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2 Fundamental Physical Aspects of Carbon Nanotube Transistors X Fundamental Physical Aspects of Carbon Nanotube Transistors Zoheir Kordrostami and Mohammad Hossein Sheikhi Department of Electrical Enineerin, Shiraz University Nanotechnoloy Research Institute, Shiraz University Shiraz, Iran 1. Introduction The increasin demand for ultra-hih speed processors, smaller dimensions and lower power consumption of interated circuits has made the technoloy scalin of the electronic components a challenin issue for device desiners. In past few decades, miniaturization of transistors has always obeyed the moore's law: the number of transistors that can be placed inexpensively on an interated circuit has doubled approximately every two years. Nanoscale field effect transistors in the sub-10 nm reime, suffer from short channel effects such as direct tunnelin from source to drain, increase in ate-leakae current and punchthrouh effect. These effects have posed severe problems for miniaturized transistors and directed the recent research toward better alternative semiconductors than silicon. Semiconductin carbon nanotubes (CNTs) because of their properties like lare mean free path, excellent carrier mobility and improved electrostatics at nanoscales as the result of their non-planar structure, have been known as the best ideal replacement for silicon. In particular, they exhibit ballistic transport over lenth scales of several hundred nanometers (Heinze, et al., 2006). Absence of the danlin bond states at the surface of CNTs and purely one-dimensional transport properties improve ate control while meetin ate leakae constrains and allows for a wide choice of ate insulators. That s why CNTs suppress the short channel effects in transistor devices. Symmetry of the conduction and valence bands makes CNTs advantaeous for complementary applications. CNTs are very attractive for nanoelectronic applications and can be used to achieve hih speed ballistic carbon nanotube field effect transistors (CNTFETs). Theoretically, CNTFETs could reach a hiher frequency domain (terahertz reime) than conventional semiconductor technoloies. CNTFETs, only a few years after the initial discovery of CNTs in 1991 by Sumio Iijima (Iijima, 1991) were first demonstrated in 1998 by Dekker et al. at Delft University (Tans, et al., 1998) and soon after by roups at IBM (Martel, et al., 1998) and Stanford University (Soh, et al., 1999). Intensive research has led to a sinificant proress in understandin the fundamental properties of CNTs. By usin a sinle-wall CNT as the channel between two electrodes which work as the source and drain contacts of a FET, a coaxial CNTFET can be fabricated. Coaxial devices are of special interest because their eometry allows for better electrostatics because the ate contact wraps all around the channel (CNT) and has a very ood control on carrier

3 170 Carbon Nanotubes transport. Type of Metal-CNT contacts plays crucial role in the output characteristics of the transistor. Heavily doped semiconductors because of the ability to form Ohmic contacts can be used as ideal electrodes but they suffer from hih parasitic resistance. Existence of potential barrier at the metal-cnt interface, chanes the device to a CNTFET resemblin to Schottky barrier MOSFETs. However, heavily doped CNT contacts can be used to et to a behavior similar to conventional MOSFETs. Understandin CNTFETs from electronic point of view requires a deep insiht for mesoscopic physics. For modellin a CNTFET, a powerful methodoloy with the abilities of solvin Schrödiner equation under non-equilibrium conditions in the presence of selfconsistent electrostatics and treatin couplin of the channel to contacts is needed. The nonequilibrium Green s function (NEGF) formalism provides a sound basis for quantum device simulations. In this chapter we aim to introduce different types of CNTFETs, their electrostatics and output characteristics. The chapter is oranized as follows: section 2 introduces different types of CNTFETs and typical desin parameters, section 3 explains different physical issues involved with CNTFETs and ultimately in section 4 we briefly describe the NEGF formalism and WKB treatment for simulatin CNTFETs. We use these methods to plot the curves for implyin physical aspects throuh the chapter. While explainin the important issues, new challenes for achievin a well-desined CNTFET will be explained. After readin this chapter, the reader should be able to distinuish how a specific type of CNTFETs behaves (both DC analysis and hih frequency response) and why. 2. Common CNTFET Desins 2.1 Types of CNTFETs There are two main types of CNTFETs that are bein currently studied, differin by their current injection methods. CNTFETs can be fabricated with Ohmic or Schottky contacts. The type of the contact determines the dominant mechanism of current transport and device output characteristics. CNTFETs are mainly divided into Schottky barrier CNTFETs (SB- CNTFETs) with metallic electrodes which form Schottky contacts and MOSFET-like CNTFETs with doped CNT electrodes which form Ohmic contacts. In SB-CNTFETs, tunnelin of electrons and holes from the potential barriers at the source and drain junctions constitutes the current. The barrier width is modulated by the application of ate voltae, and thus, the transconductance of the device is dependent on the ate voltae (Raychowdhury, et al., 2006). The other type of the CNTFETs takes advantae of the n-doped CNT as the contact. Potassium doped source and drain reions have been demonstrated and the behavior like MOSFETs have been experimentally verified (Javey, et al., 2005). In this type of transistors a potential barrier is formed at the middle of the channel and modulation of the barrier heiht by the ate voltae controls the current. 2.2 Typical Desins At this point, we consider two typical desins for two types of the CNTFETs and use them for investiation and comparison of the transistors behaviors. Let s consider two coaxial CNTFETs with a (13,0) ziza CNT as the channel which corresponds to a bandap of about 0.83 ev and a diameter of 1 nm. A 2 nm thick ZrO 2 with a relative permittivity of 25

4 Fundamental Physical Aspects of Carbon Nanotube Transistors 171 separates the ate electrode and the CNT. The ate is 10 nm lon and wrapped around the channel. The ate thickness is assumed 6 nm. SB-CNTFET employs an intrinsic CNT and 20 nm lon metallic contacts as the source and drain. A typical structure and the distribution of the potential enery on the channel are shown in fiure 1. The ate voltae makes the barriers near the source/drain thinner and increases the tunnelin current. Enery (ev) (a) S Gate voltae increase Gate Dielectric Dielectric Gate D V G =0.25 V V G =0.8 V Conduction Band V D =0.5 V Valence Band (b) Position (nm) Fi. 1. Schottky Barrier CNTFET, a) 2-D cross section of the coaxial structure with intrinsic CNT as the channel and metal source/drain contacts, b) Enery band diaram obtained from Poisson equation. The metal Fermi level is taken to be at the midap of the CNT. On the other hand the MOSFET-like CNTFET benefits from heavily doped ends of the CNT which act as Ohmic source and drain contacts. A typical structure of the MOSFET-like CNTFETs with 20 nm doped sections and its enery band diaram is shown in fiure 2. The source/drain dopin is 10-9 m -1 (~0.01 dopant per atom). In the followin sections of this chapter these two transistors will be compared to each other and the results can be eneralized to other desins. The curves of this chapter have been obtained usin the described methods in section Important Aspects of CNTFETs 3.1 Ambipolarity One of the important aspects of nanotube transistors is the ambipolarity or unipolarity of their current-voltae characteristics. SB-CNTFETs exhibit stron ambipolar behavior. For hih enouh ate voltaes the tunnelin probability of electrons throuh the source

5 172 Carbon Nanotubes Enery (ev) (a) S (b) Doped Reion Conduction band 0 Valence band Gate Dielectric Dielectric Gate Gate voltae increase Position (nm) V G =0 V V G =0.5 V D Doped Reion V D =0.5 V Fi. 2. MOSFET-like CNTFET, a) 2-D cross section of the coaxial structure with intrinsic CNT as the channel and doped CNT sections as source/drain contacts, b) Enery band diaram obtained from Poisson equation. Schottky barrier in the conduction band is hih and for the low and neative ate voltaes, a Schottky barrier is formed at the drain in valence band and tunnelin of holes increases the current. The enery bands for low and hih ate voltaes and the Schottky barriers are shown in fiure 1(b). However for MOSFET-like CNTFETs only positive ate voltaes because of lowerin the barrier in the channel increase the current. The enery bands for low and hih ate voltaes and the potential barrier in the channel are shown in fiure 2(b). It is clear from the fiures why the characteristics of these two transistors differ. The currentvoltae characteristics of the devices have been compared in fiure 3. Ambipolar behavior of the SB-CNTFETs constraints the use of these transistors in conventional CMOS loic families. We have proposed some methods to reduce or eliminate the ambipolarity in CNTFETs. Usin asymmetrical contact types we have introduced Schottky-Ohmic CNTFET which has a Schottky barrier at the CNT-metal interface at the source and an Ohmic contact at the drain at the channel-doped CNT interface (Kordrostami, et al., 2008). However, the device still suffers from band to band tunnelin. That is, in low or neative voltaes the electrons from the valence band can tunnel to the conduction band and contribute to the total current of the transistor and cause increase of the current in neative voltaes. We have compared the potential enery distribution of the channel of the transistors in fiure 4 (Kordrostami, 2007).

6 Fundamental Physical Aspects of Carbon Nanotube Transistors 173 Current (µa) V D =0.5 V SB-CNTFET MOSFET-like CNTFET Gate Voltae (V) Fi. 3. Comparison between the current-voltae characteristics of the SB and MOSFET-like CNTFETs. Ambipolar behavior of the SB-CNTFET and the unipolar characteristic of the MOSFET-like CNTFET can be seen from the curves. Source K-doped reion E Intrinsic reion MOS-CNTFET K-doped reion (a) Drain SB Metal contact SB Metal contact Schottky barrier E Schottky barrier E Intrinsic CNT Schottky barrier SB-CNTFET Intrinsic reion (b) Drain Metal contact (c) Drain SO-CNTFET K-doped reion Fi. 4. Potential enery distribution alon the carbon nanotube in three different CNTFET structures, a) MOSFET-like, b) Schottky barrier, c) Schottky-Ohmic (Kordrostami, 2007)

7 174 Carbon Nanotubes Double-ate structures can also reduce the ambipolarity while the CNT is still intrinsic. In the structure shown in fiure 5 the first ate controls carrier injection at the source and the second one acts like an electrostatic dopin and controls carrier injection at the drain which can be used to suppress the hole tunnelin current, for example by applyin the same voltae as the drain voltae to the second ate (Pourfath, et al., 2005). In this case, at any drain voltae the band ede profile near the drain contact will be flat like the band diaram in fiure 4(c). On the other hand, an advantae of an ambipolar SB-CNTFETs is that they can be used as either an n-type and p-type FET in a CMOS application (Guo and Lundstrom, 2009). In MOSFET-like CNTFETs with heavily doped source and drain reions, when applyin a neative ate voltae, the band to band tunnelin may lead to ambipolarity. For suppressin this effect in MOSFET-like CNTFETs we proposed a non-uniform dopin profile as shown in fiure 6 (Hassaninia, et al., 2008b). This reduces the radient of the channel potential barrier at each interface between the intrinsic and doped sections of the CNT and suppresses the band to band tunnelin and ambipolar conduction. (b) S 15 nm Gate 1 Gate 1 Carbon Nanotube 30 nm Oxide Oxide 15 nm Gate 2 Gate 2 Fi. 5. Double ate CNTFET (Hassaninia, et al., 2008a). The first ate modulates the source tunnelin current and the second ate in fact eliminates the Schottky barrier at the drain (path for tunnelin of the holes in the valence band). D Fi. 6. a) Cross section of the CNTFET. (b) Step and linear dopin profiles for the MOSFETlike structure versus position.

8 Fundamental Physical Aspects of Carbon Nanotube Transistors Hih Permittivity Gate Dielectrics The relatively low dielectric constant of SiO 2 (at 3.9) limits its use in transistors as ate lenths scale down to tens of nanometers. As the device dimensions approach the 10 nm reime, stron electrostatic couplin of source/drain electrodes arises fundamental challenes especially about the ate control over the channel. Enhancin the ate efficiency requires thinner ate oxides. However, due to excessive direct tunnelin leakae currents throuh the ultra-thin dielectric, the ate dielectric layer is already approachin its limit (~ 1nm) and the only feasible way is to use hih-κ ate dielectrics which afford hih ate capacitance without relyin on ultra-small film thickness (Javey, et al., 2002). The most common ate oxides for nanotransistors are Zirconium 1 (ZrO 2 ) and Hafnium dioxide (HfO 2 ) with relative permittivities about 25 and 16 respectively. The current versus the dielectric constant of the oxide in both types of transistors has been shown in fiure ZrO ZrO 2 6 HfO 2 HfO Al 2 O 3 1 Al 2 O MOSFET-like CNTFET 0.8 SB-CNTFET SiO 2 (a) (b) Dielectric Constant Dielectric Constant SiO 2 Fi. 7. The effect of the dielectric constant on the on current of the transistors in the on state. (V G =V D =0.5 V). Current (µa) Current (µa) 3.3 On/Off Current Comparison shows that MOSFET-like CNTFETs due to absence of Schottky barriers have a lower off leakae current. On the other hand, in the on state, there is no barrier at the sourcechannel junction and hence, the device demonstrates sinificantly hiher on current. The minimum current of the SB-CNTFETs occurs when the contribution of the electron and hole tunnelin currents to the total current becomes equal, that is when V G = V D /2 where the enery band diaram is symmetric. For each power supply voltae (V D ), the off-current is defined at the minimal leakae point V G (off)=v D /2 and the on-current is defined at V G (on)=v G (off)+v D. There exists a trade-off: reducin the off-current by lowerin the power supply voltae derades the on-current. 3.4 Transconductance The transconductance ( m ) of the SB-CNTFETs is severely limited by the Schottky barriers in the on state. The transconductances of the devices as a function of the ate voltae have 1 The name Zirconium oriinates from the Persian word 'zarun' meanin old-like.

9 176 Carbon Nanotubes been compared to each other in fiure 8(a). It is seen that transconductance of the MOSFETlike CNTFET is hiher than the SB-CNTFET. Fiure 8(b) plots CNTFET on state transconductance versus the ate dielectric constant for four types of widely used ate insulators. The hih-κ ate insulator improves the CNTFET performance by reducin the selfconsistent potential produced by the chare on the tube (Guo, 2004). For the SB-CNTFETs the transconductance tends to saturate when the ate insulator dielectric constant is lare. The reason is that the self-consistent potential produced by the chare on the tube is already small and further improvin the ate dielectric constant does not help to sinificantly reduce the Schottky barrier thickness and the transistor performance (Guo, 2004). Transconductance (µs) SB-CNTFET MOSFET-like CNTFET V D =0.5 V Transconductance (µs) SiO 2 (a) (b) Gate Voltae (V) Dielectric Constant Fi. 8. Transconductance of the CNTFETs, a) m versus the ate voltae, b) m variations with respect to the dielectric constant V G =V D =0.5 V HfO 2 Al 2 O 3 SB-CNTFET MOSFET-like CNTFET ZrO Gate Capacitance For a one dimensional nanotube FET, the total capacitance between the ate and the channel (ate capacitance) depends both on the eometry and the density of states (DOS). If the electrostatic potential on the channel (V S ) is uniform, then the voltae drop over the ate oxide is also uniform and the ate voltae V G is the summation of the channel potential and the voltae drop over the ate oxide. This can be modeled as the potential division between two capacitances which means the ate capacitance (total capacitance) can be modeled as the series of the electrostatic (eometry dependent) capacitance and the quantum (DOS dependent) capacitance as shown in fiure 9. Gate capacitance can be calculated from: C = Q (1) V G where Q is the total chare on the ate electrode (which has the same manitude but an opposite sin as the total net chare of the CNT) and V G is the ate voltae. The ate oxide plays the role of the dielectric between the plates of the oxide (electrostatic) capacitance which can be expressed as: C = 2 πεl ox t ox + r Ln r CNT CNT (2)

10 Fundamental Physical Aspects of Carbon Nanotube Transistors 177 where t ox is the ate dielectric thickness, ε is the dielectric constant of the ate insulator, L is the ate lenth and r CNT is the nanotube diameter. Since the chare on the channel (Q) can be derived from CNT density of states and the Fermi level for a iven channel potential, the quantum capacitance can be calculated from: C = Q (3) q V s where V s is the channel potential. For MOSFET-like CNTFETs V s and Q can be defined as the potential enery and chare density at the top of the channel potential barrier. Gate Metal Channel Potential C ox C q Fi. 9. Series combination of the oxide and quantum capacitances and the voltae drop on the capacitances. The above derivation is strictly valid only when V S is position-independent, otherwise the definition of the oxide capacitance breaks down (Guo, et al., 2007). The ate capacitances of the transistors versus the dielectric constant are shown in fiure 10. Gate Capacitance (af) V G =V D =0.5 V V G =V D =0.5 V ZrO HfO Al 2 O HfO 2 ZrO Al 2 O SB-CNTFET MOSFET-like CNTFET SiO (a) SiO 2 (b) Dielectric Constant Dielectric Constant Fi. 10. Gate capacitance versus the dielectric constant a) SB-CNTFET b) MOSFET-like CNTFET. Gate Capacitance (ff) The enery at the top of the barrier in the channel depends on the ate voltae and on the chare at the top of the barrier. When C is small, the ate voltae controls the chare at the top of the barrier (which is independent of the drain voltae) and for lare ox C ox, the ate voltae controls the potential at the top of the barrier, which is independent of the drain

11 178 Carbon Nanotubes voltae (Guo and Lundstrom, 2006). In bulk devices like conventional MOSFETs, the quantum capacitance is larer than the oxide capacitance, however, CNTFETs usually work in the quantum capacitance limit (C q <<C ox ). We have compared the effect of the ate thickness on the ate capacitances of a completely ated and a partially ated SB-CNTFET. The results are shown in fiure 11 and imply that the thicker ate increases C in the partially ated and has no effect in a completely ated SB-CNTFET (Kordrostami and Sheikhi, 2009b) Gate Capacitance (F) C Complete Gate P Partial Gate V G =0.4 V V D =0.4 V Gate Thickness (nm) Fi. 11. Gate capacitances for two types of SB-CNTFET. The channel is an intrinsic 50 nm lon (19,0) ziza CNT, the ate lenth in partially ated SB-CNTFET is 5 nm and the dielectric is SiO 2 with a permittivity of Frinin Capacitance The frinin fields between the ate metal and source and drain contacts result in capacitances which are called frinin or parasitic capacitances. Frinin field becomes more important when the channel lenth of a CNTFET is reduced. This could be comparable to the intrinsic device capacitances, and hence must be considered. In practice, appropriate electrode eometries are required to minimize the frinin field parasitic capacitances so that the parasitic capacitance due to frinin fields become neliible compared to the ate capacitance required to modulate the conductance, for example, by usin 1-D metallic electrodes or nanotubes themselves as the electrodes (Yu, et al., 2006). We have investiated the effect of the contact eometry on the electrostatics of both SB-CNTFETs and MOSFETlike CNTFETs which clearly verifies the contribution of the frinin field to the device electrostatics and frequency response (kordrostami and Sheikhi, 2009a). 3.7 Cutoff Frequency The unity current ain cutoff frequency (the frequency at which the current ain falls to unity) is usually used to describe hih-frequency performance of a transistor. The cutoff frequency of the intrinsic CNTFET is called the intrinsic and the cutoff frequency of the CNTFET with inclusion of the parasitic capacitances is called the extrinsic cutoff frequency. When the parasitic capacitances are small the extrinsic cutoff frequency approaches the

12 Fundamental Physical Aspects of Carbon Nanotube Transistors 179 intrinsic cutoff frequency. We compute the cutoff frequency (f T ) usin the quasi-static approximation. The quasi-static treatment works well when the sinal varies slowly compared to the time constant determined by the intrinsic ate capacitance and the channel inductance (Guo, et al., 2005). The small-sinal circuit model for a CNTFET based on the quasi-static approximation, which includes the equivalent capacitive and resistive elements, but omits the equivalent inductive elements, is shown in fiure 12. Gate Drain C (parasitic) C s C d (parasitic) DC m V s d R d Drain R s Source Fi. 12. Small-sinal circuit model for a nanotube transistor includin intrinsic and extrinsic elements. By usin the circuit model, we can enerally write for the cutoff frequency: 1 2πf T = ( R S + R D ) C d + 1 m ( C + C s + C d ) + d m ( R S + R D )( C + C s + C d ) (4) Where m is the transconductance, intrinsic ate capacitance and R S and R D are the parasitic resistances, C is the C s and C d are the parasitic capacitances (Burke, 2004). Calculation of the parasitic capacitances between the ate and source (drain) electrode requires separate electrostatic computations. If the parasitic resistances are excluded from the calculations we have: f T 1 = 2π C + C m s + C d (5) The cutoff frequency of the intrinsic transistor without parasitic capacitances is: f T = 1 2π C m (6) In CNTFETs the intrinsic cutoff frequency can be determined in terms of the ratio of the chane in the current to the chane in the channel chare: 1 1 V m m G D f T = VD = V ( on) = D VD = V 2π C 2π C VG 2π QCh 1 I D ( on) (7)

13 180 Carbon Nanotubes Where, I D is the source drain current and iven as (Yoon, et al., 2006): Q Ch is the total chare in the CNT channel and is Q Ch = q L Ch 0 N ( x) dx (8) e where N e (x) is the electron density as a function of the channel position and L Ch is the channel lenth. The intrinsic cutoff frequencies of the transistors with respect to the channel lenth have been compared in fiure 13. The shorter the channel the larer the cutoff frequency of the CNTFET. Cutoff Frequency (Hz) 4 x SB-CNTFET V G =V D =0.5 V Cutoff Frequency (Hz) 5 x 109 (a) (b) Gate Lenth (nm) Gate Lenth (nm) Fi. 13. Intrinsic cutoff frequency versus the channel lenth calculated for the typical desins in section 2.2, for a) SB-CNTFET, b) MOSFET-like CNTFET. x MOSFET-like CNTFET V G =V D =0.5 V Cutoff Frequency (Hz) C Complete Gate P Partial Gate V G =0.4 V V D =0.4 V Gate Thickness (nm) Fi. 14. Cutoff frequency of two types of SB-CNTFETs. The channel is an intrinsic 50 nm lon (19,0) ziza CNT, the ate lenth in partially ated SB-CNTFET is 5 nm and the dielectric is SiO 2 with a permittivity of 3.9.

14 Fundamental Physical Aspects of Carbon Nanotube Transistors 181 From the equation 5 it can be concluded that the cutoff frequency is inversely proportional to the frinin capacitances. For example the partially ated SB-CNTFET has a hiher cutoff frequency than the complete ate SB-CNTFETs because the frinin capacitances between metal electrodes are lower in partially ated SB-CNTFETs (Kordrostami and Sheikhi, 2009b). fiure 14 shows the reduction of the cutoff frequency when the ate metal is thicker because of the increase of the frinin capacitances. 3.8 Intrinsic Delay The main limitation to a faster intrinsic delay time is the ate capacitance resultin from frinin fields throuh the hih-κ dielectric directly from the ate to source and ate to drain (Khairul and Roer, 2006). The intrinsic delay is one of the most important performance metrics for diital electronic applications and characterizes how fast a transistor switches. The switchin delay can be calculated from: Where Qon Qoff τ = (9) I On Q on is the total chare of the channel at on state (V G = V D(on) ) and Q off is the total chare of the channel at off state (V G =0, V D = V D(on) ) and I On is the on current (Yoon, et al., 2006). The calculated intrinsic delays of the typical transistors are shown in fiure 15. Intrinsic Delay (ps) SB-CNTFET MOSFET-like CNTFET Gate Lenth (nm) Fi. 15. Intrinsic delays of the SB-CNTFET and the MOSFET-like CNTFET versus the ate lenth. Loner channel lenth leads to larer latency. The on state is defined when V G =V D =0.5 V and the off state is defined as V G =0 V and V D =0.5 V. 4. Physical modellin of the CNTFETs As electronic devices are bein downscaled to nanometer rane, the validity of the conventional modelin approaches becomes questionable. The quantitative simulation methods for nanoscale devices should incorporate an understandin of both atomistic structures and quantum mechanical effects.

15 182 Carbon Nanotubes In the followin sections, we describe two common methodoloies which can simulate the CNTFETs efficiently: NEGF formalism and WKB method for SB-CNTFETs. 4.1 NEGF Formalism The non-equilibrium Green s function (NEGF) formalism provides a sound basis for modelin quantum devices, due to the followin reasons: - Atomic-level description of the channel. - Channel-contact interfaces can be treated by describin open boundary conditions for the Schrödiner equation. - Dissipative scatterin processes and other phenomena like liht emission can be modeled. - Considerin quantum mechanical tunnelin throuh Schottky barriers at the metal/nanotube contacts, tunnelin and reflection at barriers in the nanotube channel and band to band tunnelin. In this section, we ive a brief summary of the NEGF simulation procedure and how to apply the approach to a nanotransistor. The procedure is as follows (Guo and Lundstrom, 2009): - Identify a suitable basis set and Hamiltonian matrix (H) for an isolated channel. - Compute the self-enery matrixes ( ΣS and Σ D ). - Compute the retarded Green s function (G). - Determine the physical quantities of interest from the Green s function. - Solve NEGF transport equation iteratively with the Poisson equation until selfconsistency is achieved. - Calculate the drain current. The procedure starts with an initial uess for the potential of the channel and then the chare density is calculated from the NEGF equations. For a iven chare density, the Poisson equation is solved to obtain the electrostatic potential in the nanotube channel. Next, the computed potential profile is used as the input for the NEGF transport equation, and an improved estimate for the chare density is obtained. The iteration between the Poisson equation and the NEGF transport equation continues until self-consistency is achieved. At this time, all the physical quantities are exact and the current can be calculated. The Green s function can be calculated from: + [( E + i0 ) I H Σ Σ ] 1 G (10) = S D Where H is the Hamiltonian matrix and Σ S, D is the self enery matrixes for the source/drain interfaces. The device and the methodoloy are shown in fiure 16. Table 1 describes the NEGF formalism and how the physical quantities can be calculated. By choosin appropriate self-enery matrixes, the procedure can be implemented to model both types of the CNTFETs. The computationally expensive part of the procedure is calculatin the Green s function. Usin the real space basis set for calculation of the Hamiltonian matrix for the CNT channel leads to a matrix whose size is the total number of carbon atoms in the nanotube. In CNTFETs the mode space basis which uses the periodic boundary condition around the circumference of the nanotube is the appropriate approach

16 Fundamental Physical Aspects of Carbon Nanotube Transistors 183 for calculation of the Hamiltonian of the channel because this approach sinificantly reduces the size of the Hamiltonian matrix. After calculation of the transmission coefficient (Table 1), the current of the transistor can be computed from Landauer-Buttiker formula: 4e I = T E [ fs E fd E ]de h ( ) ( ) ( ) (11) Where fs, D( E) is the Fermi function at the source/drain contact. Poisson Given Chare Density Self-Consistent potential E F Source [Σ S ] Iterate until self-consistency Gate CNT [H] NEGF Given Self-Consistent potential Chare Density E F -V DS [Σ D ] Drain Fi. 16. The iterative method (NEGF) for calculatin the potential and the chare density of the channel and the structure of the CNTFET under investiation. Physical Quantity Broadenin function Spectral function Correlation function Γ Matrix Computation + + S = i [ ΣS ΣS ], ΓD = i [ ΣD ΣD] + [ AS ( E) ] = GΓS G + [ A ( E ] = GΓ G D ) n D [ A S ] f S [ A D ] f D [ G ( E)] = + Description Related to the broadened density of states in the channel. Diaonal elements ive the LDOS at enery E in that representation: 1 D( E) = Trace[ A( E) ] 2π The matrix version of the electron density per unit enery. From [ G n ( E)] density matrix can be calculated. Density 1 Diaonal elements ive the electron [ ρ ] = [ G n ( E)] de Matrix 2π density: n ( x) = Dia[ ρ] Transmission + Transmission probability from source T( E) = Trace[ ΓSGΓ DG ] Coefficient to drain. Table 1. NEGF formalism and the description of physical quantities.

17 184 Carbon Nanotubes 4.2 WKB Treatment of Schottky Barrier CNTFETs The Wentzel-Kramers-Brillouin (WKB) approximation can be used to solve Schrödiner equation and find the tunnelin probability at the source/drain-metal interface. This method is assumed a semi-classical approach. The channel potential distribution can be found by solvin Laplace equation. The potential of the channel found from Laplace equation is valid as far as the device works in the quantum capacitance limit. At this limit, the nanotube quantum capacitance is very small and the associated accumulated chare is close to zero (Jiménez, et al., 2006 ). That s why the Laplace equation is valid under this condition. By usin WKB method and solvin Schrödiner equation, the transmission coefficient of the channel (probability of tunnelin throuh the Schottky Barriers) is achieved which is a function of the potential enery: z T ( E) = exp 2 2 k( z) dz (12) z1 Where z 1 and z 2 are classical turnin points and k(z) is the wave number which can be calculated from: 2 2 E k( z) = [ E + ev( z) ] 2 (13) 3aV 0 2 where a = nm, E = 0.83 ev and V 0 = 2.5 ev are the Carbon Carbon bond lenth, the CNT band ap and the tiht-bindin parameter, respectively. V(z) is the electrostatic potential alon the CNT and is obtained by solvin the Laplace equation. The computed potential profile is used as the input for the transmission coefficient and ultimately the current can be calculated from equation 11 (Kordrostami and Sheikhi, 2009c). The Schottky barrier, potential variations and the classical turnin points are shown in fiure 17. Carbon nanotube Metal Conduction band T(E) V(z) Schottky Barrier Valence band z 2 z 1 Fi. 17. Schottky barrier, potential variations near the channel-contact interface and the classical turnin points which show the tunnelin path.

18 Fundamental Physical Aspects of Carbon Nanotube Transistors Conclusion The current-voltae characteristic of the SB-CNTFETs is ambipolar and the MOSFET-like CNTFETs exhibit unipolar characteristics. Asymmetric contact types and double ate structures are two ways for suppressin the ambipolarity of the SB-CNTFETs. Linear dopin profile in MOSFET-like CNTFETs can reduce the band to band tunnelin leakae current in neative voltaes. Hih permittivity dielectrics are needed to ensure the proper control of the ate on the channel at the oxide thickness limit. The cutoff frequency of the CNTFETs severely depends on the ate and frinin capacitances. 1-D contacts can make the frinin capacitances as small as possible. Partially ated SB-CNTFET has smaller frinin capacitances and thus hiher cutoff frequency in comparison with completely ated SB- CNTFET. Two simulation methodoloies are reliable for modellin CNTFETs: NEGF transport equation self-consistently with Poisson equation for both types of the transistors and semi-classical WKB method for calculatin the tunnelin current throuh Schottky barriers in SB-CNTFETs. By usin the simulation methods we discussed some trade-offs between different parameters of a particular CNTFET desin. In order to achieve a welldesined nanotransistor, a compromise is always needed between different parameters. 6. References Burke, P. J. (2004). AC performance of nanoelectronics: towards a ballistic THz nanotube transistor. Solid-State Electronics, 48, 10-11, , Guo, J. (2004). Carbon Nanotube Electronics: Modelin, Physics, and Applications. Purdue University. Guo, J., Hasan, S., Javey, A., Bosman, G. and Lundstrom, M. (2005). Assessment of hih frequency performance potential of carbon nanotube transistors. IEEE transactions on Nanotechnoloy, 4, 6, , X. Guo, J. and Lundstrom, m. (2006). Nanoscale Transistors: Device Physics, Modelin and Simulation, Spriner, , United States of America. Guo, J. and Lundstrom, M. (2009). Device Simulation of SWNT-FETs, In: Carbon Nanotube Electronic, Javey, A. and Kon J., (Ed.), Spriner, , New York, USA. Guo, J., Yoon, Y. and Ouyan, Y. (2007). Gate Electrostatics and Quantum Capacitance of Graphene Nanoribbons. Nano Letters, 7, 7, , Hassaninia, I., Kordrostami, Z., Sheikhi, M. H. and Ghayour, R. (2008a). Non-Equilibrium Green s Function Calculations for Double Gate Coaxial Carbon Nanotube FETs. Proceedins of Nanostructures. March 2008, Kish Island, Iran. Hassaninia, I., Sheikhi, M. H. and Kordrostami, Z. (2008b). Simulation of carbon nanotube FETs with linear dopin profile near the source and drain contacts. Solid-State Electronics, 52, 6, , Heinze, S., Tersoff, J. and Avouris, P. (2005). Carbon Nanotube Electronics and Optoelectronics, In: Introducin Molecular Electronics, Cuniberti, G. and Richter, K., (Ed.), , Spriner, , Netherlands. Iijima, S. (1991). Helical microtubules of raphitic carbon. Nature, 354, 6348, Javey, A., Kim, H., Brink, M., Wan, Q., Ural, A., Guo, J., McIntyre, P., McEuen, P., Lundstrom, M. and Dai, H. (2002). Hih-[kappa] dielectrics for advanced carbonnanotube transistors and loic ates. Nat Mater, 1, 4, ,

19 186 Carbon Nanotubes Javey, A., Tu, R., Farmer, D. B., Guo, J., Gordon, R. G. and Dai, H. (2005). Hih Performance n-type Carbon Nanotube Field-Effect Transistors with Chemically Doped Contacts. Nano Letters, 5, 2, , Jiménez, D., Cartoixà, X., Miranda, E., Suñé, J., Chaves, A. F. and Roche, S. (2006 ). A drain current model for Schottky-barrier CNT-FETs Computational Electronics, 5, 4, Khairul, A. and Roer, K. L. (2006). Dielectric scalin of a zero-schottky-barrier, 5 nm ate, carbon nanotube transistor with source/drain underlaps. Journal of Applied Physics, 100, 2, Kordrostami, Z. (2007). Circuit Modelin of Carbon Nanotube Transistors and Interconnects for Interated Circuit Applications. Electrical and Electronic Enineerin. Shiraz University. Kordrostami, Z., Hassaninia, I. and Sheikhi, M. H. (2008). Unipolar Schottky-Ohmic carbon nanotube field effect transistor. Proceedins of 3rd IEEE International Conference on Nano/Micro Enineered and Molecular Systems, , , Jan kordrostami, Z. and Sheikhi, M. H. (2009a). Contact Geometry dependent Electrostatics of Carbon Nanotube Transistors, Nanoscience and Technoloy, September 2009, Beijin, China. Kordrostami, Z. and Sheikhi, M. H. (2009b). Hih Speed Switchin Performance of Carbon Nanotube Field Effect Transistors, Nanotech Europe, September 2009, Berlin, Germany. Kordrostami, Z., Sheikhi, M. H. (2009c). Schottky Barrier Field Effect Transistors with a Strained Carbon Nanotube Channel. Journal of Computational and Theoretical Nanoscience, 7, 6, , Martel, R., Schmidt, T., Shea, H. R., Hertel, T. and Avouris, P. (1998). Sinle- and multi-wall carbon nanotube field-effect transistors. Applied Physics Letters, 73, 17, Pourfath, M., Unersboeck, E., Gehrin, A., Kosina, H., Selberherr, S., Park, W.-J. and Cheon, B.-H. (2005). Numerical Analysis of Coaxial Double Gate Schottky Barrier Carbon Nanotube Field Effect Transistors. Journal of Computational Electronics, 4, 1, 75-78, Raychowdhury, A., Keshavarzi, A., Kurtin, J., De, V. and Roy, K. (2006). Carbon Nanotube Field-Effect Transistors for Hih-Performance Diital Circuits; DC Analysis and Modelin Toward Optimum Transistor Structure. IEEE Transactions on Electron Devices, 53, 11, , Soh, H. T., Quate, C. F., Morpuro, A. F., Marcus, C. M., Kon, J. and Dai, H. (1999). Interated nanotube circuits: Controlled rowth and ohmic contactin of sinlewalled carbon nanotubes. Applied Physics Letters, 75, 5, , Tans, S., Verschueren, A. and Dekker, C. (1998). Room-temperature transistor based on a sinle carbon nanotube. Nature 393, 49 52, Yoon, Y., Yijian, O. and Jin, G. (2006). Effect of phonon scatterin on intrinsic delay and cutoff frequency of carbon nanotube FETs., IEEE Transactions on Electron Devices, 53, 10, , Yu, Z., Rutherlen, C. and Burke, P. J. (2006). Microwave nanotube transistor operation at hih bias. Applied Physics Letters, 88, 23, ,

20 Carbon Nanotubes Edited by Jose Mauricio Marulanda ISBN Hard cover, 766 paes Publisher InTech Published online 01, March, 2010 Published in print edition March, 2010 This book has been outlined as follows: A review on the literature and increasin research interests in the field of carbon nanotubes. Fabrication techniques followed by an analysis on the physical properties of carbon nanotubes. The device physics of implemented carbon nanotubes applications alon with proposed models in an effort to describe their behavior in circuits and interconnects. And ultimately, the book pursues a sinificant amount of work in applications of carbon nanotubes in sensors, nanoparticles and nanostructures, and biotechnoloy. Readers of this book should have a stron backround on physical electronics and semiconductor device physics. Philanthropists and readers with stron backround in quantum transport physics and semiconductors materials could definitely benefit from the results presented in the chapters of this book. Especially, those with research interests in the areas of nanoparticles and nanotechnoloy. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the followin: Zoheir Kordrostami and Mohammad Hossein Sheikhi (2010). Fundamental Physical Aspects of Carbon Nanotube Transistors, Carbon Nanotubes, Jose Mauricio Marulanda (Ed.), ISBN: , InTech, Available from: InTech Europe University Campus STeP Ri Slavka Krautzeka 83/A Rijeka, Croatia Phone: +385 (51) Fax: +385 (51) InTech China Unit 405, Office Block, Hotel Equatorial Shanhai No.65, Yan An Road (West), Shanhai, , China Phone: Fax:

21 2010 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial- ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the oriinal is properly cited and derivative works buildin on this content are distributed under the same license.