J Comput Electron (2012) 11:166–171DOI 10.1007/s10825-012-0391-1

Regional approach to model charges and capacitances of intrinsiccarbon nanotube field effect transistors

Nune V. Sudheer · Anjan Chakravorty

Published online: 11 February 2012© Springer Science+Business Media LLC 2012

Abstract Estimation of total channel charge (QT ot ) of car-bon nanotube field effect transistor from the self-consistentlycomputed charge density (QT op) at the top of conductionband subband minima, is found to be inaccurate. A regionalapproach based on extended ballistic transport theory is pro-posed to model QT ot , and its partitioning into source anddrain components. The models for charges and subsequentlyderived capacitances are validated with the numerically sim-ulated data obtained using semi-classical technique. Themodel agreement with numerical data shows the superiorityof our regional approach compared to ones obtained fromonly the information of QT op.

Keywords Carbon nanotube FET · Regional approach ·Charge partitioning · Transcapacitances · Ballistictransport · Subthreshold charge

1 Introduction

Carbon nanotube field effect transistor (CNFET) is one ofthe serious contender to replace the conventional metal ox-ide semiconductor field effect transistor (MOSFET) in fu-ture integrated circuits since with CNFET it is possible toachieve near-ballistic operation and significantly high perunit width on-currents [1–3]. Researchers are putting seri-ous efforts to understand the operation of CNFET throughexperiments [4–6] as well as through numerical simulation[7, 8]. Our investigation is based on the data obtained us-ing a numerical simulator MOSCNT 1.0 [9] for the gate-all-around ballistic CNFET of Figs. 1(a) and (b). In the n-i-n

N.V. Sudheer · A. Chakravorty (�)Dept. of Electrical Engineering, IIT Madras, Chennai 600036,Indiae-mail: anjan@ee.iitm.ac.in

structure, channel charge only due to the excess electronsentering from source (or drain) extension region overcom-ing the source-channel (or drain-channel) barrier is consid-ered. Since the band-to-band tunneling [10] is not consid-ered in this study, instead of non-equilibrium Green’s func-tion technique, we used semi-classical approach [11] avail-able in MOSCNT to generate the reference data. As the in-vestigated structure is a MOSFET-like conventional CNFET[12] without any Schottky-barrier (SB) [13, 14], inability ofMOSCNT to handle SB contacts does not pose any limita-tion in our study.

Although the reported model [15] based on ballistictransport theory [16, 17] is capable of accurately predictingstatic drain current and charge density, QT op , at the top ofconduction band subband minima, i.e., at xT op (Fig. 1(c)),it cannot model the total channel charge (QT ot ), required todevelop dynamic model [18] of intrinsic CNFET. Assum-ing uniform carrier distribution over the channel [19], weobtain a model (hereafter called as uniform charge densitybased model) where total channel charge is given by QUni

T ot =QT opL. Simple analytical charge formulations that are use-ful in developing compact models [12, 20], rely on this uni-form charge density based model. However, the computedQUni

T ot is insufficient to predict actual QT ot especially in sub-threshold operation, since the non-zero bias-dependent sub-threshold charge resides near the source extension region ofthe channel, and does not reach the top of subband minima.Figure 2 demonstrates that QUni

T ot underestimates the totalchannel charge, QNum

T ot , in subthreshold operation (see ‘mis-match’ region in Fig. 2), where QNum

T ot is obtained from thenumerically simulated (using MOSCNT) positional infor-mation of channel charge density, QNum

ch (x) as,

QNumT ot =

∫ L

0QNum

ch (x)dx. (1)

J Comput Electron (2012) 11:166–171 167

Fig. 1 Schematics of theinvestigated CNFET withchirality (25,0): (a) sideviewand (b) cross-section.Source/drain contacts are ohmic.Gate metal work function issuitably chosen to obtain zeroflat-band potential. Illustrationof the regional approach used incalculating (c) charge densityand (d) surface potential at xi .Position dependent conductionband first subband minima,ESUB(x) is half the nanotubebandgap �1 at VGS = 0 V.Maxima of ESUB(x) occurs atx = x3 = xT op , where thecharge density isQ3,avg = QT op . Also secondand third subbands are used incharge calculations

Another essential issue for dynamic transistor model de-velopment is the channel charge partitioning into source anddrain components. Application of celebrated Ward-Duttonpartitioning [21] provides the actual bias-dependent source(QNum

S ) and drain charges (QNumD ) from QNum

ch (x) as,

QNumS =

∫ L

0

(1 − x

L

)QNum

ch (x)dx, (2)

QNumD =

∫ L

0

x

LQNum

ch (x)dx. (3)

Note that (2) and (3) result from the first moment of con-tinuity equation [22] irrespective of any transport model.Figure 2 also compares QUni

S (= 0.55QUniT ot ) and QUni

D

(= 0.45QUniT ot ) with QNum

S and QNumD , respectively. Al-

though, the model fit is acceptable at gate-to-source volt-ages (VGS ) between 0.3 V to 0.6 V, significant mismatchis observed at low VGS . Moreover, if we assume equiparti-tioning of charge, i.e., QUni

S = QUniD = 0.5QUni

T ot [19], themodeling results would be worse. Also the opposite trendof bias-dependent QNum

S and QNumD cannot be captured by

partitioning QUniT ot in any way. Effects on the resulting trans-

capacitances are even worse (see Figs. 7 and 8). Therefore,the simple ballistic model that calculates charge density atxT op to find the drain current, is insufficient to model thetotal channel charge, its source-drain components, and re-sulting trans-capacitances. To accurately predict the a.c. be-havior of CNFET, we, therefore, need a better charge model.This paper addresses a simple regional approach on chargemodeling. Section 2 describes the model followed by its de-tailed verification with numerically simulated data in Sect. 3.Finally we conclude in Sect. 4.

Fig. 2 VGS dependent total channel charge (QT ot ), source charge(QS ) and drain charge (QD) at VDS = 0.4 V: comparison of uniformcharge density based model (QUni

T ot , QUniS , QUni

D ) with numerical data(QNum

T ot , QNumS , QNum

D ). Subthreshold bias region showing maximummismatch is marked

2 Regional approach to charge modeling

The QT op , computed at xT op , signifies only the amount ofcharge forming the drain current of a ballistic FET [15, 17].However, more charge can reside at points other than at xT op

in the channel influencing the overall dynamic behavior ofthe transistor. Although QT op does not represent an averagecharge density for the total channel length, as observed fromthe model inaccuracies in Fig. 2, it can certainly be adopted

168 J Comput Electron (2012) 11:166–171

as an average for a significant portion of the channel aroundxT op . Therefore, to estimate the total channel charge, we ad-ditionally need to determine the charges within the parts ofthe channel near source and drain extension regions. Notethat subthreshold charge resides within a very small regionof the channel interfacing with the highly doped source ex-tension region. In our regional approach, we, therefore, di-vide the total channel into four regions as shown in Fig. 1(c).Within each (ith) region one representative point (xi ) is cho-sen to figure out the average charge density (Qi,avg). Notethat Q3,avg in Fig. 1(c) amounts to QT op . For the investi-gated carbon nanotube with 2 nm diameter, first three en-ergy subbands are considered both in numerical charge cal-culation and our model. For the investigated CNFET, weobserved that there is no drain induced barrier lowering ef-fect and gate has complete control over the calculation ofsurface potential at x = xT op , i.e., no electrostatic capaci-tance is needed from channel to source/drain leads to cal-culate the surface potential, ψS,3 (Fig. 1(d)). In this work,the same approach is extended for other three points, i.e.,at x = x1, x2, x4. The corresponding self-consistent iterativecalculations of Qi,avg and surface potential ψS,i (at xi ) areillustrated in Figs. 1(c) and 1(d), respectively, and the modelequations (with zero flat-band potential) read

Qi,avg = αiF (ψS,i ,μS) + (1 − αi)F (ψS,i ,μD) (4)

and

ψS,i = VGS − Qi,avg − Qi,eq

Cox

(5)

where F(ψS,i ,μ) is computed for each subband (p =1,2,3) and summed up as,

F(ψS,i ,μ)

= �p

∫ ∞

0

qD0

1 + exp(

√z2+ri�

2p−ψS,i−μ

kT)

dz (6)

with q as the electron charge, k the Boltzmann constant,T the temperature in Kelvin, D0 the metallic density ofstates (a function of carbon–carbon (C–C) bonding energyand C–C bonding distance), and Cox the oxide capacitanceper unit length [20]. Due to interface with highly dopedsource extension region, local density of states at x1 and x2

(Fig. 1(c)) are increased compared to those in other regions,i.e., at x3 (xT op), or x4 [17, 23]. To model this effect, thereduction factor ri is used in (6) by which the �p is reducedfrom their conventional values [20] allowing more charge toenter from source extension region. Note that within a re-gion, the value of ri is kept same for all the subbands.

Following [11], the Qi,avg is partly in equilibrium withsource and partly with drain electrochemical potentials, μS

and μD (in eV), respectively, signifying where from the re-spective part of the charge is contributed. Allowing sucha share of equilibrium leads through a proper choice of αi

Table 1 Parameter values for regional charge based model

Region (i) Li (nm) αi βi ri Qi,eq (qC/nm)

1 1 1 0.8 0.23 4 × 106

2 3 0.58 0.51 0.85 0

3 5 0.5 0.48 1 0

4 6 0.1 0.4 1 0

(from 0 to 1) in (4) and using suitable value for equilibriumcharge density, Qi,eq , in (5), we indirectly fix xi , where theQi,avg is computed. Multiplying Qi,avg by the respective re-gion length, Li , yields the total charge of that (ith) region,i.e., QLi = Qi,avgLi . Total channel charge is obtained by

simple summation, QRegT ot = ∑

i QLi . The partitioning of theregional charge QLi into source and drain parts is done us-ing the parameter βi (ranging from 0 to 1) in

QS,i = βiQLi (7)

and

QD,i = (1 − βi)QLi (8)

leading to total source and drain charge as QRegS = ∑

i QS,i

and QRegD = ∑

i QD,i , respectively. Note that for both dopedand undoped channel CNFET, charge modeling equationsfrom (4) to (8) remain unaltered; however, the model param-eters can assume different values, since the barrier heightsand local density of states at the source-channel (and drain-channel) interfaces can be different for doped and undopedchannel CNFETs.

3 Results and discussions

Table 1 shows the optimized parameter values used to vali-date our model. These model parameter values are specificto the investigated CNFET shown in Fig. 1.

3.1 Charge modeling

Following the modeling strategies of Sect. 2 and using theparameters of Table 1, Q

RegT ot , Q

RegS , and Q

RegD are obtained.

Figure 3 (Fig. 4) shows the VGS dependent total channel(source) charge components of four regions, QL1 (QS,1),QL2 (QS,2), QL3 (QS,3), and QL4 (QS,4) along with a com-parison of Q

RegT ot and QUni

T ot (QRegS and QUni

S ) with numer-ical data QNum

T ot (QNumS ). From Fig. 3, it is observed that

the charge component QL1 within L1 is essential to modelthe total channel charge in subthreshold operation. This re-gional charge is in equilibrium only with μS (α1 = 1 in (4))and is mostly reflected by the source-channel barrier. There-fore, QL1 is completely contributed by the source terminaland a significant part of it is assigned to the source charge,

J Comput Electron (2012) 11:166–171 169

Fig. 3 VGS dependent total channel charge at VDS = 0.4 V: (a) com-parison of regional charge based model (QReg

T ot ) and uniform chargedensity based model (QUni

T ot ) with numerical data (QNumT ot ). Effects of

individual regional charge components (QL1, QL2, QL3, QL4) are alsodisplayed

Fig. 4 VGS dependent source charge at VDS = 0.4 V: comparison ofregional charge based model (QReg

S ) and uniform charge density basedmodel (QUni

S ) with numerical data (QNumS ). Effects of individual re-

gional charge components (QS,1, QS,2, QS,3, QS,4) are also plotted

QS,1 with β1 = 0.8 in (7) and (8) (Fig. 4). The equilibriumcharge density Q1,eq accurately sets the non-zero charge atVGS = 0 V. The reduction factor r1 = 0.23 and regionallength L1 = 1 nm are used to obtain the correct slope andmagnitude of VGS dependent total channel charge in sub-threshold operation.

The regional charge QL2 (QS,2) within L2 representsa significant part of the total channel (source) charge thatis mostly contributed from the source side with α2 = 0.58.The reduction factor r2 = 0.85 reflects the increase in localdensity of states in this region. For higher VGS , this part of

the channel gets significantly charge-populated. Part of thecharge is reflected by the source-channel barrier (β2 = 0.51)and part of it is transmitted towards drain side (1 − β2 =0.49). The charge component QL3 within L3 is calculatedfrom the charge density, QT op at xT op (i.e., x3). The posi-tive (negative) velocity charge carriers within this region arein equilibrium with μS (μD), hence, α3 = 0.5. Part of thecharge is being transmitted from source to drain (1 − β3 =0.52) and part of it from drain to source (β3 = 0.48) form-ing the respective drain (QD,3) and source charge (QS,3)components. Although the charges in L2 and L3 appear tohave similar bias-dependent nature (Figs. 3 and 4), impor-tance of QL2 (and subsequently QS,2) lies in its physicalcontent. A closer view in Fig. 3 (Fig. 4) reveals that bias-dependent trends of QL2 and QL3 (QS,2 and QS,3) changewithin VGS = 0.4 V to 0.6 V, which significantly impactthe capacitance models. The regional charge QL4 withinL4 is effective at high VGS and low VDS , since the source-sided electrons that easily overcome the source-channel bar-rier (due to high VGS ), are neither swept away by the drain(due to low VDS ) nor go back to the source (due to reflec-tion at drain-channel barrier) leading to charge accumula-tion therein. Figure 5 demonstrates the VDS dependent draincharge components of four regions along with a comparisonof Q

RegD and QUni

D with numerical data QNumD . The relative

importance of QD,4 (and of QL4) is better observed in thelinear region of Fig. 5. The total charge within this regionis mostly (slightly) in equilibrium with μD (μS ) leading toα4 = 0.1. Major part of QL4 is accumulated in the drainside of the channel with 1 − β4 = 0.6. Although we couldsubdivide the L4 into two sections as we did in the sourceside of the channel, it is observed from the model agree-ment with numerical data in Figs. 3, 4 and 5 that four re-gions are optimal in modeling the bias-dependent total chan-nel charge and its partitioning into source and drain compo-nents.

Figure 6 compares the VDS dependent QRegS (QUni

S ) and

QRegD (QUni

D ) with the reference data at threshold (VGS =0.2 V) and above threshold (VGS = 0.4 V, 0.6 V) opera-tions. The uniform charge density based models, QUni

S andQUni

D , can hardly be distinguished at low VGS (0.2 V) anddo not follow the trend of QNum

S and QNumD in the linear

region (low VDS ) of higher VGS (0.4 V, 0.6 V) curves. Com-paratively the accuracy of the regional charge based modelsis found to be quite high at all bias regions. However, thesource-drain symmetry is not very accurately preserved atVDS = 0 V for low VGS (= 0.2 V) curve, which may be re-stored using further optimized parameters. Note that we areyet to figure out suitable parameter extraction methodolo-gies.

170 J Comput Electron (2012) 11:166–171

Fig. 5 VDS dependent drain charge at VGS = 0.6 V: comparison ofregional charge based model (QReg

D ) and uniform charge density basedmodel (QUni

D ) with numerical data (QNumD ). Effects of individual re-

gional charge components (QD,1, QD,2, QD,3, QD,4) are also plotted

Fig. 6 VDS dependent source and drain charges at VGS = 0.2 V, 0.4 Vand 0.6 V: comparison of regional charge based model (QReg

S , QRegD )

and uniform charge density based model (QUniS , QUni

D ) with numericaldata (QNum

S , QNumD )

3.2 Capacitance modeling

From gate, source and drain charges, the trans-capacitancesare derived [18] as,

Cm,n = −∂Qm

∂Vn

(9)

for numerical data (Num), uniform charge density based(Uni with unequal charge partitioning as in Fig. 2), andregional charge based (Reg) models with m,n = G,S,D.Note that if m = n (self-capacitances), the minus sign in (9)is omitted. The gate charge is obtained from total channelcharge as QG = −QT ot .

It is observed from Fig. 7 that our regional charge basedmodels C

RegGG , C

RegDG and C

RegSG are significantly better than

Fig. 7 VGS dependence of self- and trans-capacitances CGG, CSG andCDG at VDS = 0.4 V: comparison of uniform charge density basedmodel (CUni

mn ) and regional charge based model (CRegmn ) with numerical

data (CNummn )

Fig. 8 VDS dependent self- and trans-capacitances CDD , CSD andCGD at VGS = 0.6 V: comparison of uniform charge density basedmodel (CUni

mn ) and regional charge based model (CRegmn ) with numerical

data (CNummn )

the uniform charge density based models CUniGG , CUni

DG , andCUni

SG , respectively. Unlike our models, the latter ones inac-curately predict the numerical data in the sub-threshold op-eration and yield high overall model error. Figure 8 showsthat our models C

RegDD , C

RegSD and C

RegGD closely follow the

trend of bias-dependent behavior of CNumDD , CNum

SD , andCNum

GD , respectively. On the other hand, the uniform chargedensity based capacitance models show comparatively pooraccuracy with numerical data. Due to higher accuracy, theregional charge based trans-capacitance models would bevery much useful in RF circuit design, where input and out-put impedance matching are of primary importance [24].However, as the models depend on iterative calculations of

J Comput Electron (2012) 11:166–171 171

regional charge densities, their immediate use in circuit sim-ulation is limited.

4 Conclusions

Since the uniform charge density based model that uses thecharge density at xT op , is not capable to predict the bias-dependent total channel charge of CNFET, a regional ap-proach is proposed to model the same. Significances of bias-dependent individual regional charge components to modelthe total channel charge as well as its partitioning into sourceand drain components are demonstrated. Our modeling re-sults for both charges and trans-capacitances are found to besignificantly better than the uniform charge density basedmodel. Although this regional charge based model, due toits repeated calculations of average charge densities at fourseparate regions, may not readily be used as compact modelfor circuit simulation, it provides significant insight in de-veloping the same.

References

1. McEuen, P.L., Fuhrer, M.S., Park, H.: Single-walled carbon na-notube electronics. IEEE Trans. Nanotechnol. 1, 78–85 (2002)

2. Wind, S., Appenzeller, J., Martel, R., Derycke, V., Avouris, Ph.:Vertical scaling of carbon nanotube field-effect transistors usingtop gate electrodes. Appl. Phys. Lett. 80, 3817–3819 (2002)

3. Javey, A., Kim, H., Brink, M., Wang, Q., Ural, A., Guo, J., McIn-tyre, P., McEuen, P., Lundstrom, M., Dai, H.: High dielectrics foradvanced carbon nanotube transistors and logic. Nature Materials(2002)

4. Appenzeller, J., Knoch, J., Derycke, V., Martel, R., Wind, S.,Avouris, Ph.: Field modulated carrier transport in carbon nanotubetransistors. Phys. Rev. Lett. 89(12), 126801 (2002)

5. Avouris, Ph., Appenzeller, J., Martel, R., Wind, J.S.: Carbon na-notube electronics. Proc. IEEE 91(11), 1772–1784 (2003)

6. Javey, A., Guo, J., Wang, Q., Lundstrom, M.: Ballistic carbon na-notube field effect transistors. Nature 424, 654–657 (2003)

7. Guo, J., Javey, A., Dai, H., Lundstrom, M.: Performance Analy-sis and Design Optimization of Near Ballistic Carbon NanotubeField-Effect Transistors, IEEE IEDM tech. digest, 29.6.1–29.6.4(2004)

8. Koswatta, S.O., Lundstrom, M.S., Anantram, M.P., Nikonov,D.E.: Simulation of phonon-assisted band-to-band tunneling incarbon nanotube field-effect transistors. Appl. Phys. Lett. 87(1–3),253107 (2005)

9. Guo, J., Datta, S., Lundstrom, M.S., Anantram, M.P.: Towardsmultiscale modeling of carbon nanotube transistors. Int. J. Mul-tiscale Computational Engineering, 257-276 (2004)

10. Koswatta, S.O., Nikonov, D.E., Lundstrom, M.S.: ComputationalStudy of Carbon Nanotube p-i-n Tunnel FETs, IEEE IEDM tech.digest, 518 (2005)

11. Rhew, J.-H., Ren, Z., Lundstrom, M.: A numerical study of bal-listic transport in a nanoscale MOSFET. Solid-State Electron. 46,1899–1906 (2002)

12. O’Connor, I., Liu, J., Gaffiot, F., Pregaldiny, F., Lallement, C.,Maneux, C., Gognet, J., Fregonese, S., Zimmer, T., Anghel, L.,Dang, T.-T., Leveugle, R.: CNTFET modeling and reconfigurablelogic-circuit design. IEEE Trans. Circuits Syst. I, Fundam. TheoryAppl. 54(11), 2365–2379 (2007)

13. Hazeghi, A., Krishnamohan, T., Philip Wong, H.-S.: Schottky-Barrier carbon nanotube field-effect transistor modeling. IEEETrans. Electron Devices 54(3), 439–445 (2007)

14. Najari, M., Fregonese, S., Maneux, C., Mnif, H., Zimmer, T., Mas-moudi, N.: Efficient physics-based compact model for the Schot-tky barrier carbon nanotube FET. Phys. Status Solidi C-7 (11–12),2624–2627 (2010)

15. Guo, J., Datta, S., Lundstrom, M.: Assessment of silicon MOSand carbon nanotube FET performance limits using a general the-ory of ballistic transistors, IEEE IEDM tech. digest, 29.3.1–29.3.4(2002)

16. Natori, K.: Ballistic metal-oxide-semiconductor field effect tran-sistor. J. Appl. Phys. 76, 4879–4890 (1994)

17. Rahman, A., Guo, J., Datta, S., Lundstrom, M.: Theory of ballisticnanotransistors. IEEE Trans. Electron Devices 50(9), 1853–1864(2003)

18. Arora, N.: MOSFET Modeling for VLSI Simulation: Theory andPractice. World Scientific, Singapore (2007)

19. Deng, J., Wong, H.-S.P.: A compact SPICE model for carbon-nanotube field-effect transistors including nonidealities and itsapplication. Part I. Model of the intrinsic channel region. IEEETrans. Electron Devices 54(12), 3186–3194 (2007)

20. Raychowdhury, A., Mukhopadhyay, S., Roy, K.: A circuit-compatible model of ballistic carbon nanotube field-effect tran-sistors. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.23(10), 1411–1420 (2004)

21. Oh, S.-Y., Ward, D.E., Dutton, R.W.: Transient analysis ofMOS transistors. IEEE J. Solid-State Circuits SC-15(4), 636–643(1980)

22. McDonald, N.J.: Generalized partitioned charge based bipolartransistor modeling methodology. IEEE Electron Device Lett.24(21), 1302–1304 (1988)

23. Datta, S.: Quantum Transport: Atom to Transistor. CambridgeUniv. Press, Cambridge (2005)

24. Lee, T.-H.: The Design of CMOS Radio-Frequency Integrated Cir-cuits. Cambridge Univ. Press, Cambridge (1998)