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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 1 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Circuit-level modelling of carbon nanotube field-effect transistors Tom J Kazmierski School of Electronic and Computer Science University of Southampton, United Kingdom

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 2 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Outline oIntroduction New efficient methodology for numerical CNT FET modelling based on piece-wise non-linear approximation oPNL modelling of non-equilibrium mobile charge density Two PNL approximations leading to closed-form solution of self- consistent voltage equation oDrain current calculation oEquivalent circuit oSimulation experiments demonstrating speed up and modelling accuracy oConclusion: what next?

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 3 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Introduction oCNT FET theory and operation are gradually better understood. oEarly CNT FET models simply used MOS equations – no good. oNow a physical theory of ballistic CNT transport exists. oCircuit-level models have been developed based on theory but they are very complex in terms of computational intensity. oRecently fast models appeared, based on numerical approximation. oFocus of this talk: new, efficient piecewise non-linear approximation of mobile charge three orders of magnitude faster than evaluation of physical equations, but still maintaining high accuracy. oImportant for circuit design where very large numbers of CNT devices will need to be simulated.

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 4 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Non-equilibrium mobile charge oNon-equilibrium mobile charge is injected into CNT when drain-source voltage is applied: oState densities are determined by Fermi-Dirac probability distribution: V SC – self-consistent voltage

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 5 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Self-consistent voltage equation V SC - recently introduced concept Strongly non-linear, requires Newton-Raphson iterations and calculation of integrals – standard approach to CNT FET modelling Total charge at terminal capacitances Total terminal capacitance

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 6 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Standard approaches to evaluate charge density oNewton-Raphson technique and finite integration oNon-equilibrium Green’s function (NEGF) oRecently piece-wise linear and piece-wise non-linear approximations have been proposed to obtain closed- form symbolic solutions The aim is to eliminate the need for computationally intensive iterative calculations in development of models for circuit simulators

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 7 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Total drain current If V SC is known, total drain current can be obtained form Fermi-Dirac statistics directly: Closed-form solution for Fermi-Dirac integral of order 0 exists: hence:

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 8 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Circuit model of a top-gate CNT FET If equal portions of the equilibrium charge qN 0 are allocated to drain and source, non-equilibrium charges at drain and source can be modelled as non-linear capacitances. A hypothetical inner node can be created to represent the self-consistent potential

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 9 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University New technique to accelerate VSC calculation Model 1: 3-piece non-linear approximation of charge density: solid line: theory dashed-line: approximation Linear and quadratic pieces

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 10 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University New technique to accelerate VSC calculation Model 2: 4-piece non-linear approximation: solid line: theory dashed-line: approximation Region boundaries are optimised for best fit Linear, quadratic and 3 rd order pieces

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 11 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Speed-up due to PNL approximation FETToy – reference theoretical model implemented in MATLAB CPU times for PNL Model 1 and Model 2 obtained also from a MATLAB script Model 1 runs 3500 faster and Model 2 – 1100 times

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 12 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Loss of accuracy due to PNL approximation Model 1 – dashed, FETToy - solid Typical parameters: T=300K, Ef = -0.32eV Model 2 – dashed, FETToy - solid

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 13 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University RMS errors for Ef=-0.32eV Model 2 accurate within 2%, Model 1 – 4.6%, at T=300K

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 14 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Accuracy at extreme temperatures and Fermi levels Model 1 – dashed, FETToy - solid Extreme parameters: T=150K, Ef = 0eV Model 2 – dashed, FETToy - solid

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 15 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Accuracy at extreme temperatures and Fermi levels (2) Model 1 – dashed, FETToy - solid Extreme parameters: T=450K, Ef = -0.5eV Model 2 – dashed, FETToy - solid

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 16 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University RMS errors for Ef=-0.5eV Across T and E F ranges - Model 2 is accurate within 2.8%, Model 1 – 4.8%

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 17 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University RMS errors for Ef=0eV

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T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 18 MOS-AKMunich 14 September 2007 School of Electronics and Computer Science Southampton University Conclusion oNew, fast, numerical CN FET model has been proposeds oSuitable for a direct implementation in SPICE-like circuit-level simulators oFurther evidence to support suggestions that costly Newton- Raphson iterations and Fermi-Dirac integral calculations can be avoided leading to a substantial speed-up. oTwo models proposed and tested in simulationss oFuture work will involve CN FET analysis of speed and modelling accuracy of circuit structures built of CN FETs.

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