Circuit-level modelling of carbon nanotube field-effect transistors by ypl44131

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									                 MOS-AK                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




           Circuit-level modelling of
          carbon nanotube field-effect
                   transistors
                                            Tom J Kazmierski
                              School of Electronic and Computer Science
                              University of Southampton, United Kingdom
                         tjk@ecs.soton.ac.uk, http://www.syssim.ecs.soton.ac.uk




Circuit-level modelling of carbon nanotube field-effect transistors 1                               T.J. Kazmierski
                 MOS-AK                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




                                                        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 self-
                   consistent 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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




                                                 Introduction
       o CNT FET theory and operation are gradually better understood.
       o Early CNT FET models simply used MOS equations – no good.
       o Now a physical theory of ballistic CNT transport exists.
       o 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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007



                    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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



                  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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




           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 closed-
         form 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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




                                 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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007




           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                                                 School of Electronics and Computer Science
                  Munich                                                          Southampton University
            14 September 2007



     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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



     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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007




           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                                                   School of Electronics and Computer Science
                  Munich                                                            Southampton University
            14 September 2007




       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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



                              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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



     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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



                                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                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007



                                    RMS errors for Ef=0eV




Circuit-level modelling of carbon nanotube field-effect transistors 17                               T.J. Kazmierski
                 MOS-AK                                                  School of Electronics and Computer Science
                  Munich                                                           Southampton University
            14 September 2007




                                                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 Newton-
         Raphson 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

								
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