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MOS-AK Munich 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 [email protected], http://www.syssim.ecs.soton.ac.uk Circuit-level modelling of carbon nanotube field-effect transistors 1 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Outline o Introduction New efficient methodology for numerical CNT FET modelling based on piece-wise non-linear approximation o PNL modelling of non-equilibrium mobile charge density Two PNL approximations leading to closed-form solution of selfconsistent voltage equation o Drain current calculation o Equivalent circuit o Simulation experiments demonstrating speed up and modelling accuracy o Conclusion: what next? Circuit-level modelling of carbon nanotube field-effect transistors 2 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Introduction o o o o CNT FET theory and operation are gradually better understood. Early CNT FET models simply used MOS equations – no good. Now a physical theory of ballistic CNT transport exists. Circuit-level models have been developed based on theory but they are very complex in terms of computational intensity. o Recently fast models appeared, based on numerical approximation. o Focus 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. o Important for circuit design where very large numbers of CNT devices will need to be simulated. Circuit-level modelling of carbon nanotube field-effect transistors 3 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Non-equilibrium mobile charge o Non-equilibrium mobile charge is injected into CNT when drain-source voltage is applied: o State densities are determined by Fermi-Dirac probability distribution: VSC – self-consistent voltage Circuit-level modelling of carbon nanotube field-effect transistors 4 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Self-consistent voltage equation VSC - 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 Circuit-level modelling of carbon nanotube field-effect transistors 5 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Standard approaches to evaluate charge density o Newton-Raphson technique and finite integration o Non-equilibrium Green’s function (NEGF) o Recently piece-wise linear and piece-wise non-linear approximations have been proposed to obtain closedform symbolic solutions The aim is to eliminate the need for computationally intensive iterative calculations in development of models for circuit simulators Circuit-level modelling of carbon nanotube field-effect transistors 6 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Total drain current If VSC 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: Circuit-level modelling of carbon nanotube field-effect transistors 7 T.J. Kazmierski MOS-AK Munich 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 qN0 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 Circuit-level modelling of carbon nanotube field-effect transistors 8 T.J. Kazmierski MOS-AK Munich 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: Linear and quadratic pieces solid line: theory dashed-line: approximation Circuit-level modelling of carbon nanotube field-effect transistors 9 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University New technique to accelerate VSC calculation Model 2: 4-piece non-linear approximation: Linear, quadratic and 3rd order pieces solid line: theory dashed-line: approximation Region boundaries are optimised for best fit Circuit-level modelling of carbon nanotube field-effect transistors 10 T.J. Kazmierski MOS-AK Munich 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 Circuit-level modelling of carbon nanotube field-effect transistors 11 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Loss of accuracy due to PNL approximation Model 1 – dashed, FETToy - solid Model 2 – dashed, FETToy - solid Typical parameters: T=300K, Ef = -0.32eV Circuit-level modelling of carbon nanotube field-effect transistors 12 T.J. Kazmierski MOS-AK Munich 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 Circuit-level modelling of carbon nanotube field-effect transistors 13 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Accuracy at extreme temperatures and Fermi levels Model 1 – dashed, FETToy - solid Model 2 – dashed, FETToy - solid Extreme parameters: T=150K, Ef = 0eV Circuit-level modelling of carbon nanotube field-effect transistors 14 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Accuracy at extreme temperatures and Fermi levels (2) Model 1 – dashed, FETToy - solid Model 2 – dashed, FETToy - solid Extreme parameters: T=450K, Ef = -0.5eV Circuit-level modelling of carbon nanotube field-effect transistors 15 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University RMS errors for Ef=-0.5eV Across T and EF ranges - Model 2 is accurate within 2.8%, Model 1 – 4.8% Circuit-level modelling of carbon nanotube field-effect transistors 16 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University RMS errors for Ef=0eV Circuit-level modelling of carbon nanotube field-effect transistors 17 T.J. Kazmierski MOS-AK Munich 14 September 2007 School of Electronics and Computer Science Southampton University Conclusion o New, fast, numerical CN FET model has been proposeds o Suitable for a direct implementation in SPICE-like circuit-level simulators o Further evidence to support suggestions that costly NewtonRaphson iterations and Fermi-Dirac integral calculations can be avoided leading to a substantial speed-up. o Two models proposed and tested in simulationss o Future work will involve CN FET analysis of speed and modelling accuracy of circuit structures built of CN FETs. Circuit-level modelling of carbon nanotube field-effect transistors 18 T.J. Kazmierski
