scale representation of the energy- and position-dependence of the electroniccharge in the nanotube at VGS = 0.4V and VDS = 0.5V. The uniform columns to the left andright 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


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 = 0.5V.

The various components of the capacitance are shown in Fig. 8.4. Considering, firstly, the tube

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

Gate voltage (V)

Cap

acita

nce

(aF

)

CSE

CDE

CST

CDT

Figure 8.4: Components of the source and drain capacitances.

capacitance Cst due to charge injected from the source, the peak at around VGS = 0.35V

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


nanotubes for the source and drain. The total capacitances associated with each electrode are

shown in Fig. 8.5(a).

0

0.1

0.2

0.3

0.4

Tra

nsco

nduc

tanc

e (4

q2 /h)

6

8

10

12

Cap

acita

nce

(aF

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

Gate voltage (V)

Fre

quen

cy (

TH

z)

CSG

CDG

(a)

(b)

(c)

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 VGS is negligible. Secondly, it can

be seen that gm reaches a maximum, and then decreases as VGS increases. This phenomenon has

been reported elsewhere [12,13]. The overall reduction in gm 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


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, gm 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=300 K, 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.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

Tra

nsco

nduc

tanc

e (4

q2 /h)

Gate voltage (V)

6

8

10

12

14

16

Cap

acita

nce

(aF

)

T=300KT=4K

CSG

T=300K

CDG

T=300K

CSG

T=4K

CDG

T=4K

(a)

(b)

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 lowertemperature.

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


example of a 20 nm tube, as used here, ft peaks at about 600GHz. 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 Cde, 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

Lu should be advantageous in a negative-barrier device because the electrostatic electrode

capacitance will be reduced without a degradation in gm. 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


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 VGS 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.






120




[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).




[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).



(2006).



121


[14] Yannis P. Tsividis, Operation and Modeling of the MOS Transistor (McGraw-Hill, Toronto,

1987).



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).






[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).




[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 9High-frequency Capability ofSchottky-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, fT and fmax, 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 tins, wrapped by a cylindrical

gate of thickness tg, and capped at the ends by planar source and drain contacts of radius

tins+tg+Rt. These end contacts are separated longitudinally from the gate by a distance Lu.

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

Insulator

Wrap-around gate

Source DrainSemiconducting nanotube

Lt

Lg

Lutins

tg

Rt

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 fT , and also to consider fmax. 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 fT and fmax 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.


The model device in this study employs a (16,0) nanotube of radius 0.63 nm, length 30 nm,

bandgap 0.62 eV, unity relative permittivity, and effective mass 0.06m0, where m0 is the free

124


Cgs

Cgd

Rs

RdRggds

s

dg

S

DG

gmvgs Cmvgs

Csd

S

Figure 9.2: Full small-signal equivalent circuit for the CNFET. The transcapacitance Cm =Cdg − Cgd.

electron rest mass. Moreover, tins is 2.5 nm, tg is 3 nm, the insulator permittivity is 25 as is

appropriate for zirconia [8], and Lu is 5 nm 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 gm 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 fT and fmax for our model CNFET. Fig. 9.5(a) shows fT for

Rs = Rd set to 1 kΩ, 10 kΩ, and 100 kΩ, 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 kΩ. Note that the contact resistance is automatically included in our self-consistent

125


0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

Gate voltage (V)

Cap

acita

nce

(aF

)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

Gate voltage (V)

Tra

nsco

nduc

tanc

e (4

q2 /h)

Cgs

Cgd

Csd

Cm

Figure 9.3: Capacitances and transconductance for the model device.

0 5 10 15 20 25 30−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

z (nm)

Ene

rgy

(eV

)

Figure 9.4: Charge density for the model device subject to a gate-source voltage of 0.38V.Brighter patches indicate higher charge density, while a portion of the conduction band edge isshown 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 fT occurs at the first peak in gm. Turning now to fmax, shown in Fig. 9.5(b), we focus on

the effect of Rg, and show results for Rg set to 100Ω, 1 kΩ, and 10 kΩ, with Rs = Rd = 10 kΩ.

Again, a pronounced peak coincides with the first peak in gm.

126


0 0.2 0.4 0.6 0.8 10

200

400

600

800

Gate voltage (V)

f T (

GH

z)

(a)

0 0.2 0.4 0.6 0.8 10

200

400

600

800

Gate voltage (V)

f max

(G

Hz)

(b) 100 Ohm1 kOhm10 kOhm

1 kOhm10 kOhm100 kOhm

Figure 9.5: Extrapolated figures of merit: (a) fT with Rs = Rd set to 1 kΩ (solid), 10 kΩ(dashed), and 100 kΩ (dotted), and (b) fmax with Rs = Rd = 10 kΩ for Rg set to 100 Ω (solid),1 kΩ (dashed), and 10 kΩ (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, fT and fmax. At

the resonance peaks, these frequencies may reach the THz level.

References



(2006).





127


[4] 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).


1987).




[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).








New York, 1995).

128

Chapter 10Extrapolated fmax for CNFETs

10.1 Introduction

The frequency fmax, 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 fmax 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 (CNFETs), 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 fmax. Moreover, because of the small

size of CNFETs, 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 fmax.

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 fmax

for Carbon Nanotube Field-Effect Transistors,” Nanotechnology , 17(1), 300–304 (2006). Online athttp://www.iop.org/journals/nano.

129

Chapter 10. Extrapolated fmax for CNFETs

10.2 Modeling Procedures

The small-signal, extrinsic z-parameters for the equivalent circuit previously shown in Fig. 9.2

are given by the standard expressions [2, p.440]:

z11e = y22/Y + Rsg

z12e = −y12/Y + Rs

z21e = −y21/Y + Rs

z22e = y11/Y + Rsd

Y = y11y22 − y12y21 ,

(10.1)

where Rsg = Rs + Rg, Rsd = Rs + Rd, and the intrinsic y-parameters are [3, p.378]:

y11 = jω(Cgs + Cgd)

y12 = −jωCgd

y21 = gm − jω(Cm + Cgd)

y22 = gds + jω(Csd + Cgd) .

(10.2)

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

ωT 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 -20 dB/decade, follows,

namely [2, p.441]:1

ωT=

1ωt

[1 + gdsRsd] + RsdCgd , (10.3)

where the intrinsic “cut-off” frequency is given by

ωt =gm

(Cgs + Cgd). (10.4)

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


the following restrictions:

ω2 ¿ g2m/(Cm + Cgd)2 (10.5)

ω2 ¿ gm/(ARs) (10.6)

ω2 ¿ g2m/(Cdg + BRs)2 (10.7)

ω2 ¿ (Cgg + BRsd)2/(ARsd)2 , (10.8)

where

A = CdgCgd − CggCdd

B = Cgggds + Cgdgm

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]:

U =|z21e − z12e|2

4 [<(z11e)<(z22e)−<(z12e)<(z21e)]. (10.9)

One further restriction on the extrapolation frequency is required to obtain an expression for

U with the desired dependence on ω−2:

ω2 ¿ B2/A2 . (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]:

U =ωT

4ω2(RC)eff, (10.11)

where (RC)eff is an effective time constant. Because the assumptions we wish to make in

order to simplify the expression for fmax affect both ωT and (RC)eff we elect not to isolate the

131


expression for the latter [1], but, instead, to work on the expression for the reciprocal cyclic

frequency τeff , where

τ2eff =

(RC)effωT

, (10.12)

and

fmax =1

4πτeff. (10.13)

With the assumptions made so far, τeff is given by

τ2eff,1 = B

g2mRg(Cgg + 2RcB) + Rc [Cgg − Cm

+2Csd + RcB + AB (gm + 2gds)

],

(10.14)

where, for convenience, we have assumed similar source and drain contacts, and set Rs = Rd =

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:

τ2eff,2 =

C2gd

gm

(1 +

2gds

gm

)(2Rg + Rc)[1 + Rc(gm + 2gds)] . (10.16)

Finally, in the interests of further simplification, we suggest:

gm À 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


result of this additional assumption is a very compact expression:

τ2eff,3 =

C2gd

gm(2Rg + Rc)(1 + gmRc) . (10.18)

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.


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 Lg = 50 nm and of thickness tg = 20 nm, an insulator

relative permittivity of 16 (HfO2), and Pd end-contacts of radius tc = 4nm. Unless otherwise

stated, the contact length is Lc = 100 nm, the gate underlaps are Lus = Lud = 5 nm, and the

contact resistances are computed from a Pd resistivity of 0.48 kΩ·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, Rg = 1 kΩ. With the dimensions listed above, for example, Rc ≈ 0.9 kΩ. 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


108

109

1010

1011

1012

1013

0

50

100

Frequency (Hz)U

nila

tera

l pow

er g

ain

(dB

)

1011

1012

−10

0

10

Frequency (Hz)

U (

dB)

Figure 10.1: Unilateral power-gain for Device 1 using Eq. (10.9) (solid), and Eq. (10.11) withEq. (10.14) (dots), Eq. (10.16) (circles), and Eq. (10.18) (crosses). The inset magnifies the curvesnear 0 dB. VGS = VDS = −0.5 V.

holes of −0.04 eV [9]. The combination of low barrier height and an insulator thickness of

tins = 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

VGS = VDS = −0.5V, 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 VGS , as illustrated in

Fig. 10.2. It can be seen that the lowest τeff is ≈ 0.16 ps, which corresponds to fmax ≈ 500GHz.

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 τeff at the most negative bias

considered. The effect of assumption Eq. (10.17) is more severe at high bias because gm 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 gm. Moreover, gds rises in

that bias range, ultimately yielding a ratio 2gds/gm ≈ 1 near VGS = −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 tins of 8 nm and shorter contacts Lc of 30 nm. The higher barriers

and thicker insulator will reduce gm below that of Device 1. These features should lead to a

lower fmax than predicted for Device 1. However, this should be mitigated somewhat by lower

134


−0.8 −0.7 −0.6 −0.5 −0.4 −0.3

0.1

0.2

0.3

0.4

0.5

0.6

Gate voltage (V)

τ eff (

ps)

Figure 10.2: τeff 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 = −0.5V.

capacitances Cgs and Cgd, due to the larger tins and smaller Lc. The results shown in Fig. 10.3

show that fmax is, indeed, significantly lower than for Device 1. Interestingly, τeff ,2 is a better

approximation to the true τeff in this case, which is perhaps unexpected, given that the shorter

Lc and thicker tins should reduce the inter-electrode capacitances that would otherwise help

to equalize Cgs and Cgd. 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 τeff ,3 is no worse an approximation,

relatively speaking, than it was for Device 1.

One of the reasons for the low fmax 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 Rc ≈ 0.9 kΩ and ≈ 0.3 kΩ for Devices 1 and 2, respectively,

and Rg = 1kΩ. To examine the effect of parametrically changing these values, results are

135


−0.8 −0.7 −0.6 −0.5 −0.4 −0.30

50

100

150

200

Gate voltage (V)

f max

(G

Hz)

0 10 20

−0.3−0.2−0.1

00.10.20.3

z (nm)

Ene

rgy

(eV

)

Figure 10.3: fmax estimates for Device 2, at VDS = −0.5 V, using Eq. (10.9) (solid) andEq. (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) and2 (solid) at VGS = VDS = −0.5V. 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 fmax

was examined, after making each of the assumptions leading to the three expressions for τeff .

For the first two, the error in fmax 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 Rc. This is because the approximated term,

(gm + 2gds) → gm in simplifying Eq. (10.16), is multiplied by the square of Rc, whereas Rg

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 fmax. 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 τeff on Cgd. By contrast, τeff has a lesser dependence on transcon-

ductance. One way to trade-off gm against Cgd would be to increase tins. Ways to reduce Cgd

directly would be to shorten the drain contact Lc, and to increase the gate-drain underlap Lud.

Although the functional dependencies of gm and Cgd on tins 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


101

103

105

101

103

105

0

0.5

1

Rg (Ohms)R

c (Ohms)

(a)

Err

or (

%)

101

103

105

101

103

105

8

10

12

14

Rg (Ohms)R

c (Ohms)

(b)

Err

or (

%)

Figure 10.4: Error in fmax prediction for Device 1, incurred by the use of: (a) Eq. (10.16); (b)Eq. (10.18). VDS = −0.5V.

−0.8 −0.7 −0.6 −0.5 −0.4 −0.3100

200

300

400

500

600

Gate voltage (V)

f max

(G

Hz)

Figure 10.5: fmax for Device 1 and fmax improvement for Device 1 computed from Eq. (10.9)without assumptions, both at VDS = −0.5V. Solid line: Device 1 as originally specified; dottedline: Device 1 with tins = 8 nm, Lc = 30 nm, Lus = 5 nm, and Lud = 15nm.

peak value of fmax is raised by about 15 % to 580 GHz.

10.4 Conclusions

From this study of the extrapolated fmax in Schottky-barrier CNFETs it can be concluded that:

1. compact expressions for fmax 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. fmax values in excess of 0.5 THz should be realizable.

137


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).


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 fT for carbon nanotube FETs,” IEEE Trans. Nanotechnol., 4(6), 699–704

(2005).




[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).




[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


[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.



139

Chapter 11Conclusion and Recommendations forFuture 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 I-V -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


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 1D 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

fT and fmax, 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 fT and fmax. These oscillations were correlated to

142


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

fmax were derived, from which the importance of the gate-drain capacitance and the transcon-

ductance were ascertained. Extrinsic fT and fmax 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


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 AComplete 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

LV = Q + B (A.1)

Q = q∑

b

Db

∫ Emax

Emin

GS,b [1− u− fS ] + GD,b [1− u− fD] dE (A.2)

I =2q

h

∑

b

Db

∫ Emax

Emin

(fS − fD)T dE (A.3)

Hψ = Eψ , (A.4)

where L 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, Gc,b = Gc,b(z,E) = (ψ†ψ)c,b

is the local density of states associated with injection from contact c, ψ is the wavefunction

of a carrier with energy E, fc = f(E − µc) is the contact Fermi function with Fermi level µc,

145

Appendix A. Complete Schrodinger-Poisson Model

T = T (E) is the transmission function, andH is the effective-mass Hamiltonian in Schrodinger’s

equation. The unit step function, referenced to the midgap energy E0 (also known as the charge

neutrality level), is specified by

u = u(E − E0) =

1 , E > E0

0 , E < E0 .

To address Eq. (A.1), 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 = L−1(Q + B). In building the operator L 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 ρ ≈ 0.6 nm in Fig.A.1); (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.1);

(3) a Laplace solution is carried out for all other interior points (in set Si). Thus,

εins∂V

∂ρ

∣∣∣∣R+

t

− εt∂V

∂ρ

∣∣∣∣R−t

= − Q

2πRtε0∀ (ρ, z) ∈ St

∂V

∂ρ= 0 ∀ (ρ, z) ∈ Sa

∂2V

∂z2+

1ρ

∂V

∂ρ+

∂2V

∂ρ2= 0 ∀ (ρ, z) ∈ Si ,

where Rt is the tube radius, εins and εt are, respectively, the relative permittivities of the in-

sulator and nanotube, ε0 is the permittivity of free-space, and the 1D charge density Q has

been uniformly smeared around the tube circumference. Open boundaries (dotted lines in

Fig.A.1) 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


0 10 20 30 40 50

0

5

10

15

20

z (nm)

ρ (n

m)

Figure A.1: Simulation space for CNFET with metal contacts. Neumann boundary conditionsare specified on dotted perimeter, and Dirichlet conditions on edges of contacts. Linear chargespecified on nanotube surface (solid line at ρ ≈ 0.6 nm).

perimeter of the simulation space.

The solution of Eq. (A.1) is effected via the finite-difference scheme, chosen primarily for sim-

ulation speed. This involves discretizing the two-dimensional solution space of Fig. A.1 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 1D time-independent

Schrodinger’s equation (Eq. (A.4)). The general 1D effective-mass Hamiltonian is given by

H = − ~2mb

∂2

∂z2+ Epot ,

where mb is the carrier effective mass in sub-band b (equal for electrons and holes, and constant

across the device), ~ 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

∂2ψ

∂z2= −k2ψ ,

where ψ(z, E) is the electron wavefunction. The wavevector k = k(z, E) in the nanotube is

given by

k2 =2mb

~2

|E − E0| −∆b , |E − E0| > ∆b

|E − E0|2 −∆2b

2∆b, |E − E0| < ∆b ,

(A.5)

where ∆b is the energy distance between the bottom of sub-band b and E0, and the two energy

ranges distinguish, respectively, between carriers outside and inside the bandgap (also defined

147


as propagating and evanescent modes). The intra-gap wavevector employs a correction for the

non-parabolicity of the ε-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 ε-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 −E0| > ∆ = Eg/2 may be

rewritten as a double requirement E − E0 > ∆ and E − E0 < −∆, or E > E0 + ∆ = Ec and

E < E0 −∆ = Ev.

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Ene

rgy

rela

tive

to E

0 (eV

)

Wavevector

← imaginary k

↑electron

hole↓

real k →

Figure A.2: Complex bands in semiconducting nanotube: comparison between nearest-neighbour tight binding (solid), effective mass (dashed) and energy-dependent effective masswithin bandgap, as per Ref. [3] (circles). The agreement between the energy-dependent effectivemass 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 ε-k relation, i.e., at a given position in the device

channel, the solution ψ†ψ is counted as an electron if E > E0 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


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 ε-k relation is given by

k2 =2mb

~2(Eref − |E − µc|) , |E − µc| < Eref , (A.6)

where mb is the effective mass of the nanotube sub-band into which the carriers are being

injected, and Eref is a fitting parameter related to the type of metal being used. A half-full

metal band is assumed, such that Eref = Etop − µc = µc − Ebottom and µc = E0,c (the contact

charge neutrality level). A typical value used in the simulations was 5.5 eV, as estimated for

various metals commonly used in CNFETs from the data in Ref. [5]. The metal ε-k relation

and density of states are illustrated in Fig.A.3.

0 1 2 3−6

−4

−2

0

2

4

6

Density of states (nm−1)0 1 2 3

−6

−4

−2

0

2

4

6

Wavevector (nm−1)

Ene

rgy

(eV

)

Figure A.3: Dispersion relation and density of states in 1D metal contact.

149


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

ψ(z, E) = Aeik(z,E)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

IP (E) = iq~2m

[(∂ψ

∂z

)†(ψ)−

(∂ψ

∂z

)(ψ)†

]=

q~km|A|2 , (A.7)

and the Landauer current as

IL(E) =q

π~f(E − µS)T (E) , (A.8)

where the transmission function, defined as a ratio of outbound to inbound particle currents, is

T (E) =k(Lt, E)k(0, E)

|A(Lt, E)|2|A(0, E)|2 .

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

|A(0, E)|2 =m

π~2k(0, E)f(E − µS) . (A.9)

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Note that this normalization condition is related to the 1D density of states in the metal contact

(assuming parabolic bands):

g(E) =2π

∂k

∂E=

2m

π~2k, (A.10)

where the apparent discrepancy by a factor of 2 is due to the fact that g(E) includes both

±k-states, while |A|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 1D 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 ψ), 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 Gc,b (computed from the appropriate ψ’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).

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We normally solve the non-linear Schrodinger-Poisson system for the potential and charge

density using the Picard iterative scheme. The (k+1)-th iteration is given by

Vk+1 = Vk − αL−1rk

rk = LVk − [Q(Vk) + B] ,(A.11)

where rk is the residual of the k-th iteration, and 0 < α ≤ 1 is a damping parameter. Note

that if α = 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 k to 1;

2. compute V0 by solving Laplace’s equation (Q = 0 in Eq. (A.1));

3. solve Schrodinger’s equation for all relevant E to get corresponding ψ’s;

4. integrate ψ’s to obtain carrier density Qk(Vk−1);

5. compute new Vk by solving Poisson’s equation with new Qk in Eq. (A.11);

6. check that ||rk||∞ 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 µV. 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 1 nA. The choice of α 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, α ≈ 0.2 virtually guarantees convergence,

but often at prohibitive simulation times; in most cases, a value of α ≈ 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

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Qapprox is specified by an equation identical to Eq. (A.2), except the functions Gc,b are replaced

by the position-independent 1D 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 ∂Qapprox/∂V required

for the Jacobian operator is analytic. The algorithm is given by the following steps:

1. build matrix L and vector B; make initial guess for EqF ;

2. inner loop to find V (Qapprox(EqF )), employing Newton’s method: compute Qapprox(EqF )

and analytic ∂Qapprox/∂VCS , update potential with

Vupdate = −(L − ∂Qapprox/∂V )−1(LV −Qapprox −B), and loop until tolerance reached;

3. compute new Q(V ) using potential from step 2 and Eq. (A.2);

4. check if V = L−1(Q + B) has reached desired tolerance relative to last iteration of outer

loop; if not, find new EqF by inverting Q(EqF ) 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.1 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


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 1(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

ε-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 ε-k relation is parabolic (top component of Eq. (A.5))

for all energies, F=1 when the ε-k relation is described exactly by Eq. (A.5), and F=2 when the

ε-k relation has an energy-dependent effective mass (bottom component of Eq. (A.5)) for all

energies, i.e., for both |E −E0| < ∆b and |E −E0| > ∆b. 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.

−1 −0.5 0 0.5

10−8

10−7

10−6

10−5

Gate voltage (V)

−I D

(A

)

−0.4 −0.3 −0.2 −0.1 0

−16

−12

−8

−4

0

Drain voltage (V)

I D (

µA)

(a) (b)

Figure A.4: Agreement between Schrodinger-Poisson model and data from experimental deviceof Ref. [9]: (a) I-VGS plot for VDS = −0.3V, and (b) I-VDS plot for (from top to bottom offigure) VGS = −0.43, −0.73 and −1.03V (these values correspond to VGS = −0.1, −0.4 and−0.7V in Ref. [9]; see text for explanation). Solid lines are experimental results, while symbolsare 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

(Eg=0.52 eV), a self-aligned gate with Lg = 50 nm (underlap Lu = 0) and thickness tg = 4 nm,

an insulator of thickness tins = 8 nm and relative permittivity of 15 (HfO2), and Pd end-contacts

(φ=5 eV) of radius tc = 6nm and length Lc = 4 nm. We set Eref = 1 eV for the metal contacts,

154


and F=2. Note also that all the simulation curves in Fig. A.4 are offset in the gate bias by

0.33V 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.

−0.2 0 0.2 0.4 0.6 0.8

10−7

10−6

10−5

Gate Voltage (V)

Dra

in c

urre

nt (

A)

F=0F=1F=2NEGF

Figure A.5: Comparison of the solver described in this section to the atomistic simulations ofRef. [10], in modeling the I-V characteristics of a p-i-n device. Refer to text for parametersemployed in simulation and an explanation of figure caption.

In another comparison, we illustrate in Fig. A.5 the simulated I-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, Lg = 7 nm, Lu = 0, Lc = 10nm, tg = 0.1 nm, tins = 2nm, εins = 3.9, and

VDS = 0.4V. 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 ε-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


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).



[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:

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[10] David Llewellyn John, Simulation studies of carbon nanotube field-effect transistors, Ph.D.

thesis, University of British Columbia, Vancouver, Canada (2006).

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