Circuit-level modelling of carbon nanotube field-effect transistors

1. 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

2. Outline • Introduction • New efficient methodology for numerical CNT FET modelling based on piece-wise non-linear approximation • PNL modelling of non-equilibrium mobile charge density • Two PNL approximations leading to closed-form solution of self-consistent voltage equation • Drain current calculation • Equivalent circuit • Simulation experiments demonstrating speed up and modelling accuracy • Conclusion: what next?

3. Introduction • 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. • Recently fast models appeared, based on numerical approximation. • 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. • Important for circuit design where very large numbers of CNT devices will need to be simulated.

4. Non-equilibrium mobile charge • Non-equilibrium mobile charge is injected into CNT when drain-source voltage is applied: • State densities are determined by Fermi-Dirac probability distribution: VSC – self-consistent voltage

5. Self-consistent voltage equationVSC - 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

6. Standard approaches to evaluate charge density • Newton-Raphson technique and finite integration • Non-equilibrium Green’s function (NEGF) • 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

7. 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:

8. 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

9. 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

10. 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

11. 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

12. Loss of accuracy due to PNL approximation Model 1 – dashed, FETToy - solid Model 2 – dashed, FETToy - solid Typical parameters: T=300K, Ef = -0.32eV

13. RMS errors for Ef=-0.32eV Model 2 accurate within 2%, Model 1 – 4.6%, at T=300K

14. Accuracy at extreme temperatures and Fermi levels Model 1 – dashed, FETToy - solid Model 2 – dashed, FETToy - solid Extreme parameters: T=150K, Ef = 0eV

15. 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

16. RMS errors for Ef=-0.5eV Across T and EF ranges - Model 2 is accurate within 2.8%, Model 1 – 4.8%

17. RMS errors for Ef=0eV

18. Conclusion • New, fast, numerical CN FET model has been proposeds • Suitable for a direct implementation in SPICE-like circuit-level simulators • Further evidence to support suggestions that costly Newton-Raphson iterations and Fermi-Dirac integral calculations can be avoided leading to a substantial speed-up. • Two models proposed and tested in simulationss • Future work will involve CN FET analysis of speed and modelling accuracy of circuit structures built of CN FETs.