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A Three-Dimensional Simulation Study of the Performance of Carbon

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A Three-Dimensional Simulation Study of the Performance of Carbon Università di Pisa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three-dimensional simulation study of the performance of carbon nanotube field-effect transistors with doped reservoirs and realistic geometry,ƞ IEEE Transactions on Electron Devices, 53 , 8, pp.1782-1788 (2006). # # 1782 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 53, NO. 8, AUGUST 2006 A Three-Dimensional Simulation Study of the Performance of Carbon Nanotube Field-Effect Transistors With Doped Reservoirs and Realistic Geometry Gianluca Fiori, Giuseppe Iannaccone, Member, IEEE, and Gerhard Klimeck, Senior Member, IEEE Abstract—This paper simulates the expected device perfor- induced barrier lowering (DIBL), it has been demonstrated that mance and scaling perspectives of carbon nanotube (CNT) field- the downscaling of device dimensions has to follow particular effect transistors with doped source and drain extensions. The rules, like maintaining the ratio between the channel length and simulations are based on the self-consistent solution of the three- dimensional Poisson–Schrödinger equation with open boundary the oxide thickness larger than 18 [5]. conditions, within the nonequilibrium Green’s function formal- To alleviate these problems, different solutions have been ism, where arbitrary gate geometry and device architecture can be proposed to achieve channel modulation of the barrier. considered. The investigation of short channel effects for different Javey et al. [6] and Nosho et al. [7] have respectively shown, gate configurations and geometry parameters shows that double- for example, that ohmic contacts can be obtained by choosing gate devices offer quasi-ideal subthreshold slope and drain- induced barrier lowering without extremely thin gate dielectrics. Pd or Ca as the metal for the contacts. Inducing charge in the Exploration of devices with parallel CNTs shows that ON currents source and drain regions is another possible solution, reported, per unit width can be significantly larger than the silicon counter- for example, in [8] and [9]. part, while high-frequency performance is very promising. Channel modulation has also been obtained [10], [11] by Index Terms—Ballistic transport, carbon nanotubes (CNTs), definition of multiple gates able to fix independent potentials nonequilibrium Green’s function (NEGF), technology computer- both in the reservoirs and in the channel as well as transparent aided design (CAD). Schottky barrier at the contacts. In a scenario where many geometries are feasible, adequate I. INTRODUCTION physical models and simulation tools are necessary not only to provide explanations to experimental results but also to ARBON nanotubes (CNTs) represent a promising alter- define device guidelines for the fabrication of CNT FETs, with native to conventional silicon technology [1] for future C performance benchmarked against their mainstream silicon nanoelectronics at the end of the International Technology counterpart. Roadmap for Semiconductors (ITRS) [2]. Since the first work Gate-all-around CNT FETs have been studied [12], [13], on the topic by Iijima [3], significant improvements have been where the three-dimensional (3-D) Poisson equation has been achieved from the point of view of both technology and physi- reduced to two dimensions because of the cylindrical symmetry cal modeling. In particular, Heinze et al. [4] have demonstrated of the electrostatic potential, and transport has been computed Schottky barrier CNT-FETs, where the modulation of the cur- through the Landauer formalism. Such a coaxial geometry has rent is mainly determined by the field-induced modulation of also been adopted in [14], where the Poisson equation has been the nanotube band structure at the CNT ends. Such a working coupled with the nonequilibrium Green’s function (NEGF) principle, however, strongly limits the device performance. The formalism using the uncoupled mode space approach, which ambipolar behavior, the poor control of the channel, and the enables the computation of transport for a small number of possible degradation of electrical properties like I /I ratio on off electron subbands with a small computational cost. deeply affect such kind of devices, especially nanotubes with However, planar gate structures are more attractive because large diameters, as shown in [5]. To obtain acceptable drain- of their simpler fabrication technology, and all the experimental data discussed above are on planar geometries. Afull3-Dapproachhasbeenfollowedin[15],where Manuscript received December 12, 2005; revised April 7, 2006. This work was supported in part by EU SINANO NoE under Contract 506844, by MIUR- the Poisson equation has been solved using the method of PRIN “Innovative Architectures and Models for Nano-MOSFETs,” and in part moments. Such a method provides the advantage of requir- by the National Science Foundation (NSF) under Grant EEC-0228390. The ing the computation of the Poisson equation only in regions review of this paper was arranged by Editor M. Reed. G. Fiori and G. Iannaccone are with the Dipartimento di Ingegneria where charge is not zero, with the drawback that it is prac- dell’Informazione, Università di Pisa, 56100 Pisa, Italy. tically impossible to treat more than two different dielectric G. Klimeck is with the Network for Computational Nanotechnology, School constants. of Electrical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: g.fi[email protected]). In this paper, we focus on realistic and experimentally Digital Object Identifier 10.1109/TED.2006.878018 relevant CNT-FETs with doped source and drain extensions, 0018-9383/$20.00 © 2006 IEEE FIORI et al.:3-DSIMULATIONSTUDYOFCNT-FETsWITHDOPEDRESERVOIRSANDREALISTICGEOMETRY 1783 and evaluate their performance against the requirements at the given that there are no fully confined states, the electron end of the ITRS. To this purpose, we have developed a code able concentration is to solve full band Schrödinger equations with open boundary + conditions in the NEGF framework. Such a module has been ∞ n("r)=2 dE ψ (E,"r) 2 f(E E ) included in our 3-D Poisson solver NANOTCAD ViDES [16], | S | − FS which can deal with very general structures, since it does not E#i $ take advantage of particular symmetries and can consider struc- 2 + ψD(E,"r) f (E EF ) (3) tures in which both CNTs and conventional semiconductors are | | − D % simultaneously present. while the hole concentration is Our realistic simulations show that CNT-FETs are very at- tractive for 1) their capabilities of suppressing short channel Ei 2 effects; 2) driving high ON currents per unit length; 3) providing p("r)=2 dE ψS(E,"r) [1 f(E EF )] | | − − S Ion/Ioff ratio required by the ITRS for the 15-nm gate length, # & and 4) their potential for terahertz applications. On the other −∞ + ψ (E,"r) 2 [1 f(E E )] (4) hand, we shall show that the subthreshold slope deteriorates for | D | − − FD agatevoltageclosetozeroduetothefillingofholestatesin ' where "r is the coordinate of the carbon site, f is the the channel. 2 2 Fermi–Dirac occupation factor, ψS ( ψD ) is the probability that states injected by the source| (drain)| | reach| the carbon site II. SIMULATION APPROACH ("r),andEFS (EFD )istheFermilevelofthesource(drain). The current has been computed as The potential profile in the 3-D simulation domain obeys the Poisson equation + 2q ∞ I = dE (E)[f(E E ) f(E E )] (5) h T − FS − − FD + # [!("r) φ("r)] = q p("r) n("r)+ND ("r) NA−("r)+ρfix −∞ ∇ ∇ − − − (1) ! " where q is the electron charge, h is Planck’s constant, and (E) T where φ("r) is the electrostatic potential, !("r) is the dielec- is the transmission coefficient computed as [17] + tric constant, ND and NA− are the concentrations of ionized = Tr Σ Σ† G Σ Σ† G† (6) donors and acceptors, respectively, and ρfix is the fixed charge. T − S − S D − D The electron and hole concentrations (n and p,respectively) $( ) ( ) % are computed by solving the Schrödinger equation with open where Tr is the trace operator. We have to point out that with boundary conditions by means of the NEGF formalism [17]. the present model, we only deal with the one-dimensional (1-D) transport between source and drain reservoirs, while the leakage Atight-bindingHamiltonianwithanatomistic(pz orbitals) real-space basis [18] has been used with a hopping parameter gate current has not be taken into account. For the considered t = 2.7 eV. devices with channel length of a few nanometers, it can be Green’s function can then be expressed as shown that the gate current is negligible with respect to the drain current. 1 From a numerical point of view, Green’s function is G(E)=[EI H Σ Σ ]− (2) − − S − D computed by means of the recursive Green’s function (RGF) technique [20], [21]. Particular attention must be put in the def- where E is the energy, I the identity matrix, H the Hamiltonian inition of each self-energy matrix, which can be interpreted as a of the CNT, and ΣS and ΣD the self-energies of the source and boundary condition of the Schrödinger equation. In particular, drain, respectively. As can be seen, transport is assumed here to in our simulation, we have considered a self-energy for semi- be completely ballistic. infinite leads as boundary conditions, which enables to consider The considered CNTs are all zigzag nanotubes, but the the CNT as connected to infinitely long CNTs at its ends. proposed approach can be easily generalized to nanotubes with We have to point out that the computation of the self-energy agenericchiralitysincetherequiredchangesinvolveonlythe is quite demanding.
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