The research paper published by IJSER journal is about Stability Analysis for Carbon Nanotube based Field Effect Transistors 1

ISSN 2229-5518

Stability Analysis for Carbon Nanotube based

Field Effect Transistors

Saeede Afsharmehr, Mehrnoosh Zeinalkhani, Mohammadali Ghorbani, and Alireza Heidari

Abstract—We present Nyquist stability criterion for small-signal equivalent circuit model of carbon nanotube field effect transistors (CNTFETs). To the best of our knowledge, this is the first instance that such analysis has been done for CNTFETs so far. In this analysis , the dependence of the degree of relative stability in CNTFETs on the length and diameter of channel tube is acquired. It is shown that increasing both the tube length and diameter causes higher stability.

Index Terms—Carbon nanotube field effect transistor (CNTFET), Nyquist Stability.

1 INTRODUCTION

—————————— ——————————
resistance of source/drain contact[5].
A planar gate structure which is closer to the reality than the
igh mobility, low-defect structure, direct band gaps, symmetric bands, and intrinsic-nanometer scale of carbon nanotubes (CNTs) has led to an intense
research effort into the viability of utilizing CNT field-effect transistors (CNTFETs) as a promising replacement for, or complement to traditional silicon MOSFETs [1], [2]. Semiconducting CNTs have been instead used to demonstrate field-effect transistors (FETs) having quasi-1-D channels [2]. The CNTFET can be characterized by: the highest carrier mobility at room temperature of any known material, nearly two orders of magnitude better than existing semiconductor technology, highly scalable, low-power, and low leakage current relatively to the applied voltages and low inverse subthreshold slope. With ultra-long (~ 1µm) mean free path (MFP) for elastic scattering, a ballistic or near-ballistic transport in the channel can be obtained with an intrinsic carbon nanotube (CNT) under low voltage bias leading to higher transconductance compared to any other materials. Due to near ballistic transport and the high band-structure- limited velocity in CNTs, CNTFETs also promise good potential for radio-frequency (RF) applications. The quasi-1-D
structure provides better electrostatic control over the

coaxial gate structure is used in the modeling. A complete compact circuit model is illustrated in Fig. 2 where CGB is the coupling capacitance between gate and substrate that is derived as CGB=CBG=LgCsubCox/(Ctot+CQs+CQd). The substrate-to- gate capacitance Csub can be calculated with the simple equation Csub=2k20/ln(2Hsub/d). The parameters Hsub and k2 represent the bulk dielectric material thickness and dielectric constants respectively. Cox, the capacitance between the gate and channel, and Ctot=Cox+Csub+Cc, Cc is the capacitance between the channel and external drain [4]. Cxy/Cyx is the transcapacitance pairs. LMS is the magnetic inductance which is three orders smaller than LKS, the quantum inductance of CNT [7].The kinetic inductance is estimated as LKS=hl/4q2vF, where vF is the carrier velocity [6]. In this figure RS and RD represent series resistance due to scattering. Cgs/Csg includes the geometrical capacitance (CES Electrostatic capacitance) in series with the quantum capacitance;the quantum capacitance is multiplied by four because of the band structure degeneracy. Therefore:




1  1  1 (1)

channel region than 3-D devices (bulk CMOS) and 2-D device (fully depleted SOI) structures [3], [4], [5], [6]. A circuit- compatible SPICE model for single-wall nanotube FET

Cgs

CES

2

CQ

2

CES 1


 (2)
(SWNT-FET) is used to model ac characteristics and gain
Transfer Function of the SWNT-FET. This Transfer Function
achieved from SWNT-FET configured as a common-source
cosh (2h / d ) ln(h / d )
where h and d denote CNT’s height and diameter,
amplifier.[5] Briefly it is a physics-based model that also
respectively [9].

Csb Csg (Csub / Cox )

and Cdb Cdg (Csub / Cox ) .
includes device nonidealitics such as the quantum
The Rds of CNTFETs is generally expressed as
confinement effect in the circumferential and the channel

R R

 (h / 4e2 )(l / ) where RSB is the contact resistance, l is
length directions, the capacitance and resistance of the doped source/drain SWNT region, as well as the possible SB

————————————————

Saeede Afsharmehr, Institute for Advanced Studies, Tehran 14456-63543, Iran.

Mehrnoosh Zeinalkhani, Institute for Advanced Studies, Tehran 14456-

63543, Iran.

Mohammadali Ghorbani, Institute for Advanced Studies, Tehran 14456-

63543, Iran.

Alireza Heidari, Institute for Advanced Studies, Tehran 14456-63543, Iran,

E-mail: Prof.Alireza.Heidari@InstituteforAdvancedStudies.us

ds SB

channel length and is the electron mean free path [10], [11], [12].
On the other hand, the Nyquist diagram that is a complex plane of coordinates with a real (horizontal) axis and an imaginary (vertical) axis can be a powerful tool for investigating the system relative stability [8], [13], [14], [15]. Point (-1,0), in this complex plane, is the critical point for stability. When inside the diagram, for the system to be more stable, the point must move toward outside the diagram, as

IJSER © 2012

http://www.ijser.org

The research paper published by IJSER journal is about Stability Analysis for Carbon Nanotube based Field Effect Transistors 2

ISSN 2229-5518

its parameters are varied. Otherwise, the farther the point moves from the diagram, the more stable is the system. In this letter, using the Nyquist diagram as a criterion, we examine the relative stability of a SWNT-FET common source configuration that is shown in Fig. 2.

Gate

diagram for all three given diameter. However, as d increases, the critical point (-1,0) moves toward outside. Fig. 4 shows similar Nyquist diagram for CNTFET with d = 5 nm and l =18,
50, and 100 nm. Fig. 4 also shows that the critical point (-1,0) also moves toward outside. In other hand, this means, as d or LCH increases, that the relative stability of the CNTFET increases. In fact as d increases CES, Csb, Cdb, and CGB increase. Due to an increase in CES, Cgs, Cgd increases. As l increases, Lk,

Source

Oxide

Substrate

CNT

Drain

LM, RSDB, and RSB increases. Note that, while CES, CGB, Cgs, Cgd,

RSDB, and RSB are the dominant parameters in determining the switching delay, the effects of variations in LK and LM on the

switching delay are negligible. Hence, as increase in either of CGB, Cgs, Cgd, RSDB, and RSB gives rise to the CNTFET switching delay. As the switching delay increases, the step response of the CNTFET tends to damp more rapidly, and the system becomes more stable.

Fig. 1. The 2D cross-sectional layout of fabricated SW NT-FET.

Nyquist Diagram

Gate

C

C /C

Cgd/Cdg

300

200

D=1nm

D=3nm

GB

Source

gs sg

LMS LKS RS

IDS f (VGS,VDS)

Drain

RD

100

0

D=5nm

Csb / Cbs

RSB

RSDB

RSDB

Cdb / Cbd

RDB

-100

-200

-300

-150 -100 -50 0 50 100 150

Real Axis

Fig. 2. Small-signal SW NT-FET device model. Cxy=∂Qx/∂Vy. Cgs/Csg, Cgd/Cdg, Csb/Cbs, and Cdb/Cbd are transcapacitance pairs.

Fig. 3. The Nyquist diagrams for SW CNTFETs configuration of Fig. 2 for

l=50 nm and 1≤ d ≤5 nm.

2 RELATIVE STABILITY

In the configuration shown in Fig. 2, an SWNT-FET of channel length ( l ) and carbon nanotube (CNT) diameter ( d ) that is represented by a coupling capacitance (CGB) , magnetic inductance (LMS) and kinetic inductance (LKS) , transcapacitance pairs (Cxy/Cyx), source/drain resistance (RS/RD), (RSB/RDB), and (RSDB) (all in per length units). Using these elements, the input-output transfer function becomes

20

15

10

5

0

-5

-10

Nyquist Diagram

lch=18nm

lch=50nm

lch=100nm

H (s)  Vo (s) / Vi (s) 

4

i 0

ai s / bi s

(3)

-15

-20

-10 -5 0 5 10

Real Axis

where s = jis the complex frequency and coefficients ai and

bi are given in the appendix.

By varying the nanotube dimensions (18 ≤ l ≤100 nm and 1 ≤ d
≤ 5 nm) and generating various Nyquist diagrams, we have
studied the effect of SWNT-CNTs geometry on the relative
stability of the configuration shown in Fig. 2. In this analysis,
we have assumed that RS = RD =3 k, =700 nm and h=140 nm.
Fig. 3 shows the Nyquist diagram for the configuration of Fig.
2, for nanotube of l =50 nm and d =1, 3, and 5 nm. As can be
seen in Fig. 3, the critical point (-1,0) is located inside the

Fig. 4 The Nyquist diagrams for SW CNTFETs configuration of Fig. 2 for d=1

nm and 18≤ l ≤100 nm.

3 CONCLUSION

Using small signal model with Nyquist stability diagrams, the relative stability for CNTFET has been studied. We have shown that with increasing either the diameter or length of channel of the CNTFET, the relative stability increases, and hence, the system becomes more stable. This is because an increase in either parameter gives rise to switching delay and,

IJSER © 2012

http://www.ijser.org

The research paper published by IJSER journal is about Stability Analysis for Carbon Nanotube based Field Effect Transistors 3

ISSN 2229-5518

hence, its step response tends to damp faster, and as a result, the system become more stable.

APPENDIX

Coefficients ai and bi in (3) are as follows:

a0 RD g m C gs [ X 1 X 2 X 3 ]

a1 RD [(g m C gs ( RSDB (Cdb X 2 X 4 X 1 )  Csb )

RSDB

REFERENCES

[1] ChangxinChen,Dong Xu,Eric Siu-Wai kong,and Yafei Zhang , "Multichannel Carbon-Nanotube FETs and Complementary Logic Gates With Nanowelded Contacts," IEEE Electron Device Lett.,vol.27,no.10

[2] Zahra Arefinia and Ali A.Orouji, Quantum Simulation Study of a New Carbon Nanotube Field-Effect Transistor With Electrically Induced Source/Drain Extension,”IEEE Trans. on Device and Materials Reliability,vol.9,no 2.

[3] S.DHamieh,P.Desgreys,and J.F.Naviner,”Scattering Effects on the

C gs C gd ((1 

RDB

)( X 2 X 3 )  C gs CGB X 3 )]

Performance of Carbon Nanotube Field Effect Transistor in A

Compact Model,”Eur. Phys. J. B.

[4] Jie Deng, and H.-S.Philip Wong,”A Compact SPICE Model for Carbon-

a2 RD [Cgs Cgd ( RSDB ( X 2Cdb X 4 X 1 )  Csb )

g m Cgs RSDB (Cdb X 4 RSDBCGBCsb X 1 )  CGBCsb Cgs

Csb Cdb (CGB Cgs Cgd )]

a3   RD [Cgs Cgd RSDB (Cdb X 4

Nanotube Field-Effect Transistors Including Nonidealities and Its Application---Part Ι:Model of the Intrinsic Channel Region,”IEEE Trans. on Electron Devices,vol.54,no.12,pp.,2007 .

[5] Islamshah Amlani, King F.Lee,Jie Deng and H.-S.Philip Wong,”Measuring Frequency Response of a Single-Walled Carbon Nanotube Common-Source Amplifier,”IEEE Trans.

R C

C X )  g C

R 2 SDBC C C )

Nanotechnology,vol.8,no.2,pp.,2009.

SDB GB sb 1

m gs

2

db GB sb

[6] Jing Guo,Sayed Hasan,Ali Javey,Gijs Bosman and Mark

Lundstrom,”Assessment of High-Frequency Performance Potential

a4  RDCgsCgd RSDBCGBCsbCdb

b0 C gs RSDB X 2 X 1 C gs X 3

b1 Cgs RSDB[Cdb X 2 X1 X 4 )]  CgsCsb

RDCgs (Cgd Cdb )((RSDB X 2 X1 )  X 3 )

b2  Cgs RSDB (Cdb X 4RSDBCGBCsb X1 )  RDCgs (Cgd Cdb )

2

SDB db 2 1 4 sb db SDB gs D 2

Csb RDCdb (Cgd Cdb )

of Carbon Nanotube Transistors,”IEEE Trans. on Nanotechnology,

vol.4,no.6,pp.,2005.

[7] Jie Deng and H.-S.Philip wong,”A Circuit-Compatible SPICE model for

Enhancement Mode Carbon Nanotube Field Effect Transistors,”IEEE .

[8] R. C. Dorf, and Robert H. Bishop, “ Modern Control System,” 11th Ed.,

2008, Pearson Prentice-Hall, New Jersey.

[9] Peter J.Burke,”AC performance of nanoelectronics:towards a ballistic

THz Nanotube transistors,”Solid-State Electronics

[10] Guangyu Xu,Fei Liu,Song Han,Koungmin Ryu,Alexander Badmaev,Bo

Lei,Chongwu Zhou and Kang L.Wang,”Low-frequency noise in top-

gated ambipolar carbon nanotube field effect transistors,”APPLIED

PHYSIC LETTERS 92 ,23114 (2008)

[11] A. Heidari, A. Nabatchian, M. Godarzvand Chegini, M.

Zeinalkhani, M. Ghorbani, “A Novel Analytical Approach to

Superconductivity in Single-Walled Carbon Nanotubes Using the

b3  C gs

2

SDB

CGBCsb

Cdb

RD

RSDBC gs

(C gd

Cdb )

Green’s Function: A Mathematical Study”, Journal of Mathematics

(Cdb X 4

RSDB

CGBCsb

X )  C 2 R

SDB

RD C gs X 4

Research 4 (2), 86 (2012)

[12] A. Heidari, N. Heidari, R. Amiri, F. Khademi Jahromi, M. Zeinalkhani,

F. Ghorbani, A. Piri, S. Kumar, M. Ghorbani, “A new approach to

2

4 GB gs gd sb db SDB D

studying and investigating hydrogen storage in carbon nanostructures”, International Journal of Scientific & Engineering Research 3 (3), (2012)

X 1

X 3

1

RSDB

1

RSDB

 1

RDB

 1

RSB

, X 2

 1

RSDB

 2

RSB

[13] E. Hosseini Nezhad, N. Heidari, M. Ghorbani, A. Heidari, “A New Approach to Synthesizing NiO-YSZ Nanocomposite as Solid-Oxide Fuel-Cell Anode Functional Layers by Electrophoretic Deposition”, Journal of Materials Science Research 1 (2), (2012)

[14] A. Heidari, N. Heidari, F. Khademi Jahromi, R. Amiri, M. Zeinalkhani,

X 4 CGB

 2Csb


RSDBCGB CGBCsb

F. Ghorbani, A. Piri, M. Ghorbani, “A new and numerical approach to

RSB

ACKNOWLEDGEMENT

C gs

corrosion resistance of electrodeposited nanostructured Ni-Co coatings using a Watts bath under pulsed current”, International Journal of Scientific & Engineering Research 3 (3), (2012)

[15] M. Ghorbani, A. Heidari, N. Heidari, A. Yıldırım, “Conformational,

Structural and Aromatic Features of (8,8) Close-Ended Carbon

The work described in this paper was fully supported by grants from the Institute for Advanced Studies of Iran. The authors would like to express genuinely and sincerely thanks and appreciated and their gratitude to Institute for Advanced Studies of Iran.

nanotube 7/5/7 Ring Arrangement: A Theoretical Ab Initio Study”,

14th International Conference on the Union of Pure and Applied

Chemistry within Polymers and Organic Chemistry, 2012, Doha,

Qatar.

IJSER © 2012

http://www.ijser.org