MOSFET-Like Nanotube Field Effect Model

Mohammad Taghi Ahmadi, Yau Wei Heong, Ismail Saad, Razali Ismail

Faculty of Electrical Engineering Universiti Teknologi Malaysia 81310 Skudai, Johor, Malaysia, [email protected]

ABSTRACT curiosity towards ballistic nature of the carriers is elevated. Initially, it was in the work of Arora[1] that the possibility An analytical model that captures the essence of of ballistic nature of the transport in a very high electric physical processes in a CNTFET’s is presented. The model field for a nondegenerate was indicated. In covers seamlessly the whole range of transport from drift- this article, the work of Arora is extended to embrace diffusion to ballistic. It has been clarified that the intrinsic degenerate domain in the where speed of CNT’s is governed by the transit time of electrons. have analog type classical spectrum only in one direction Although the transit time is more dependent on the while the other two directions are quantum confined or saturation velocity than on the weak-field mobility, the digital in nature. When only the lowest digitized quantum feature of high- mobility is beneficial in the sense state is occupied (quantum limit), a carbon nanotube shows that the drift velocity is maintained always closer to the distinct one-dimensional character. saturation velocity, at least on the drain end of the transistor where electric field is necessarily high and controls the 2 DISTRIBUTION FUNCTION saturation current. The results obtained are applied to the modeling of the current-voltage characteristic of a carbon In one dimensional nanotube with diameter around nanotube field effect transistor. The channel-length nanometer (see Figure. 1), only one of the three Cartesian modulation is shown to arise from drain velocity becoming directions is much larger than the De - Broglie wavelength closer to the ultimate saturation velocity as the drain (Taken is x direction). Since the current is independent of voltage is increased. band structure a MOSFET-like CNTFET in the quantum limit can be described by the same theory for 1 INTRODUCTION semiconductor nanowires . For the Carbon nanotube near the k = 0 band structure is parabolic and the There is an intensive search and research for high-speed energy spectrum is analog-type only in x-direction. devices is an ongoing process. Two important factors that determine the speed of a signal propagating through a = kE 22 G x (1) conducting channel. One is the transit-time or gate delay E +≈ * that depends on the length of the channel and the other is 2 2m the wire delay that is due to finite RC time constants. The two factors are intertwined as in each one the ultimate with saturation of velocity plays a predominant role. The higher mobility brings an electron closer to saturation as high 2 ta ⎛ i −−− )1(36 i ⎞ electric field is encountered, saturation velocity remaining −cc ⎜ ⎟ (2) EG ×≈ ⎜ ⎟ i = 1,2,3,..... the same [Mohammad]. The reduction in conducting d ⎝ 4 ⎠ channel length of the device results in reduced transit-time- delay and hence enhanced operational frequency. There is no clear consensus on the interdependence of saturation Here EG is the band gap for ith sub bands and a −cc is velocity on low-field mobility that is scattering-limited. In the Carbon –Carbon band in the quantum limit of a any solid state device, it is very clear that the band structure nanotube and d is the diameter of the Carbon nanotube, and parameters, doping profiles (degenerate or nondegenerate), t=2.7 (eV) is the nearest neighbor C-C tight binding overlap and ambient temperatures play a variety of roles in energy [5, 6].In the y, z-direction where the length determining performance behavior. The outcome that the L ,zy << λD , the De - Broglie wavelength λD with a higher mobility leads to higher saturation is not supported by experimental observations [1, 2]. This paper focuses on typical value of 10 nm. kx is the momentum wave vector in * the process controlling the ultimate saturation. It has been the x-direction. = .0 05mm o [7]. confirmed in a number of works that the low-field mobility is a function of quantum confinement [3, 4]. In the following, the fundamental processes that limit drift velocity are delineated. As devices are being scaled down in all dimensions, the

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1 G (Ef k ) = G (6) Fk +− qEE ε .A e BTk +1

This distribution has a simple interpretation [8, 9] as given in the tilted band diagram In an extremely large electric field, virtually all the electrons are traveling in the positive x-direction (opposite to the electric field). This is Figure 1. A prototype CNT’s (9, 2) with R << λ and D what is meant by conversion of otherwise completely . L >> λD random motion into a streamlined one with ultimate velocity per electron equal to the intrinsic velocity v . The quantum wave in the x-direction is a traveling wave i and that in the y (z) - direction is the standing wave. Hence the ultimate velocity is ballistic independent of scattering interactions. The ballistic motion in a mean-free path is interrupted by the onset of a quantum emission of j(k .x) ⎛πy ⎞ ψ (x,y,z)= 2 e x sin sin⎛πz ⎞ (3) energy =ω . This quantum may be an optical or a k Ω ⎜ L ⎟ ⎜ L ⎟ 0 ⎝ y ⎠ ⎝ z ⎠ photon or any digital energy difference between the quantized energy levels with or without external stimulation In carbon nanotube near the minimum band energy present. The mean-free path with the emission of a similar to one dimensional nanowire we obtain following quantum of energy is related to A (zero-field mean free equation for density of state (DOS). 0 path) by an expression. 1 * 1 − Δnx 1 EG 2m 2 2 EQ AQ DOOS = (E −= () 2 ) (4) − − ΔEL 2π 2 = qεA A x = AA 1[ − e 0 A 1[] −= e 0 ] (7) 0 0

The distribution function of the energy E is given by the Fermi-Dirac distribution function qεA Q = Q = (NE 0 + )1 =ω0 (8)

1 1 f (Ek ) = N = Ek −EF (5) 0 =ω0 (9) k T e B +1 e BTk −1 where EF is the one-dimensional Fermi energy at which the Here N0 + )1( gives the probability of a quantum probability of occupation is half and T is the ambient emission. N is the Bose-Einstein distribution function temperature. In non- degenerately doped o determining the probability of quantum emission. The the ‘1’ in the denominator of Eq. (5) is neglected as compared to the large exponential factor. In this degraded mean free path A is now smaller than the low- approximation, the distribution is Maxwellian, which is field mean free path A o . ≈ AA 0 in the Ohmic low-field used in determining the transport parameters extensively. regime as expected. In the high electric field This simplification is good for a nondegenerately-doped semiconductor. However, most nanoelectronic devices ≈ AA Q = Q qE ε . The inelastic scattering length during these days are degenerately doped. Hence any design based which a quantum is emitted is given by .Obviously on the Maxwellian distribution is not strictly correct and A = ∞ in zero electric field and will not modify the often leads to errors in our interpretation of the Q experimental data. In the other extreme, for strongly traditional scattering described by mean free path A 0 as degenerate carriers, the probability of occupation is 1 when A Q >> A 0 . The low-field mobility and associated drift Ek < EF and is zero if Ek > EF.Arora modified the equilibrium distribution function of Eq. (4) by replacing EF motion is therefore scattering-limited. The effect of all (the chemical potential) with the electrochemical potential possible scattering interactions is now buried in the mean G G G E + qε .A ε free path A . This by itself may be enough to explain the F where is the applied electric field, q the 0 G degradation of mobility μ in a high electric field; [10] electronic charge and A the mean free path. Arora’s distribution function is given by

575 NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 qτ qA qA Q 2 Tk N μ = c = ≈ B c * * * (10) vi = * × ℑ0 ()ηF (16) m m vi vm i πm n

Here τ c is the mean free time (collision time) during With which the electron motion is ballistic. v i is the mean 1 * 2 intrinsic velocity for 1D carbon nanotube that is discussed ⎛ 2 BTkm ⎞ N = ⎜ ⎟ (17) in the next section. c ⎜ π= 2 ⎟ ⎝ ⎠ 3 INTRINSIC VELOCITY where Nc is the effective density of states for the The intrinsic velocity is given by conduction band with m* now being the density-of-states

effective mass. n1 is the carrier concentration per unit 1 ℑ0 ()η F vi = vth (11) length. Figure 2 indicates the ultimate velocity as a function π ℑ 1 ()ηF of temperature for (9,2) CNT with effective mass − * 2 m =0.099m0 . Also shown is the graph for nondegenerate approximation. The velocity for low carrier concentration With follows T 2/1 behavior independent of carrier concentration. However for high concentration (degenerate carriers) the E E − G velocity depends strongly on concentration and becomes 2 Tk F independent of the temperature. The ultimate saturation v = B , η = 2 (12) th m* F k T velocity is thus the thermal velocity appropriate for 1D B carrier motion:

1 ∞ x j 1 2 Tk ℑ η)( = dx (13) v = = vv = B (18) j ∫ x−η)( iND thd th * Γ( j + 1) 0 e + 1 π πmd

Here, ℑ j η)( is the Fermi-Dirac (FD) integral of order j and j +Γ )1( is a Gamma function. Its value for an integer j is )1( =+Γ jj Γ = jj !)( . For half integer values, it is ⎛ 3 ⎞ 1 ⎛ 1 ⎞ 1 Γ⎜ ⎟ Γ= ⎜ ⎟ = π . In the strongly degenerate ⎝ 2 ⎠ 2 ⎝ 2 ⎠ 2 regime, the FD integral transforms to

1 η j+1 η j+1 ℑ η)( ≈ = (14) j Γ( j + ) j + 11 Γ j + 2)( Figure.2. Velocity versus temperature for carbon nanotube for various concentrations. The carrier concentration per unit length n1 is given by

Figure 3 shows the graph of ultimate intrinsic velocity n Nc ℑ= 1 (η F ) (15) − as a function of carrier concentration for three temperatures 2 (T= 4.2 K, 77 K, and 300 K). As expected, at a low temperature, carriers follow the degenerate statistics and For quasi-one-dimensional carbon nanotubes (CNT) or hence their velocity is limited by an appropriate average of nanowires, the ultimate average velocity per electron is a the Fermi velocity that is a function of carrier function of temperature and doping concentration concentration. When degenerate expression for the Fermi energy as a function of carrier concentration is utilized, the ultimate saturation velocity is given by

NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 576 = With Eq. (20) the drain current I as a function of the v = (n π ) (Degenerate) (19) D iD 4m* gate voltage VGS and drain voltage VD is obtained as The ultimate velocity in a quasi-one-dimensional carbon nanotube (CNT) or nanowire may become lower when [2 −VVV 2 ] quantum emission is considered. The inclusion of the I = β GT D D (21) D V quantum or optical phonon or any other similar emission 1+ D will change the temperature dependence of the saturation Vc velocity. with μ C β = lf G (22) 2L V = −VV GT GS T (23) v sat (24) Vc = L μAf

where CG is the gate capacitance per unit length, and L the

effective channel length. VT is the threshold voltage . The value of μ = 60000 cm2 /V.s is taken. v = v is the Figuure 3 Velocity versus doping concentration for T=4.2 K Af sat i1 (liquid helium), T = 77K (liquid nitrogen) and T=300 K intrinsic velocity of Eq. (19). In the absence of quantum (room temperature). In (9,2) semiconducting CNT. emission for (9, 2) semiconductor carbon nanotube the 13− 3 intrinsic carrier concentration is n1 =×7921. 10 m High mobility does not influence the saturation velocity [11,12]. Because of the unknown nature of the quantum of the carriers, although it may accelerate the approach emission, it is ignored in this calculation. With the simple towards saturation as high fields are encountered in nano- geometry of the carbon nanotube transistor of Figure 4, the length channels. gate capacitance is given by CG [13].For single wall carbon nanotube (SWCNT) channel in CNTFETs, the gate-channel 4 CARBON NANOTUBE FIELD EFFECT capacitance can be expressed as CG=(Ce CQ)/( Ce CQ ), TRANSISTOR where Ce is the electrostatic gate coupling capacitance of the gate oxide and CQ is the quantum capacitance of the gated SWCNT. The quantum capacitance CQ arises from the fact that the electron wave function vanishes at the carbon nanotube (CNT)-insulator interface [14]. For a carbon nanotube of length, L and radius, r capacitance comes out to be [15]. 2πε Ce ≈ L [25] ⎡ rt ++ 2 + 2 rtt ⎤ ln⎢ 0x ox ox ⎥ r ⎣⎢ ⎦⎥ Figure 4 The schematic of a carbon nanotube transistor where ε, t , and r are the dielectric constant and thickness with gate dielectric. 0x of oxide, and the radius of SWCNTs, respectively [16, The velocity saturation effect can conveniently be 17,18]. Using ε = 20ε0 , t0x = 80 nm, L = 250 nm and implemented in the modeling of a carbon nanotube r = 1 .7 nm. which for typical experimental setups is in the transistor if the empirical relation given below is used. order of tens of aF/μm. To add to this, the quantum capacitance in carbon nanotubes is of considerable significance. The quantum capacitance can be write as [19] μ0 E 6 vD = (20) E CQ = 2e vF where, vF ≈10 m s is the Fermi velocity 1+ of electrons in the CNT [20]. Numerically C comes out to Ec Q be 76.5 aF/μm shows thereby that both electrostatic as well

577 NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 as the quantum capacitances play important role in the 5 CONCLUSIONS CNTs. [21,22]. At the onset of current saturation, all Intrinsic speed of carriers in CNT’s is governed by the carriers leave the channel at the saturation velocity where transit time of electrons. Although the transit time is more electric field is extremely high. Therefore, the saturation dependent on the saturation velocity than on the weak-field current is given by mobility, the feature of high- is beneficial

in the sense that the drift velocity is maintained always I Dssat = G (VC GT − Dsat )vV sat (26) closer to the saturation velocity, at least on the drain end of the transistor where electric field is necessarily high and When Eqs. (21) and (26) at the onset of current controls the saturation current. saturation are consolidated, the drain voltage at which the In many ways, the distribution function reported is current saturates is given by. similar to what is presented by Buttiker [24]. A series of ballistic channels of length A comprise a macro-channel of ⎡ ⎤ length L. The behavior is well understandable as we 2VGT VDssat = Vc ⎢ 1+ −1⎥ (27) consider ballistic low-field mobility where A is replaced by ⎣⎢ Vc ⎦⎥ L. However, it does not affect the velocity saturation in high electric field that is always ballistic. The ends of each mean free path thus can be considered Buttiker’s With this value of V substituted in Eq. (26), the Dsat thermalizing virtual probes, which can be used to describe saturation current is given by transport in any regime. In high electric field, the electrons are in a coordinated relay race, each electron passing its 2 velocity to the next electron at each virtual probe. The I Dssat = βVDsat (28) saturation velocity is thus always ballistic whether or not

device length is smaller or larger than the mean free path. Figure 5 shows the current-voltage characteristics of a The ballistic saturation velocity is always independent of transistor with L = 250 nm. In this regime, VGT > Vc . The scattering-limited low-field mobility that may be degraded steps in I-V characteristics tend to be equally spaced as by the gate electric field. The relation between mobility V becomes much larger thanV . An analysis of the and the mean free path has deep consequences on the GT c understanding of the transport in a nanoscale device. velocity distribution in the channel indicates that the The importance of developing FET structures with high velocity throughout the length of the channel is lower than electron concentrations and high current-drive capabilities that on the drain end where carriers are moving with the is emphasized. The current drive can be enhanced by an maximum permissible velocity. The equally-spaced step array of CNT’s particularly for high speed behavior of I-V characteristics is in direct contrast to the digital applications. The switching speed is usually behavior expected from long channel transistors ( determined by the charging process of capacitive loads, VGT << Vc ), when steps separation varies quadratic ally including interconnects. The presence of a tradeoff relation with the adjusted gate voltageV . In fact, if Eqs. (21) is between high mobility and high doping concentration is GT required to optimize performance of the CNT’s transistor expanded to second order in VGT , one can easily notice the following the recommendations of Sakaki [25]. The complete absence of pinch off effect[23]. channels can be designed for the benefit of utilizing selectively doped double-heterojunctions and other FET nanostructures. The role of the quantum engineering by the gate-field induced mobility degradation in determining low- field electron mobility is indicated. Also, the role of quantum waves in determining the correct capacitance to make quantitative interpretations of the gate capacitance is required. Although the technical difficulty encountered in the preparation of a CNT transistor is far greater than that for 2D nanostructures, there appear to be several bright prospects if one develops novel schemes of microfabrications, including the use of edge of quantum- well nanostructures as the theory guides the experimenters toward the design of CNT transistors and allows them to Figure .5. Current -Voltage characteristics of nanowire. assess performance accordingly.

NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 578 ACKNOWLEDGEMENT Transistors With Randomly Doped Reservoirs Iannaccone2006 IEEE The authors would like to thank Malaysian Ministry of [12] Jose M. Marulanda, Ashok Srivastava1 Carrier Science, Technology and Industry (MOSTI) for a research Density and Effective Mass Calculations for Carbon grant for support of postgraduate students. The work is Nanotubes 2007 IEEE supported by UTM Research Management Center (RMC). [13] S. Ilani*, L. A. K. Donev, M. Kindermann and P. L. Mceuen* Measurement of the Quantum Capacitance of Interacting Electrons in Carbon REFERENCES Nanotubes Nature Physics Vol 2 October 2006 [14]Changxin Chen , Zhongyu Hou, Xuan Liu, Eric Siu- [1] V. K. Arora, “High-Field Distribution And Mobility Wai Kong, Jiangping Miao, Yafei Zhang In Semiconductors,” Japanese Journal Of Applied Fabrication And Characterization Of The Physics ,Jpn. J. Appl. Phys 24, P. 537, 1985. Performance Of Multi-Channel Carbon-Nanotube [2] Hui Houg Lau, M.Taghi Ahmadi*, Razali Ismail, Field-Effect Transistors 0.1016/J. Physleta. 2007 ,Vijay K. Arora “Current-Voltage Characteristics [15] Jose M. Maruland, Ashok Srivastavand Siva Of A Nanowire Transistor” Proceedings Of Yellampalli,” Numerical Modeling of the I-V The Wra-Ldsd 7-9 April (2008) Nottingham Characteristic ofCarbon Nanotube Field Effect [3] Vijay K. Arora, “Quantum Well Wires: Electrical Transistors” 40th Southeastern Symposium on And Optical Properties,” J.Phys. C:, Vol.18, Pp. System Theory, March 16-18, 2008 3011-3016, 1985. [16] J.Q. Li, Q. Zhang, D.J. Yang, J.Z. Tian, Carbon 42 [4] Vijay K. Arora, Michael L. P. Tan, Ismail Saad, (2004) 2263. And Razali Ismail, “Ballistic Quantum Transport In [17] A. Javey, H. Kim, M. Brink, Q. Wang, A. Ural, J. A Nanoscale Metal-Oxide-Semiconductor Field Guo, P. Mcintyre,P. Mceuen, M. Lundstrom, H. Effect Transistor,” Applied Physics. Letters, Vol. Dai, Nature Mater. 1 (2002) 241. 91, Pp. 103510-103513, 2007. [18] R. Martel, T. Schmidt, H.R. Shea, T. Hertel, P. [5] Vijay K. Arora, “Failure Of Ohm's Law: Its Avouris, Appl. Phys.Lett. 73 (1998) 2447 Implications On The Design Of Nanoelectronic [19] P.J. Burke. An Rf Circuit Model For Carbon Devices And Circuits,” Proceedings Of The Ieee Nanotubes. Ieee Trans , 2(1):55– International Conference On Microelectronics, May 58, March 2003. 14-17, 2006, Belgrade, Serbia And Montenegro, Pp. [20]Y.-M. Lin, J. Appenzeller , Z. Chen , Z.-G. Chen, 17-24. H.-M. Cheng, and Ph. Avouris,” Demonstration of a [6] R. A. Jishi, D. Inomata, K. Nakao, M. S. High Performance 40-nm-gate Carbon Nanotube Dresselhaus, And G. Dresselhaus, “Electronic Field-Effect Transistor” 0-7803-9040-2005 IEEE And Lattice Properties Of Carbon Nanotubes,” [21]Sandeep K. Shukla, R. Iris Bahar,” Nano, Quantum J.Phys.Soc.Jap., Vol. 63, No. 6, Pp. 2252–2260, And Molecular Computing “Kluwer Academic 1994. Publishers,2004 [7] A. M. T. Fairus And V. K. Arora, “Quantum [22] J. W. G. Wild¨Oer, L. C. Venema, A. G.Rinzler, R. Engineering Of Nanoelectronic Devices: The Role E. Smalley, And C. Dekker, Electronic Structure Of Of Quantum Confinement Onmobility Atomically Resolved Carbon Nanotubes,” Nature Degradation,” Microelectronics Journal, Vol. 32, (London), Vol. 391, No. 6662, Pp. 59–62, 1998. Pp. 679-686, 2000. [23]H. Sakaki, Scattering Suppression And High- [8] Mark S. Lundestorm, And, Jing Guo Nanoscale Mobility Effect Of Size Quantized Electrons In Transistors: Device Physics, Modeling and Ultrafine Semiconductor Wire Structures,” Jpn. J. Simulation. Spriger, New York, 2006. Appl. Phys., Vol. 19, Pp. L735-L738, 1980. [9] V. K. Arora, “Quantum Enginering Of [24] M. Buttiker, “Role Of Quantum Coherence In Nanoelectronic Devices: The Role Of Quantum Series Resistors,” Phys. Rev. B, Condens. Matter, Emission In Limiting Drift Velocity And Diffusion No. 33, Pp. 3020–3026, 1986. Coefficient,” Microelectronics Journal, Vol. 31, [25] H. Sakaki, “Velocity Modulation Transistor (Vmt)- No.11-12, Pp 853-859. 2000. A New Field-Effecttransistorconcept,” Jpn. J. Appl. [10] H. Ishii, N. Kobayashi, K. Hirose, Contact And Phys., Vol. 21, Pp. L381-L383, 1982. Phonon Scattering Effects On Quantum Transport Properties Of Carbon-Nanotube Field-Effect Transistors, Applied Surface Science (2007), Doi:10.1016/J.Apsusc.2008.01.116 [11]Gianluca Fiori,Giuseppe ,Threshold Voltage Dispersion And Impurity Scattering Limited Mobility In Carbon Nanotube Field Effect

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