T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 1 MOS-AKMunich 14 September 2007 School of Electronics and Computer

T.J. KazmierskiT.J. KazmierskiCircuit-level modelling of carbon nanotube field-effect transistors Circuit-level modelling of carbon nanotube field-effect transistors 11

MOS-AKMOS-AKMunichMunich

14 September 200714 September 2007

School of Electronics and Computer ScienceSchool of Electronics and Computer ScienceSouthampton UniversitySouthampton University

Circuit-level modelling of carbon nanotube field-effect

transistors

Tom J KazmierskiSchool of Electronic and Computer ScienceUniversity of Southampton, United Kingdom

[email protected], http://www.syssim.ecs.soton.ac.uk





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 calculationo Equivalent circuito Simulation experiments demonstrating speed up and modelling

accuracyo Conclusion: what next?





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.





Non-equilibrium mobile chargeo 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





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





Standard approaches to evaluate charge density

o Newton-Raphson technique and finite integrationo 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





Total drain currentIf 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 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 asnon-linear capacitances.

A hypothetical inner node can be created to represent

the self-consistent potential





New technique to accelerate VSC calculationModel 1: 3-piece non-linear approximation of charge density:

solid line: theory

dashed-line: approximation

Linear and quadratic pieces





New technique to accelerate VSC calculationModel 2: 4-piece non-linear approximation:

solid line: theory

dashed-line: approximation

Region boundaries are optimised for best fit

Linear, quadratic and 3rd order pieces





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





Loss of accuracy due to PNL approximation

Model 1 – dashed, FETToy - solid

Typical parameters: T=300K, Ef = -0.32eV






RMS errors for Ef=-0.32eV

Model 2 accurate within 2%, Model 1 – 4.6%, at T=300K





Accuracy at extreme temperatures and Fermi levels


Extreme parameters: T=150K, Ef = 0eV






Accuracy at extreme temperatures and Fermi levels (2)


Extreme parameters: T=450K, Ef = -0.5eV






RMS errors for Ef=-0.5eV

Across T and EF ranges - Model 2 is accurate within 2.8%, Model 1 – 4.8%





RMS errors for Ef=0eV





Conclusion

o New, fast, numerical CN FET model has been proposedso Suitable for a direct implementation in SPICE-like circuit-level

simulatorso 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 simulationsso Future work will involve CN FET analysis of speed and modelling

accuracy of circuit structures built of CN FETs.

Documents

T.J. Kazmierski Circuit-level modelling of carbon nanotube field-effect transistors 1 MOS-AKMunich 14 September 2007 School of Electronics and Computer