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High-frequency limits of carbon nanotube transistors Chen, Li 2008

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High-frequency limits of carbonnanotube transistorsbyLi ChenB.E. (Electrical Engineering), The Wuhan University, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October, 2008c Li Chen 2008AbstractThis thesis is focused on the high-frequency performance of carbon nanotube  eld-e ect transistors (CNFETs). Such transistors show their promising performancein the nanoscale regime where quantum mechanics dominates. The short-circuit,common-source, unity-current-gain frequency fT is analyzed through regional signal-delay theory. An energy-dependent e ective-mass feature has been added to anexisting Schr odinger-Poisson (SP) solver and used to compare with results from aconstant-e ective-mass SP solver. At high drain bias, where electron energies con-siderably higher than the edge of the  rst conduction sub-band may be encountered,fT for CNFETs is signi cantly reduced with respect to predictions using a constante ective mass.The opinion that the band-structure-determined velocity limits the high-frequencyperformance has been reinforced by performing simulations for p-i-n and n-i-n CN-FETs. This necessitated incorporating band-to-band tunneling into the SP solver.Finally, to help put the results from di erent CNFETs into perspective, a meaningfulcomparison between CNFETs with doped-contacts and metallic contacts has beenmade. Band-to-band tunneling, which is a characteristic feature of p-i-n CNFETs,can also occur in n-i-n CNFETs, and it reduces the fT dramatically.iiContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvStatement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Carbon Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Modeling Coaxial CNFETs . . . . . . . . . . . . . . . . . . . . . . . 4iiiContents1.2.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Energy-dependent E ective Mass Model (EEM) . . . . . . . . . . . . 101.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Speci c Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Examination of the High-frequency Capability of Carbon NanotubeFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Schottky-barrier CNFETs . . . . . . . . . . . . . . . . . . . . 242.3.2 Doped-contact CNFETs . . . . . . . . . . . . . . . . . . . . . 262.3.3 Comparison of Doped-contact- and SB-CNFETs . . . . . . . 282.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34ivContents3 Comparison of p-i-n and n-i-n Carbon Nanotube FETs RegardingHigh-frequency Performance . . . . . . . . . . . . . . . . . . . . . . . 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 Energy-dependent E ective-mass Model (EEM) . . . . . . . . 393.2.2 Maximum Band Velocity vmax for Zigzag CNTs . . . . . . . . 403.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Summary, Conclusions and Further Work . . . . . . . . . . . . . . . 52AppendicesA Matrix Method for Heterostructure Transport . . . . . . . . . . . . 55A.1 Scatter Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.2 Interface Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B Zigzag CNT Band Velocity . . . . . . . . . . . . . . . . . . . . . . . . 60B.1 Maximum Band Velocity . . . . . . . . . . . . . . . . . . . . . . . . 60B.2 The First Sub-band vmax . . . . . . . . . . . . . . . . . . . . . . . . 62vContentsB.3 Choice of Parameter p . . . . . . . . . . . . . . . . . . . . . . . . . . 62B.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65C Issues of Band-to-band Tunneling in CNFETs (BTBT) . . . . . . 66C.1 Modeling without Band to Band Tunneling . . . . . . . . . . . . . . 66C.2 Modeling with Band to Band Tunneling . . . . . . . . . . . . . . . . 68C.3 n-i-n CNFET: with and without BTBT . . . . . . . . . . . . . . . . 70References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72viList of Tables2.1 Maximum band-structure-limited velocity, and the energy above theedge of the  rst conduction sub-band at which it is attained. The Sidata is for a [100] nanowire of diameter 1.36nm, as inferred from datain Ref. [14]. The InAs data is for a [100] nanoribbon of cross-section13 13nm2, as inferred from data in [15]. . . . . . . . . . . . . . . . . 232.2 Comparison of small-signal parameters of SB- and doped-contact-CNFETs having the properties listed in the text. CGG is the totalgate capacitance, CGD is the gate capacitance due to a change in VDS,gds is the drain conductance, Rc is the resistance of each of the sourceand drain contacts. The extrinsic fT is computed from Ref. [30]. . . . 31viiList of Figures1.1 Pictorial representation of: (A) Graphene monolayer [13]; and (B)Electronic structure [14] . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Pictorial representation of: (A) Graphene sheet (Courtesy of RichardMartel, IBM); and (B) Single-walled carbon nanotube lattice struc-tures. a1 and a2 are the lattice vectors of graphene. ja1j = ja2j = p3a,where a is the carbon-carbon bond length (Adapted from D.L.John etal. [15]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Sketch of coaxial CNFET [17] . . . . . . . . . . . . . . . . . . . . . . 51.4 2D simulation space. Axes z and   are also shown. . . . . . . . . . . 71.5 Energy dependence of vband for the  rst sub-band of (11,0) tube . . . 101.6 Drain-bias dependence of the ratio of fT for the energy-dependente ective-mass case to that for the constant e ective-mass case. VGS =0:5V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Gate-bias dependence of fT for a SB-CNFET, for constant- and energy-dependent-e ective mass. VDS = 0:5V, gate length =20nm. . . . . . 131.8 Dependence of maximum, band-structure-limited velocity on chiralityfor zig-zag nanotubes. . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 fT dependence on chirality for a n-i-n CNFET. VDS = 0:7V, VGS =0:5V, gate length=7 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 14viiiList of Figures1.10 Record measured fT values for various transistors [28{30], and pre-dicted results for n-i-n CNFETs. Open-square data: increasing thegate metal thickness (0.1, 1, 10nm) reduces fT . Cross data: increas-ing the contact resistance (0, 5, 50 k ) further reduces fT . . . . . . . 152.1 Experimental data from high-frequency transistors. CNFETs - squares [2,9{11]. SiCMOS - circle [12]. HBT - diamond [13]. The \ultimate"curve is from Eq. (2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Coaxial Schottky-barrier CNFET with wrap-around gate, showingsome of the pertinent structural parameters [20]. . . . . . . . . . . . . 242.3 Summary of simulation results for SB-CNFETs. E ect of variousparameters on fT: CGe [20, 21]; oxide permittivity and nanotube chi-rality [22]; contact resistance [23]; contact size [24]; gate-drain under-lap [25]; phonon scattering [26]. Arrows indicate increasing parameter. 252.4 Doped-contact CNFET with double-gate [3, 4]. . . . . . . . . . . . . 272.5 Summary of simulation results for doped-contact-CNFETs. Double-gate devices:  lled circles [5], open circle [31]. Coaxial devices - dia-monds, squares, triangles, crosses [32]. . . . . . . . . . . . . . . . . . 272.6 Regional signal delay for doped-contact n-i-n CNFETs with gate-metal thicknesses of either 0.1 or 10nm. In each case the gate iscentrally located and LG = 7 nm. Other common parameters are: chi-rality (11,0), oxide thickness=2nm, source and drain contacts=70nm. 292.7 Comparison of regional signal delays for SB- and doped-contact-CNFETswith properties described in the text. . . . . . . . . . . . . . . . . . . 303.1 Dependence of maximum, band-structure limited velocity on chiralityfor zig-zag nanotubes. . . . . . . . . . . . . . . . . . . . . . . . . . . 41ixList of Figures3.2 Energy dependence of vband, as computed from a Tight-binding calcu-lation (dotted line), and Hamiltonians using either an energy-dependente ective mass (solid line), or a constant e ective mass (dashed line).Results are for the  rst sub-band of a (22,0) tube. . . . . . . . . . . . 433.3 Drain-bias dependence at VGS = 0.4V of the ratio of fT for the EEMcase to that for the CEM case. The e ect of including BTBT in then-i-n device is also shown. . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Energy band diagams at VGS=0.4V and VDS=0.6V for doped-contactCNFETs: p-i-n (left), n-i-n (right). On the right, the thick dot-dashed lines are band edges when BTBT is included. The currentspectrum in this case is shown by the thin dot-dashed lines. In bothdiagrams the dashed lines are the quasi-fermi levels in the contacts. . 443.5 Comparison of the drain characteristics at VGS=0.4V. . . . . . . . . . 443.6 Comparison of the drain-bias dependence of fT at VGS=0.4V. . . . . 453.7 Comparison of the gate characteristics at VDS = 0.4 V. . . . . . . . . 453.8 Comparison of the gate-bias dependence of fT at VDS = 0.4 V . . . . 463.9 Current-energy spectrum for the p-i-n CNFET at VGS=0.5V, VDS=0.4V. The dashed lines are the Fermi levels in the doped contacts,and the solid lines are the band edges. . . . . . . . . . . . . . . . . . 463.10 Current-energy spectrum for the p-i-n CNFET at VGS=0V, VDS =0.4V.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.11 fT dependence on chirality for n-i-n and p-i-n CNFETs. VGS = 0.5 V,VDS = 0.4V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.12 Energy band diagrams at VGS=0.5V and VDS=0.4V for p-i-n CN-FETs made from tubes of chirality (10,0) (dotted lines) and (22,0)(solid lines). The dashed lines are the Fermi energies in the contacts. 49xList of FiguresA.1 Tunneling in multilayer system . . . . . . . . . . . . . . . . . . . . . 56C.1 p-i-n CNFET band structure, VGS=0.6V, VDS=0.7V. The gate lengthis 16 nm. The dotted horizontal line illustrates carriers interbandtunnelling at energy 0.02 eV, the dashed line between the conductionand valence band is E0 and the  uctuant solid line is ldos. . . . . . . 68C.2 The source- and drain-originating components of the spatial chargedistribution. The total charge (solid line), QS (dashed line), QD (dot-ted line). The left  gure is the result from equation (C.7) and theright is generated by moving the division line to the valence band edge. 70C.3 Change in the source- and drain-originating components of the spatialcharge for @VGS = +5 mV at VGS = VDS = 0:7 V. (11,0) CNFET,without (left) and with (right) interband tunnelling simulation. . . . 71C.4 Change in the source- and drain-originating components of the spatialcharge for @VGS = +5 mV at VGS = 0:4 V; VDS = 0:6 V. (22,0) CN-FET, without (left) and with (right) interband tunnelling simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71xiList of Symbolsk Wavevectorz Position along nanotube axis of symmetry  Radial distance from nanotube axis of symmetryfT cut-o  frequency t Electron a nity of semiconducting nanotube  WorkfunctionE EnergyEc Conduction band edgeEv Valence band edgeE0 Mid-gap energy level  Wavefunctionq Magnitude of electronic chargeh Planck's constant (~ = h2  is Dirac's constant) 0 Permittivity of free spaceBTBT band-to-band tunnelingDOS density of statesQM quantum mechanicsxiiList of SymbolsCEM constant e ective mass modelEEM Energy-dependent e ective mass modelxiiiAcknowledgementsI would like to acknowledge my colleagues Mohammad Ali Mahmoudzadeh andGeorge B. Abadir, for many valuable discussions and comments. I'm grateful toDrs. Leonardo Castro and David John for their patient help and encouragement.I don't think I could have made an impressive achievement without their teaching,especially in the understanding of the Schr odinger-Poisson (SP) solver.It is with great pleasure and honor that I thank my mentor Professor David Pulfrey,whose profound knowledge and insightful guidance is always a source of inspiration.xivDedicationI want to express my deepest gratitude to John Madden, Anne Coates and HyoshinKim, without their kindly suggestions I would not have had the opportunity to cometo UBC. Thanks to my friends Michael Lee and Michael Chen who helped me settlein the new environment with many fruitful and interesting discussions. My workwould not have been possible without the love and support of my family and friends.They are the inspiration in all of the aspects of my life.xvStatement of Co-AuthorshipThe research presented in the body of this thesis was performed within the UBCNanoelectronics Group between 2006 and 2008. The simulation code and methodsare based on the developments of David L. John and Leonardo C. Castro. Thisstatement describes the author's role in the research reported in this thesis. Chapters1 and 4 and the Appendices were written by the author. Chapter 2 is a paperwith my supervisor that has been accepted for publication. My contribution was toperform the research and prepare the  gures for the new results that were added toa review written by my supervisor. Chapter 3 is a paper with my supervisor thathas been submitted for publication. I performed the derivations, simulations andanalysis; I generated and prepared the  gures; I wrote the  rst draft of the paper,and participated in the collaborative e ort that led to the  nal version of the paper.xviChapter 1IntroductionThe discovery of carbon nanotubes (CNT) by Sumio Iijima in 1991 [1] attracted muchattention, from not only fundamental science but also from the perspective of applications.Studies of CNTs have demonstrated their unique nanostructure and remarkable electronicand mechanical properties [2, 3]. An ideal CNT can be considered as a hexagonal web ofcarbon atoms rolled up to make a seamless hollow cylinder. In 1992, Saito et al. proposeda tight-binding model to derive the energy band structure and electronic properties forCNTs [4, 5]. Through di erent atomic arrangements, CNTs could possess either metallicor semiconducting behaviour. As semiconductors, such one-dimensional nanostructuresexhibit a diameter-dependent bandgap and high conductivity.There are many possible applications for CNTs, such as interconnects [6] and sensors [7].One of the thrilling possibilities is to build nano-scale transistors, which could be applied forfuture high-speed VLSI circuits with higher density, and lower power dissipation. The car-bon nanotube  eld-e ect transistor (CNFET) employs semiconducting carbon nanotubesas the channel material between two contacts acting as source and drain electrodes, andmakes use of the gate voltage to control the essentially one-dimensional transport current[8, 9]. Theoretically, the CNFET could reach a higher frequency domain (terahertz regime)than conventional semiconductor technologies [10], suggesting high-speed wireless applica-tions. In 2006, Chen et al. [11] demonstrated a CNFET-based ring oscillator, which isthe  rst complete circuit on one individual CNT [9]. In 2007, Rutherglen and Burke [12]reported successful experimental results for a CNT-based demodulator in an actual AMradio-receiver demodulating high- delity music. Although the CNFET has shown some ofits potential abilities, the technology, such as fabrication, integration of nanosystems, etc.is still in its infancy when compared to state-of-the-art bulk-silicon CMOS devices.11.1. Carbon NanotubeThis thesis is a report on the study of the high-frequency capability of single-wall coaxialCNFETs between 2006 and 2008 with the UBC Nanoelectronics Group. Through di erenttypes of contact to the CNT channel, three di erent structures, n-i-n, Schottky-barrier(SB) and p-i-n CNFETs are investigated and compared. The remainder of this chapterwill mention some CNT properties, will cover characteristics relevant to high-frequencyperformance in short-channel CNFET devices, and will describe the new energy-dependent-e ective-mass model, followed by an outline of the thesis.1.1 Carbon NanotubeStarting from the nearest neighbour tight-binding approximation, the energy dispersion forgraphene ( gure 1.1.A) is expressed in the Cartesian coordinate system as:E(kx;ky) =   [1 + 4 cos(p3akx2 ) cos(aky2 ) + 4cos2(aky2 )]0:5; (1.1)where a is the lattice constant and   is the nearest-neighbour transfer integral [4]. Thisbandstructure is plotted in  gure 1.1.B as a function of wavevectors kx and ky. Thusgraphene is described as a semi-metal: the valence and conduction bands overlap but onlyat six points at the corners of the Brillouin zone.(A) (B)Figure 1.1: Pictorial representation of: (A) Graphene monolayer [13]; and (B)Electronic structure [14]21.1. Carbon NanotubeTreating the CNT as a rolled sheet of graphene,  rst we quantize the wavevector in thecircumferential direction: !k    !C = kxCx + kyCy = 2 p; (1.2)where C is shown in  gure 1.2.A and p is an integer. Equation (1.2) indicates that, byrolling the graphene sheet, states in CNT are determined by the particular values of Cx,Cy and p. Therefore electrons are con ned in di erent longitudinal lines intersecting thegraphene bandstructure. This could also be the reason that CNTs have less scattering thanother semiconducting materials, and that the current transport could be ballistic. Anotherimportant property of this formation is that CNTs could be metallic or semiconducting,depending on whether the lines pass through the graphene Fermi points [14].(A)  !(B)Figure 1.2: Pictorial representation of: (A) Graphene sheet (Courtesy of RichardMartel, IBM); and (B) Single-walled carbon nanotube lattice structures. a1 and a2are the lattice vectors of graphene. ja1j = ja2j = p3a, where a is the carbon-carbonbond length (Adapted from D.L.John et al. [15]).Substituting kx, ky from equation (1.1) with the axial direction wavevector k and C, pcomponents aforementioned, the CNT band structure is expressed by:E(k) =   [1 + 4 cos(3Cxka2C   3 paCyC2 ) cos(p3Cyka2C +p3 paCxC2 )+4 cos2(p3Cyka2C +p3 paCxC2 )]0:5; (1.3)31.2. Modeling Coaxial CNFETswhere Cx = ap3(n + 0:5m) and Cy = 1:5am. The integer sequence (n;m) is called thechirality, which is used to di erentiate CNTs. For example, (n;0) is labelled as a zigzagtube. When jn   mj = 3   I where I is an integer, the CNT's energy band crosses theFermi points, and thus it behaves metallically.From equation (1.3), the propagation velocity of electrons carrying the conduction current,which is also called the band-structure-determined velocity vb (=@E=~@k), plays an im-portant role for ballistic FETs in high-frequency performance [16]. The evidence shows themaximum band velocity vmax in carbon nanotube channels outperforms other materials inthe form of nanocylinders [18].1.2 Modeling Coaxial CNFETsBased on the CNT's extraordinary properties, such as high mobility, ultrathin body chan-nel, etc., CNFETs use semiconducting CNTs as the channel material between two elec-trodes, which are either doped contacts, or metals forming a Schottky barrier. Figure1.3 depicts the coaxial CNFET structure. A gate completely surrounds the dielectric toreduce short channel e ects like drain-induced barrier lowering. Meanwhile, the coaxialgeometry maximizes the capacitive coupling between the gate electrode and the nanotubesurface, thereby inducing more channel charge at a given bias than other geometries. Froma computational point of view, this structure reduces simulation dimensions by one due toits azimuthal symmetry.41.2. Modeling Coaxial CNFETsFigure 1.3: Sketch of coaxial CNFET [17]Work in the UBC Nanoelectronics Group by D. L. John and L. C. Castro has developeda self-consistent Schr odinger-Poisson solver [19], upon which this author's work is based.This thesis introduces an energy-dependent e ective mass m (E) based on Flietner's work[20], but extends it to apply to the entire band structure, rather than only to the bandgapregion. Applying charge-control theory [21] to FETs, the overall, source-drain signal delaytime  SD relates to the unity-current-gain frequency fT , and to the change in input chargethat is required to bring about a change in output current SD = 12 fT= @Q@I : (1.4)For a FET, @Q is the change in charge either on the gate or the tube (@QG    @QCNT =@Qin), @I is the change in drain current. The signal velocity in the CNFET channel,de ned as vsig(z) =h@Q(z)@Ii 1, does not exceed the highest propagation velocity attainedin the drain [16].51.2. Modeling Coaxial CNFETs1.2.1 ElectrostaticsThe axial symmetry of the coaxial structure in  gure 1.3 reduces the simulation spaceto 2D, as shown in  gure 1.4. Therefore, Poisson Equation in this cylindrical system isexpressed as:@2V@2  + 1  +1 @ @   @V@  +@2V@2z =  Qv( ;z)  ; (1.5)where V ( ;z) is the potential within the device cylinder, Qv( ;z) is the volumetric chargedensity,   is the permittivity. Boundary conditions are applied at the interface of eachpair of di erent materials. The Dirichlet boundary conditions on the surface of the source,drain, and gate terminals are given byV ( ;0) =   S=q (1.6)V ( ;Lt) = VDS    D=q (1.7)V (Rg;z) = VGS    G=q; (1.8)where   is the work function of each electrode, VGS and VDS are the gate- and drain-sourcevoltages, respectively. In the Schottky-barrier CNFET case, the electron barrier height atthe source/drain-channel interfaces is: Bn =  M    CNT ; (1.9)and the hole barrier height is Bp = Eg    Bn: (1.10)61.2. Modeling Coaxial CNFETs CNT is the electron a nity of a CNT with bandgap Eg, and since we are using intrinsicCNTs with symmetric conduction and valence band, its relationship with the workfunctionis expressed as: CNT =  CNT   Eg=2: (1.11)Due to the discontinuity in permittivity   at the nanotube-insulator and insulator-gateinterfaces, a matching condition is applied to the electric  ux from Gauss's law: ins @V@      =R+t  t @V@      =R t=  q(p   n)2 Rt 0; (1.12)Z zDzSrG @V@      =Rtdz +Z rG+tgrG @V@z    z=zSd   Z rG+tgrG @V@z    z=zDd  = QG2  ins 0: (1.13)Figure 1.4: 2D simulation space. Axes z and   are also shown.p, n are the hole and electron concentrations respectively, within the CNT circumferenceat distance z, QG is the gate electrode charge, and the relative permittivity of the CNT  tis one in this work [22]. Since there are no dangling bonds at the CNT surface, there is71.2. Modeling Coaxial CNFETsless restriction on the choice of dielectric as gate insulator. Various dielectrics have beenproposed in the literature, from the usual silicon dioxide ( ins  = 3:9) [23] to high-  HfO2 lms ( ins  = 20) [24].The fringing electric  elds between the gate and other terminals, determined by equa-tion (1.13), induce the parasitic capacitance. This could be comparable to the intrinsicdevice capacitances, and hence must be considered [10]. A notable example is that anincreased gate-metal thickness tg causing larger parasitic capacitance will reduce CNFEThigh-frequency performance dramatically [18].1.2.2 TransportBecause of their molecular uniformity and quasi-one-dimensional nature, carbon nanotubesare expected to have longer electron mean free paths than silicon. Theoretical predictionsstated they would be ballsitic for most nanotube diameters encountered in experiments[25]. Electron conduction within the small scale (between 1 and 100 nm) must be treatedquantum mechanically to account for QM re ection and tunnelling in the device. The 1DSchr odinger wave equation is employed for the charge on the CNT surface, with Poisson'sequation to establish a self-consistent relation between the surface potential V (R t ;z) andcharge within the CNFET domain.From the Landauer-B uttiker formalism, the device is treated as being composed of twocarrier reservoirs, connected by a 1D scattering region described by a transmission coe -cient T(E). Each contact region is described by its own equilibrium carrier statistics f(E),and is assumed to be also one-dimensional. Under this formalism, the electron currentI(E) =  qnv, where the charge density isn(E) = g(E)f(E)T(E)=2; (1.14)with the density of states accounting for spin-degeneracy g(E) = (2= )(@k=@E), and the81.2. Modeling Coaxial CNFETsband-structure-determined velocity is v. Therefore the current can be expressed below:I =  2qhZ[f(E    S)   f(E    D)]T(E)dE: (1.15) S and  D are the source and drain Fermi levels, respectively. Note that for each contactonly half of the electrons inject into the channel, so either +k or  k modes are considered;This is the reason the division sign appears in the charge density formula Equation (1.14).The transmission coe cient T(E) is expressed asT(E) =  D  D S  SvDvS ; (1.16)where  (E) is the electron wavefunction, * denotes the complex conjugate, v(E) is thegroup velocity and S, D are the source and drain terminals, respectively. From the one-dimensional Schr odinger equation, the wavefunction can be expressed by incident andre ected waves: (z) = azeikz + bze ikz; (1.17)where k are the wavevectors and az, bz are coe cients to be determined. Matching condi-tions on the boundary xij between adjacent layers i and j are expressed as: i(xij) =  j(xij); (1.18)1m i (E)@ i@z     z=xij= 1m j(E)@ j@z     z=xij: (1.19)Equation (1.19) includes an e ective-mass to satisfy current conservation. Use of an energy-dependent e ective mass m (E) here, and as described in the next section, is the maincontribution of this thesis. Details of using a matrix method to solve the Schr odingerequation can be seen in Appendix A.91.3. Energy-dependent E ective Mass Model (EEM)1.3 Energy-dependent E ective Mass Model(EEM)Based on Flietner's work [20], the energy-dependent e ective mass within any sub-band bcan be expressed as:m (E) = mb2 b(jE   E0j +  b); (1.20)where E0 is the mid-gap energy level,  b is one-half of the bandgap for sub-band b, and mbis the constant, parabolic-band, e ective mass for sub-band b. Combined with the formulaE = Ec+~2k22m  for states above the conduction band, the band-structure-determined velocityvband(E  Ec) is drawn in  gure 1.5 for the energy-dependent e ective mass model (dashedline). Curves from a tight-binding calculation (TBA - dotted line), in which an overlapparameter of 2.8 eV is used, and constant e ective mass (solid line) are also plotted here.The appropriate values for (11,0) tube parameters in equation (1.20) are mb = 0:122m0, b   (Ec   E0) = 0:473 eV. Clearly, the energy-dependent e ective-mass approach givessigni cantly better agreement at higher energies.Figure 1.5: Energy dependence of vband for the  rst sub-band of (11,0) tubeThe 1D DOS (density of states) g(E) = (2= )(@k=@E), can be expressed as a function of101.3. Energy-dependent E ective Mass Model (EEM)m  and E above the conduction band:g(E) = 1 ~r 2m E   Ec : (1.21)From equation (1.20), the DOS computed from EEM is larger than CEM. The chargedensities can be expressed as:Q(z;E) = qXbDb[GS;b(z;E)(u   fS) + GD;b(z;E)(u   fD)]; (1.22)where Db is the degeneracy of sub-band b, GC;b is the local density of states (LDOS) arisingfrom coupling to contact C , and fC is the Fermi function at contact C. The actual LDOSis given by the density of states and the wavefunction:GC;b(z;E) = gC;b(E) C(z;E)  C(z;E) (1.23)The parameter u is used to di erentiate between electrons and holes:u(z;E) =(0; E > E0 (electron)1; E < E0 (hole): (1.24)In  gure 1.6, the frequency fT results are presented for a doped-contact n-i-n CNFETcomprising an (11,0) nanotube in a coaxial structure with the following speci cations:gate length 7nm, gate thickness 0.1 nm, oxide thickness 2 nm, oxide relative permittivity3.9, source and drain length 30 nm, source and drain doping 0.5 /nm.In ballistic CNFETs, the electrons comprising the current attain their highest energies asthey exit the gated, intrinsic portion of the nanotube, and enter the doped portion of thedrain contact [16]. Thus, as the drain-source voltage VDS increases, one would expect tosee the constant-e ective-mass approach yielding increasingly exaggerated values of vband.111.3. Energy-dependent E ective Mass Model (EEM)Figure 1.6: Drain-bias dependence of the ratio of fT for the energy-dependente ective-mass case to that for the constant e ective-mass case. VGS = 0:5V.This, in turn, would lead to an increasing underestimate of the contribution of the above-mentioned portion of the nanotube to the signal delay and, consequently, to an overestimateof fT as VDS increases. Figure 1.6 con rms that this is indeed the case.One would expect that the switch to m (E) from a constant e ective mass would lower thetransconductance gm, as this parameter is directly related to the carrier velocity. However,in CNFETs contacted by positive Schottky barriers, gm has an interesting dependence ongate bias VGS due to internal resonances [26]. The result is shown in  gure. 1.7: the peakis shifted and is slightly higher than in the constant-e ective-mass case.gm is related to current, and higher values are obtained as the bandgap of the nanotubeis reduced [27]. The bandgap is inversely related to the diameter and chirality of the nan-otube, and it is shown in this thesis (Appendix B) that, for zig-zag tubes, the propagationvelocity (band-structure-limited velocity) has a maximum value (for the  rst sub-band) ofvb;max = 9:1   105 m/s for tubes of chirality (3i+1,0), where i is an integer, and a peakvalue that increases towards vb;max as i increases for tubes of chirality (3i+2,0). This be-haviour is illustrated in  gure 1.8 and more details are given in Appendix B. When this\oscillation" in peak velocity is combined with the improvement in gm that results fromhaving a lower source/channel barrier height in higher chirality tubes, the fT -chirality plot121.3. Energy-dependent E ective Mass Model (EEM)for doped-contact n-i-n CNFETs exhibits the interesting form shown in  gure 1.9.The results presented so far have been for structures with near-zero-thickness gate metal,and for no series resistance in the source and drain contacts. The e ects of increasing thegate metal thickness to 10 nm, and of adding some contact resistance to an n-i-n CNFEThave been examined in this thesis and are shown in  gure 1.10. It can be seen that the\headroom" over conventional transistors is not great when these practical features areadded to the simulations.Figure 1.7: Gate-bias dependence of fT for a SB-CNFET, for constant- and energy-dependent-e ective mass. VDS = 0:5V, gate length =20nm.131.3. Energy-dependent E ective Mass Model (EEM)Figure 1.8: Dependence of maximum, band-structure-limited velocity on chiralityfor zig-zag nanotubes.Figure 1.9: fT dependence on chirality for a n-i-n CNFET. VDS = 0:7V, VGS = 0:5V,gate length=7 nm.141.4. Thesis OutlineFigure 1.10: Record measured fT values for various transistors [28{30], and predictedresults for n-i-n CNFETs. Open-square data: increasing the gate metal thickness(0.1, 1, 10nm) reduces fT . Cross data: increasing the contact resistance (0, 5, 50k ) further reduces fT .1.4 Thesis OutlineIn the following chapters, two manuscripts published in, or submitted to, journals over thecourse of researching CNFETs are presented, followed by a conclusion summarizing thekey achievements and addressing future work. Chapter 2 analyzes the high-frequency per-formance of CNFETs, using a self-consistent, constant e ective-mass Schr odinger-Poissonsolver. A meaningful comparison between CNFETs with doped-contacts and metallic con-tacts has been made. Chapter 3 compares the high-frequency capabilities of p-i -n andn-i-n doped-contact CNFETs via EEM simulations, and considers the e ect of band-to-band tunneling in both devices. Other relevant work performed by the author appearsin the appendices. In Appendix A, a detailed matrix method for solving the Schr odingerequation is given. Appendix B derives the  rst-band velocity formalism for zigzag CNTsfrom Tight-Binding theory. Appendix C identi es a problem with SP modeling problem151.5. Speci c Contributionsthat is facing when including band-to-band tunneling.1.5 Speci c ContributionsThe work presented in this thesis has contributed to determining the high-frequency limitsof carbon nanotube  eld-e ect transistors. The energy-dependent e ective mass modeldeveloped here shows its superiority over the constant e ective mass method, especially asthe drain-source bias is increased. In addition, an in-depth analysis explains how the band-structure -determined velocity of zig-zag carbon nanotubes changes as chirality increases.Such behavior also in uences the CNFET's high-frequency performance. Band-to-bandtunneling (BTBT) has been incorporated into the model and shown to reduce the perfor-mance of n-i-n CNFETs. Also parasitics, in the form of gate metal thickness and contactresistance, have been included to better predict the performance of realistic device struc-tures.161.5. ReferencesReferences[1] S. Iijima, \Helical microtubules of graphitic carbon", Nature, vol. 354, pp. 56{58,1991.[2] M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and CarbonNanotubes (Academic Press, Toronto, 1996).[3] Mildred S. Dresselhaus, Gene Dresselhaus, and Phaedon Avouris, eds., Carbon Nan-otubes: Synthesis, Structure, Properties, and Applications, vol. 80 of Topics in AppliedPhysics (Springer, Berlin, 2001).[4] R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, \Electronic structure ofchiral graphene tubules", Appl. Phys. Lett., vol. 60, pp. 2204{2206, 1992.[5] R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, \Electronic structure ofgraphene tubules based on c60", Phys. Rev. B, vol. 46, pp. 1804{1811, 1992.[6] F. Kreup, A. P. Graham, M. Liebau, G. S. Duesberg, R. Seidel, and E. Unger, \Carbonnanotubes for interconnect applications", in IEDM Tech. Dig., pp. 683{686, 2004.[7] Liming Dai, Prabhu Soundarrajan, and Taehyung Kim, "Sensors and sensor arraysbased on conjugated polymers and carbon nanotubes", Pure Appl. Chem., Vol. 74,No. 9, pp. 1753{1772, 2002.[8] S. J. Tans, A. R. M. Verschueren, and C. Dekker, \Room-temperature transistor basedon a single carbon nanotube", Nature, vol. 393, pp. 49{52, 1998.[9] J. Appenzeller, \Carbon nanotubes for high performance electronics|Progress andprospect", Proc. IEEE, vol. 96, no. 2, Feb. 2008.[10] Peter J. Burke, \AC performance of nanoelectronics: towards a ballistic THz nanotubetransistor", Solid-State Electronics, Vol. 48, Iss. 10-11, 1981-1986, 2004[11] Z. Chen, J. Appenzeller, Y.-M. Lin, J.S. Oakley, A.G. Rinzler, J. Tang, S. Wind, P.Solomon, and Ph. Avouris, \An integrated logic circuit assembled on a single carbonnanotube", Science, vol. 311, 1735, 2006.171.5. References[12] Chris Rutherglen and Peter Burke, "Carbon Nanotube Radio", Nano Lett., 7(11) pp3296{3299, 2007.[13] L.C. Castro, Ph.D. Thesis. The University of British Columbia, 2006.[14] M P Anantram and F L eonard, \Physics of carbon nanotube electronic devices", Rep.Prog. Phys. 69, 507-561, 2006.[15] D.L. John, L.C. Castro, P.J.S. Pereira and D.L. Pulfrey, \A Schr odinger-Poisson Solverfor Modeling Carbon Nanotube FETs", Tech. Proc. of the 2004 Nanotechnology Conf.and Trade Show, vol. 3, pp. 65-68, 2004.[16] D.L. Pulfrey, L.C. Castro, D.L John, and M. Vaidyanathan, "Regional signal-delayanalysis applied to high-frequency carbon nanotube FETs", IEEE Trans. Nanotech-nology. vol. 6, 711-717, 2007.[17] L.C. Castro and D.L. Pulfrey, \Extrapolated fmax for Carbon Nanotube FETs", Nan-otechnology, vol. 17, pp. 300-304, 2006.[18] D.L. Pulfrey and Li Chen, \Examination of the high-frequency capability of carbonnanotube FETs", Solid-State Electronics, vol. 52, 9, pp. 1324-1328, 2008.[19] D. L. John, L. C. Castro, P. J. S. Pereira, and D. L. Pulfrey, \A Schr odinger-Poissonsolver for modeling carbon nanotube FETs", in Tech. Proc. of the 2004 NSTI Nan-otechnology Conf. and Trade Show, vol. 3, (Boston, U.S.A.), pp. 65{68, March 2004.[20] H. Flietner, Phys. Stat. Sol. (b), vol. 54, 201-208, 1972.[21] H. K. Gummel, \On the de nition of the cuto  frequency fT ", Proc. IEEE, vol. 57,no. 12, p. 2159, Dec. 1969.[22] F. Leonard and J. Terso , \Multiple Functionality in Nanotube Transistors", PhysicalReview Letters, vol. 88, Jun. 2002, p. 258302.[23] J.A. Misewich et al., \Electrically Induced Optical Emission from a Carbon NanotubeFET", Science, vol. 300, May. 2003, pp. 783-786.181.5. References[24] A. Javey et al., \Carbon Nanotube Field-E ect Transistors with Integrated OhmicContacts and High- Gate Dielectrics", Nano Letters, 4 (3), 447 -450, 2004.[25] C. T. White and T. N. Todorov, \Carbon nanotubes as long ballistic conductors",Nature, 393, 240-242 (1998).[26] D.L. John, L.C. Castro, and D.L. Pulfrey, \Quantum capacitance in nanoscale devicemodeling, J. Appl. Phys., vol. 96, 5180-5184, 2004.[27] D.L. Pulfrey, Proc. IEEE ESSDERC, pp. 234-238, 2007.[28] W. Hafez, W. Snodgrass, and M. Feng, Appl. Phys. Lett., vol. 87, 252109, 2005.[29] S. Lee et al., IEDM Tech. Digest, 241-244, 2005.[30] A. Le Louarn et al., Appl. Phys. Lett., vol. 90, 233108, 2007.19Chapter 2Examination of theHigh-frequency Capability ofCarbon Nanotube FETs2.1 IntroductionThis invited paper provides an opportunity to update a critique of the high-frequencyperformance of CNFETs presented at ESSDERC-07 [1]. The focus is on the short-circuit,common-source, unity-current-gain frequency, fT. The existing measured data is collected,and a new record value is reported [2]. The existing simulation results for Schottky-barrier(SB) CNFETs are collected, and grouped such that the e ects on fT of the following factorscan be clearly seen: oxide permittivity, tube chirality, extrinsic capacitance, contact sizeand resistance, phonon scattering. New simulation results for doped-contact n-i-n CNFETsare added to the data presented in Ref. [1]; they show the e ect on fT of: tube chirality,gate length, gate metal thickness, and contact resistance. Importantly, correction of someearlier data, which suggested an extraordinarily high fT capability [3, 4], has been noted [5].It is now observed that all the simulation results for both SB- and doped-contact-CNFETsfall below the limit imposed by the propagation velocity of electrons in the gated regionof the nanotube [6]. This may seem like an obvious result; however, its application toA version of this chapter has been published. D. L. Pulfrey and Li Chen. \Examination ofthe high-frequency capability of carbon nanotube FETs", Solid-State Electronics, vol. 52, 9, pp.1324-1328, 2008.202.2. Experimental Resultsnanoscale FETs needs to be re-asserted for at least two reasons:  rstly, to dispel doubtscaused by earlier simulation results [3, 4]; secondly, to re-a rm that \image-charge" e ects,which can lead to signal-delay times being shorter than propagation-delay times in  eldregions of bipolar transistors, are not signi cant in nanoscale FETs [7], even though apronounced  eld can exist in the gated region of doped-contact CNFETs [8].2.2 Experimental ResultsThe low current-drive and high input/output impedance of single CNFETs make it di -cult to perform direct measurements of high-frequency electrical properties, at least whenusing instrumentation based on a reference impedance of 50  . In order to make a directmeasurement of a recognized high-frequency  gure-of-merit, such as fT, it has been real-ized that CNFETs assembled from multiple nanotubes must be employed [2, 9{11]. Suchmeasurements are in their infancy, and problems of non-parallel nanotubes, the presenceof some metallic nanotubes, and excessive gate overlap capacitance need to be addressed.However, progress is being made, and the highest fT recorded thus far, after de-embedding,is 30GHz [2]. The experimental data is shown in Fig. 2.1; there is some dependence on gatelength LG, which is indicative of the success of the de-embedding procedures employed tonegate the e ect of the pad parasitics. The  gure also shows the gate-length dependenceof fT, as predicted in the \ultimate" limit of the signal delay being determined solely bythe propagation of electrons through the gated portion of the nanotube [6]. Satisfaction ofthis condition is equivalent to having no charge change (capacitance) associated either withregions of the CNFET external to the gated-portion, or with parasitic structures. Clearly,such an ideal situation cannot be attained in practice, but the comparison emphasizes thate ort should be put into making measurements on structures using shorter nanotubes. Cer-tainly, as Fig. 2.1 also shows, shorter channel lengths or basewidths have been employedto obtain record fT values for other types of transistor: Si MOSFETs (330GHz [12]) andInP/InGaAs HBTs (710GHz [13]).The ultimate limit, referred to above, proposes2 fT;ultimate = vb;highLG; (2.1)212.3. Simulation ResultsFigure 2.1: Experimental data from high-frequency transistors. CNFETs - squares [2,9{11]. SiCMOS - circle [12]. HBT - diamond [13]. The \ultimate" curve is fromEq. (2.1).where vb;high is the maximum, band-structure-limited velocity that can be attained. In thezig-zag nanotubes considered here, the value of vb;high depends on the choice of the overlapparameter used in the tight-binding approximation to get the band structure. Here we use2.8 eV, which gives the maximum velocities listed in Table 2.1 for various tubes. Note thatthe maximum propagation velocity in the carbon nanotubes is attained around 1 eV abovethe edge of the  rst conduction sub-band, and, consequently, is only likely to be reachedby electrons injected into a region of high electric  eld. The \ultimate" line in Fig. 2.1 isdrawn for vb;high = 8:8   105 m/s: this value gives a convenient  gure for fT;ultimate in THzof 140/LG, with LG in nm. This number is indicative of a fundamental limit, as opposedto a phenomenological limit, for which one proposed value is 80=LG [16].2.3 Simulation ResultsDetailed theoretical analyses involve the self-consistent solution of the equations of Schr odingerand Poisson, usually under the quasi-static approximation [17], which is appropriate as fT222.3. Simulation ResultsMaterial Chirality Bandgap Evb;high   EC Maximum velocity(eV) (eV) (105 m/s)C NT 10,0 0.98 1.22 9.1C NT 11,0 0.95 1.06 7.5C NT 13,0 0.76 1.11 9.1C NT 14,0 0.74 1.00 7.9C NT 16,0 0.62 1.03 9.1C NT 17,0 0.61 0.96 8.1C NT 19,0 0.52 0.96 9.1C NT 20,0 0.51 0.89 8.2C NT 22,0 0.45 0.90 9.1Si NW 2.85   0:6   5:8InAs NW 0.48   0:18   4:5Table 2.1: Maximum band-structure-limited velocity, and the energy above the edgeof the  rst conduction sub-band at which it is attained. The Si data is for a [100]nanowire of diameter 1.36nm, as inferred from data in Ref. [14]. The InAs data isfor a [100] nanoribbon of cross-section 13 13nm2, as inferred from data in [15].is a parameter attained by extrapolation from lower frequencies. Methods involving eitheran e ective-mass wave equation, or a Hamiltonian based on atomistic considerations, havebeen employed, and, under suitably low-bias conditions, should give similar results [18],provided the simulation space is properly bounded [19].The extrapolated fT is given by2 fT = @ID@QG  gmCGi + CGe; (2.2)where @ID and @QG are changes in output (drain) current and input (gate) charge, respec-tively, due to a change in gate-source voltage, for example; gm is the transconductance, andCGi and CGe are contributions to the total gate capacitance CGG arising from the regionunder the gate (intrinsic), and the gate-electrode regions (extrinsic), respectively.232.3. Simulation ResultsFigure 2.2: Coaxial Schottky-barrier CNFET with wrap-around gate, showing someof the pertinent structural parameters [20].2.3.1 Schottky-barrier CNFETsFig. 2.2 is illustrative of the coaxial, all-around-gate structure that is usually used insimulations of SB-CNFETs [20]. The results that have been obtained for simulation of SB-CNFETs are collected together in Fig. 2.3. The data labeled CGe comes from two devicesof di erent gate length and underlaps [20, 21]: for the LG = 2 nm case (solid diamonds),the e ect is large because of the small gate-source underlap LuS (14 nm); in the LG = 5nmcase (open diamonds), increasing the separation of source and drain electrodes (LuS andLuD) to 24nm mitigates the e ect. The results shown are for contact radii varying fromthat of the nanotube itself, to that of the nanotube plus oxide and gate thicknesses [21].The bene cial e ect of increasing LuD (from 5 to 25 nm) is also shown by the solid-circledata at LG = 50 nm [25]. The open-circle data at LG = 50 nm comes from a planarstructure [24]; the degradation of fT is again related to an increase in CGe, and is dueto changing the contact from that of a needle of radius equal to that of the nanotube,to that of a metallic strip of width 8 m. The latter was the actual electrode structureof a high-performance DC device [27], and emphasizes the need to develop  ner contactarrangements for HF devices. It is clear that CGe has a large e ect on the performance ofthese nanoscale transistors, and it must be included in simulations if predictions of fT areto be meaningful [28].The square data points at LG = 2nm show the e ect of increasing  ox from 3.9 to 25 while242.3. Simulation ResultsFigure 2.3: Summary of simulation results for SB-CNFETs. E ect of various pa-rameters on fT: CGe [20, 21]; oxide permittivity and nanotube chirality [22]; contactresistance [23]; contact size [24]; gate-drain underlap [25]; phonon scattering [26].Arrows indicate increasing parameter.252.3. Simulation Resultskeeping the insulator thickness  xed at 2.5 nm [22]. The point of these two simulations wasto assess the trade-o  between increased CGi (there were no underlaps) and increased gmdue to the stronger electrostatic coupling between gate and nanotube. Evidently, the e ectof the capacitance is greater, so fT decreased. For Schottky-barrier contacts representingpalladium, the barrier height for hole injection decreases as the nanotube chirality (anddiameter) increases [29]. This enhances gm, leading to the improved performance shown inFig. 2.3 on changing the chirality from (11,0) (diameter=0.8 nm) through (16,0) to (22,0)(diameter=1.7 nm) [22]. Fig. 2.3 also shows the e ect of considering the actual resistance ofthe source and drain contacts. Such resistances can be expected to be high when employingnanoscale needle contacts. The results shown are for Rcontact increasing from zero through10 k  to 100k  [23]. Similar degradations also apply to fmax [30]. Phonon scatteringcould be important, at least in tubes of length greater than about 10-20 nm, which is themean-free-path for optical phonons [26]. The e ect is illustrated by the downward shift ofthe \ultimate" line to that of the dashed line shown in Fig. 2.3 [26]. Phonon scatteringleads to a build-up of charge in the channel, i.e., to an increase in CGi.2.3.2 Doped-contact CNFETsIn addition to coaxial structures akin to those in Fig. 2.2, double-gate structures, of theform shown in Fig. 2.4 [3, 4], have been used in the simulation of doped-contact-CNFETs.The results that have been obtained for simulation of doped-contact-CNFETs are collectedtogether in Fig. 2.5. Earlier, very high, fT results for double-gate structures [3, 4] have nowbeen corrected [5], bringing them into good agreement with results from other workers [31].All data shown are for (11,0) nanotubes. Results for a coaxial geometry using the sametube are also shown in Fig. 2.5 [32]. The slightly inferior performance of the coaxialdevices is due principally to the increased capacitance that results from this geometricalarrangement. The bene cial e ect of increasing the chirality (from (11,0) through (16,0)to (22,0), see the diamond data points) is due to the associated reduction in bandgap (seeTable 2.1), which lowers the potential barrier at the doped-source/intrinsic-gated-regioninterface, thereby improving the transconductance.Most simulations are performed with an essentially zero-thickness gate electrode. This isconvenient from a numerical analysis point-of-view; it reduces the simulation space required262.3. Simulation ResultsFigure 2.4: Doped-contact CNFET with double-gate [3, 4].Figure 2.5: Summary of simulation results for doped-contact-CNFETs. Double-gate devices:  lled circles [5], open circle [31]. Coaxial devices - diamonds, squares,triangles, crosses [32].272.3. Simulation Resultsto contain source and drain contacts that are su ciently long to ensure charge neutralityat their ends [33]. However, it is an unrealistic situation, which is also impractical from thepoint of view of obtaining a high fmax [30]. Fig. 2.5 (square data points) shows the e ect ofincreasing the gate metal thickness from 0.1 nm through 1 nm to 10 nm. Even though thelast value may still be low for a practical device, it does indicate the deleterious e ect ofthe associated increase in CGe. If a  nite contact resistance is added to this, fT is furtherreduced: Fig. 2.5 (cross data points) shows the e ect of 5 and 50k  of resistance in thesource and drain contacts. The latter may not be unreasonable for nanoscale contacts, andit would bring the estimated value of fT down to levels that have actually been realized inanother type of transistor [13].The importance of the gate-metal thickness is emphasized in Fig. 2.6, which breaks downthe overall source-drain signal delay  SD into regional delays [8] SD =LCNQXr r = 1@ILCNQXrZr@Q(z) dz ; (2.3)where LCNQ is the length of the nanotube over which there is a change in charge, @Q(z) isthe change in local charge density integrated over energy, and @I is the change in drain cur-rent. Fig. 2.6 indicates how LCNQ is much enlarged by increasing the gate-metal thickness.2.3.3 Comparison of Doped-contact- and SB-CNFETsFig. 2.3 and Fig. 2.5 display, to the best of our knowledge, all the simulation results thathave been reported thus far for fT in CNFETs. However, a comparison between SB-and doped-contact-devices is not easily made from this collection because of the di eringdevice properties that have been used, e.g., device chirality, oxide thickness, voltage bias,and because of the di erent simulators that have been employed. To provide a meaningfulcomparison, we provide Fig. 2.7 and Table 2.2, which compare the salient high-frequencyparameters for the two types of CNFET with common parameters of: chirality (19,0), LG =7 nm, gate-metal thickness = 5 nm, oxide thickness = 2 nm, source/drain underlap = 5 nm,contact lengths = 45nm, jVGSj = 0:6 V, jVDSj = 0:7 V. The SB-CNFET used Pd contacts,282.3. Simulation ResultsFigure 2.6: Regional signal delay for doped-contact n-i-n CNFETs with gate-metalthicknesses of either 0.1 or 10 nm. In each case the gate is centrally located andLG = 7 nm. Other common parameters are: chirality (11,0), oxide thickness=2nm,source and drain contacts=70nm.292.4. DiscussionFigure 2.7: Comparison of regional signal delays for SB- and doped-contact-CNFETswith properties described in the text.whereas the doping density in the doped-contact case was 5   108 /m. The parametervalues were chosen in accordance with realizing good high-frequency performance. Notethat di erent parameter values would be needed for a CNFET more suited to high-speeddigital-logic applications [34, 35], in particular: higher bandgap to produce a reasonableON/OFF-current ratio, and a longer gate length to reduce source-drain tunneling.The Table highlights the signi cant di erence in transconductance between the two de-vices; this is due to the reduced quantum-mechanical re ection of electrons at the injectingsource/intrinsic-nanotube interface. The capacitances are only slightly higher in the SBcase, but, when taken together with the lower current, of which the lower gm is a manifes-tation, they result in signi cantly higher regional signal delays, as Fig. 2.7 shows.2.4 DiscussionAll of the data presented in this review now falls below the \ultimate" propagation limit [6].This should now remove speculation about how extraordinarily high values of fT might302.4. DiscussionContacts gm CGG fT gds CGD Rc \Extrinsic" fT( S) (aF) (THz) ( S) (aF) (k ) (THz)Pd 19.6 1.47 2.1 1.80 0.66 50 1.0C (n-type) 59.5 1.37 6.9 0.97 0.54 50 2.0Table 2.2: Comparison of small-signal parameters of SB- and doped-contact-CNFETshaving the properties listed in the text. CGG is the total gate capacitance, CGD isthe gate capacitance due to a change in VDS, gds is the drain conductance, Rc is theresistance of each of the source and drain contacts. The extrinsic fT is computedfrom Ref. [30].arise in nanoscale FETs due to fortuitous variations in local charge densities [1, 8]. Thepossibility of the propagation velocity in regions of high  eld, such as can exist in thechannel of short FETs, being exceeded by the signal velocity has also been ruled out [7].Essentially, this is because any local changes in charge in the nanotube are imaged onthe gate electrode, thereby contributing wholly to the change in input charge. The nearone-to-one correspondence of nanotube charge and gate charge arises because of the two-dimensional geometry and the close proximity of the gate electrode to the nanotube. In abipolar transistor, which is essentially a one-dimensional device, the electrostatics is muchsimpler, and it is easily shown that not all of the charge change within the semiconductingregions is imaged on the input electrode (the base) [36]. This can lead to the signal delay inthe base-collector space-charge region being less than the propagation delay in that region.Inevitably, when considering the performance of a new  eld-e ect transistor, comparisonswill be made with Si MOSFETs. This review has suggested that the signal delay in thenon-neutral regions of FETs is unlikely to be less than the band-limited propagation delay.Thus, a relevant question is: how does the band-limited propagation velocity vband forcarbon nanotubes compare with that in nanoscale Si structures? The result quoted inTable 2.1 suggests that carbon nanotubes have a slight advantage as regards the maximumvalue of vband, at least when compared to the particular Si nanowire cited. Guo et al.have suggested that vband for an ultra-thin body Si MOSFET is about 50% of that in aCNFET [24]. Thus, the ultimate fT in CNFETs would appear to be only slightly greaterthan might be achievable with nanoscale Si FETs.312.5. ConclusionsThese comparisons are for ballistic transport, and it may be argued that attainment ofballistic transport is more likely in a CNFET than in a Si MOSFET, primarily becauseof the relatively long mean-free-path associated with phonon scattering in carbon nan-otubes, but also because of the more one-dimensional form of a tube, as opposed to thatof a wire or a ribbon. However, it seems unreasonable to ignore the e ect of surfacescattering, which greatly a ects the mobility in present Si MOSFETs. The nature of theoxide/semiconductor interface is di erent in the two devices, of course, but some penetra-tion of the electron wavefunctions into the oxide of a CNFET is to be expected. There ispresently no information on this, to the authors' knowledge.We have shown that when the practical features of gate-metal thickness and contact re-sistance are included in the simulations, then fT for CNFETs can drop into the region of700-800 GHz. This is about a factor of 2 higher than values that have actually been realizedalready in planar Si MOSFETs [12]. Add this fact to the need to arrange CNFETs in par-allel to improve the current drive, and one wonders whether the small material superiorityof vband and the geometrical superiorities of a wrap-around gate and a one-dimensionalstructure, will be enough to combat the matchless technological superiority of silicon FETprocessing. Perhaps further research and development in high-frequency CNFETs shouldbe directed towards biological applications, for which silicon-based electronics may be lesscompatible?2.5 ConclusionsFrom this review of the high-frequency performance of CNFETs it can be concluded that:  experimental fT values should improve by employing multiple, parallel nanotubes ofshorter length than used hitherto;  theoretically, the e ects on fT of nanotube chirality (diameter), oxide permittiv-ity, gate-source and gate-drain underlap, source- and drain-electrode diameter andresistance, gate-metal thickness, and phonon scattering are well understood;322.5. Conclusions  doped-contact CNFETs o er better performance capability than Schottky-barrierdevices because of their superior transconductance;  the presently available simulation data indicates that the signal delay time is notless than the propagation time. This suggests that the band-structure-determinedvelocity is a key factor in assessing the high-frequency prospects for a FET material.The slight advantage that a carbon nanotube has over silicon in this regard maynot be su cient to o set the technological superiority of Si FETs when it comes toprocessing practical devices.AcknowledgementThe authors gratefully acknowledge the profound contributions made to the UBC Nano-electronics Group by Drs. Leonardo Castro and David John. Financial support was fromthe Natural Sciences and Engineering research Council of Canada.332.5. ReferencesReferences[1] Pulfrey DL. Critique of high-frequency performance of carbon nanotube FETs. ProcIEEE ESSDERC 2007;234-8.[2] Le Louarn A, Kapche F, Bethoux J-M, Happy H, Dambrine G, Derycke V, et al..Intrinsic current gain cuto  frequency of 30 GHz with carbon nanotube transistors.Appl Phys Lett 2007;90:233108.[3] Fiori G, Iannaccone G, and Klimeck G. Performance of carbon nanotube  eld-e ecttransistors with doped source and drain extensions and arbitrary geometry. IEDMTech Digest 2005;522-5.[4] Fiori G, Iannaccone G, and Klimeck G. A three-dimensional simulation study of theperformance of carbon nanotube  eld-e ect transistors with doped reservoirs and re-alistic geometry. IEEE Trans Elec Dev 2006;53:1782-8.[5] Fiori G. Private communication. Aug. 29, 2007.[6] Hasan S, Salahuddin S, Vaidyanathan M, and Alam MA. High-frequency perfor-mance projections for ballistic carbon-nanotube transistors. IEEE Trans Nanotechnol2006;5:14-22.[7] John DL. Limits to the Signal Delay in Ballistic, Nanoscale Transistors: Semi-Classicaland Quantum Results. IEEE Trans Nanotechnol 2008;7:48-55.[8] Pulfrey DL, Castro LC, John DL, and Vaidyanathan M. Regional signal-delay anal-ysis applied to high-frequency carbon nanotube FETs. IEEE Trans Nanotechnol2007;6:711-7.[9] Kim S, Choi T-Y, Rabieirad L, Jeon J-H, Shim M, and Mohammadi S. A poly-Si gatecarbon nanotube  eld-e ect transistor for high-frequency applications. Proc IEEEMTT Symp 2005;303-6.[10] Bethoux J-M, Happy H, Dambrine G, Derycke V, Go man M, and Bourgoin J-P,An 8-GHz fT carbon nanotube  eld-e ect transistor for gigahertz range applications.IEEE Electron Dev Lett 2006;27:681-3.342.5. References[11] Narita K, Hongo H, Ishida M, and Nihey F. High-frequency performance of multiple-channel carbon nanotube transistors. Physica Stat Sol A 2007;204:1808-13.[12] Lee S, Jagannathan, Csutak S, Pekarik J, Zamdmer N, et al., Record RF performanceof sub-46 nm NFETs in microprocessor SOI CMOS technologies. IEDM Tech Digest2005;241-4.[13] Hafez W, Snodgrass W, and Feng M. \12.5 nm base pseudomorphic heterojunctionbipolar transistors achieving fT=710 GHz and fmax = 340 GHz. Appl Phys Lett2005;87:252109.[14] Wang J, Rahman A, Ghosh A, Klimeck G, and Lundstrom M. Performance evaluationof ballistic silicon nanowire transistors with atomic-basis dispersion relations. ApplPhys Lett 2005;86:093113.[15] Persson MP and Xu HQ. Electronic structure of [100]-oriented free-standing InAs andInP nanowires with square and rectangular cross sections. Phys Rev B 2006;73:125346.[16] Burke PJ. AC performance of nanoelectronics: Towards a ballistic THz nanotubetransistor. Solid-State Elec 2004;48:1981-6.[17] Castro LC, John DL, Pulfrey DL, Pourfath M, Gehring A, and Kosina H. Method forpredicting fT for carbon nanotube FETs. IEEE Trans Nanotechnol 2005;4:699-704.[18] Koswatta SO, Neophytou N, Kienle D, Fiori G, and Lundstrom MS. Dependence of DCcharacteristics of CNT MOSFETs on bandstructure models. IEEE Trans Nanotechnol2006;5:368-72.[19] McGuire DL and Pulfrey DL. Error analysis of boundary-condition approximationsin the modeling of coaxially gated carbon nanotube FETs. Physica Stat Sol A2006;203:1111-6.[20] Alam K and Lake R. Performance of 2 nm gate length carbon nanotube  eld-e ecttransistors with source/drain underlaps. Appl Phys Lett 2005;87:073104.[21] Alam K and Lake R. Dielectric sensitivity of a zero Schottky-barrier, 5nm gate,carbon nanotube  eld-e ect transistor with source/drain underlaps. J Appl Phys2006;100:024317.352.5. References[22] Castro LC. Unpublished data.[23] Castro LC, Pulfrey DL, and John DL. High-frequency capability of Schottky-barriercarbon nanotube FETs. Solid-State Phenomena 2007;121-3:693-6.[24] Guo J, Hasan S, Javey A, Bosman G, and M. Lundstrom. Assessment of high-frequencyperformance of carbon nanotube FETs. IEEE Trans Nanotechnol 2005;4:715-21.[25] Pourfath M, Kosina H, Cheong BH, Park WJ, and Selberherr S. Improving DC andAC characteristics of ohmic contact carbon nanotube  eld-e ect transistors. ProcESSDERC 2005;541-4.[26] Yoon Y, Ouyang Y, and Guo J. E ect of phonon scattering on intrinsic delay and cuto  frequency of carbon nanotube FETs. IEEE Trans Elec Dev 2206;53:2467-70.[27] Javey A, Guo J, Farmer DB, Wong Q, Yenilnez E, Gordon RG, et al.. Self-alignedballistic molecular transistors and electrically parallel nanotube arrays. Nano Lett2004;4:1319-22.[28] Paul BC, Fujita S, Okajima M, and Lee T. Impact of geometry-dependent para-sitic capacitances on the performance of CNFET circuits. IEEE Electron Dev Lett2006;27:380-2.[29] Chen Z, Appenzeller J, Knoch J, Lin Y-M, and Avouris P. The role of metal-nanotubecontact in the performance of carbon nanotube  eld-e ect transistors. Nano Lett2005;5:1497-502.[30] Castro LC and Pulfrey DL. Extrapolated fmax for carbon nanotube FETs. Nanotech-nology 2006;17:300-4.[31] John DL, Pulfrey DL, Castro LC, and Vaidyanathan M. Terahertz carbon nanotubeFETs: feasible or fantastical?. Proc. 31st WOCSDICE 2007; ISBN: 978-88-6129-088-4.[32] Chen Li. Unpublished data.[33] John DL and Pulfrey DL. Issues in the modeling of carbon nanotube FETs: structure,gate thickness and azimuthal asymmetry. J Computational Electronics 2007;6:175-8.362.5. References[34] Pulfrey DL, John DL, and Castro LC. Proc 13th Int Workshop Phys SemiconductorDev 2005;7-13.[35] John DL and Pulfrey DL, Switching-speed calculations for Schottky-barrier carbonnanotube  eld-e ect transistors. J Vac Sci Tech A 2006;24:708-12.[36] Laux SE and Lee W. Collector signal delay in the presence of velocity overshoot. IEEEElec Dev Lett 1990;11:174-6.37Chapter 3Comparison of p-i-n and n-i-nCarbon Nanotube FETs RegardingHigh-frequency Performance3.1 IntroductionAggressively scaled nanowire and nanotube FETs with isotypically doped source and drainregions are predicted to exhibit large sub-threshold currents, which result from unwantedband-to-band tunneling (BTBT) at the drain end of the device [1, 2]. Contrarily, byusing di erently doped contacts, i.e., a p-type source and an n-type drain, BTBT canapparently be exploited to yield inverse sub-threshold slopes below the thermionic-emissionlimit of 60 mV/decade [3]. However, a possible drawback to such p-i-n tunnel FETs is thereduction in ON current with respect to conventional FETs, in which charge injection isby thermionic emission, rather than tunneling, is the operative injection mechanism [4].Nevertheless, high ON/OFF current ratios have been predicted, which, coupled with a lowswitching energy, have led to suggestions that tunnel FETs may be suited to ultra-lowpower applications [4, 5]. These attributes depend on the suppression of direct source-drain tunneling, either by keeping the channel length above about 15 nm, or by limitingthe drain-source bias. The desirable properties of tunnel MOSFETs have led them to beinvestigated in other semiconductor-material systems [6].A version of this chapter has been submitted for publication. Li Chen and D.L. Pulfrey,\Comparison of p-i-n and n-i-n carbon nanotube FETs regarding high-frequency performance",Solid-State Electronics, submitted, 8/14/2008.383.2. MethodHere, we explore the capability of p-i-n CNFETs for high frequency performance. Acomparison with n-i-n CNFETs, for which we include the BTBT e ect, is also given.An energy-dependent e ective-mass (EEM) model, rather than a constant e ective-mass(CEM) model, is applied to our Schr odinger-Poisson solver [7], thereby allowing achieve-ment of more accurate simulation results for devices in which high electric  elds are ex-pected to be present. This situation is likely to arise in the drain region of the device athigh drain-source bias and, if not correctly treated, could lead to an underestimate of thesignal delay time in this region [11], and to a corresponding overestimate of fT . We alsoexplore the e ect of chirality, thereby extending the work on n-i-n CNFETs that has beenpresented recently [8].3.2 Method3.2.1 Energy-dependent E ective-mass Model (EEM)Flietner's energy-dependent e ective-mass formulation is extended to apply to energieswithin the bands of a carbon nanotube, rather than merely to energies within the bandgap[9]. We write:m (E) = mb2 b(jE   E0j +  b); (3.1)where E0 is the mid-gap energy level,  b is one-half of the bandgap for sub-band b, andmb is a constant, parabolic-band, e ective mass for sub-band b.In our scattering-matrix solution to compute transmission probability [10], the boundaryconditions for the derivative of the wavefunction need to include m (E) to satisfy currentconservation: 1m i (E)@ i@x    x=xij= 1m j(E)@ j@x    x=xij; (3.2)where xij is the position of the interface between piece-wise rectangular layers i and j .393.2. MethodThe wavevector in the nanotube is given by:k =p2m (E)(jE   E0j    b)=~ : (3.3)The charge densities can by expressed as:Q(z;E) = qXbDb[GS;b(z;E)(u   fS) + GD;b(z;E)(u   fD)]; (3.4)where Db is the degeneracy of sub-band b, GC;b is the local density of states arising fromcoupling to contact C [11], and fC is the Fermi function at contact C. The parameter u isused to di erentiate between electrons and holes:u(z;E) =(0; E > E0 (electron)1; E < E0 (hole): (3.5)3.2.2 Maximum Band Velocity vmax for Zigzag CNTsIn view of the importance of the band-limited velocity in determining the upper-bound tofT in FETs [12], we examine here the maximum band velocity vmax in zigzag CNTs:vmax = 1~(dEdk )    max;b: (3.6)The energy (E)-wavevector (k) relationship in sub-band b of a zig-zag tube of chiral index(n,0) can be expressed from Tight-binding theory [14]:E =   1 + 4 cos(3ak2 ) cos( pn ) + 4cos2( pn ) 0:5; (3.7)where k is the longitudinal wavevector, a = 0:142 nm is the carbon-carbon bond length, pis an integer from 1 to 2n indicating the di erent bands, and   is the overlap parameter.From Eqns. (B.5) and (3.7), vmax in the  rst sub-band, and the energy Ea at which it isachieved, can be expressed as [13]:403.3. Results and DiscussionFigure 3.1: Dependence of maximum, band-structure limited velocity on chirality forzig-zag nanotubes.For (3i+1,0) tube, vmax = 3a2~ , Ea =  q4 cos2(2i+13i+1 )   1.For (3i+2,0) tube, vmax =  3a~   cos(2i+13i+2 ), Ea =  q1   4 cos2(2i+13i+2 ).Where i is an integer. vmax in the  rst band for zig-zag nanotubes (n;0) is drawn in Fig.3.1. It can be seen that vmax = 9:1   105 ms 1 for tubes of chirality (3i+1,0), and that themaximum value for (3i+2,0) tubes increases towards this peak as the chiral index increases.3.3 Results and DiscussionSimulation results are presented for coaxial, doped-contact CNFETs made from (22,0)nanotubes. In all cases, the gate length is 16 nm (to avoid direct source-drain tunneling[5]), the gate thickness is 1nm, the oxide thickness is 3.2nm, the oxide relative permittivityis 3.9, and the source and drain lengths are 50 nm. The source and drain contact dopingdensities are 0.5 nm 1 for both the n- and p- type regions of the n-i-n and p-i-n CNFETs413.3. Results and Discussionthat are to be compared. These speci cations are similar to those for devices used in astudy of switching performance [5], with the notable exception of the relative permittivityof the gate dielectric: we use 3.9, as opposed to the value of 16 used in [5], as this reducesthe intrinsic capacitances, thereby improving fT [12].Figure 3.2 compares the band-determined velocity dispersion relationship from the twoe ective-mass models with that calculated from a Tight-binding, nearest-neighbor calcula-tion using   = 2:8 eV. It can be seen that an energy-dependent e ective mass approach isnecessary if the velocity is to be correctly modeled at energies above about 0.1 eV. As VDSis increased, electrons will attain and exceed this energy on entering the drain. Thus, useof the constant-e ective mass model will overestimate the velocity in this region, leadingto an underestimate of the signal delay time in the drain [11], and, consequently, to anover-optimistic value of fT . This fact is demonstrated in Fig. 3.3. The e ect is more severein the p-i-n case because of the opening-up of another high-energy current path at largeVDS, as illustrated in Fig. 3.4a. Speci cally, at high bias, tunneling of electrons into thedrain at energies close to that of the conduction-band edge in the drain is facilitated. Thisphenomena can also be viewed as tunneling of holes into the i-region. The holes enterthis region at high energy, so their velocity is overestimated by the constant-e ective-massmodel. BTBT can also occur at high bias in n-i-n structures, as can be seen from theemergence of a subsidiary peak in the current spectrum in Fig. 3.4b. The onset of thiscurrent at VGS = 0:4 V is illustrated in Fig. 3.5. However, in this case, the holes injectedinto the i-region cause a charge build-up that, evidently, more severely a ects fT than doesthe increase in current, leading to a reduction in fT (= 12   I Q). This is clear from Fig. 3.6,and is also shown in Fig. 3.3. The ambipolar nature of conduction in p-i-n CNFETs iswell known [4, 5], and its e ect on the gate characteristic is illustrated in Fig. 3.7. Con-trarily, the n-i-n device shows the more usual, unipolar relationship. The ambipolaritynecessitates the re-de nition of fT as fT = 12      I Q    for the p-i-n case, with the resultthat fT drops dramatically around the point of the current minimum (see Fig. 3.8), whichoccurs in this case at VGS = VDS=2 = 0:2 V. The di erent energy paths for the majoritycarriers (electrons at VGS > 0:2 V, and holes at VGS < 0:2 V), are evident in the diagramsof Figs. 3.9 and 3.10, respectively.We now turn to the chirality-dependence of the maximum band-determined velocity vmax.The results are shown in Fig. 3.1, and the e ect on fT is shown in Fig. 3.11. For both the423.3. Results and DiscussionFigure 3.2: Energy dependence of vband, as computed from a Tight-binding calcu-lation (dotted line), and Hamiltonians using either an energy-dependent e ectivemass (solid line), or a constant e ective mass (dashed line). Results are for the  rstsub-band of a (22,0) tube.Figure 3.3: Drain-bias dependence at VGS = 0.4V of the ratio of fT for the EEMcase to that for the CEM case. The e ect of including BTBT in the n-i-n device isalso shown.433.3. Results and Discussion(a) (b)Figure 3.4: Energy band diagams at VGS=0.4V and VDS=0.6V for doped-contactCNFETs: p-i-n (left), n-i-n (right). On the right, the thick dot-dashed lines areband edges when BTBT is included. The current spectrum in this case is shownby the thin dot-dashed lines. In both diagrams the dashed lines are the quasi-fermilevels in the contacts.Figure 3.5: Comparison of the drain characteristics at VGS=0.4V.443.3. Results and DiscussionFigure 3.6: Comparison of the drain-bias dependence of fT at VGS=0.4V.Figure 3.7: Comparison of the gate characteristics at VDS = 0.4V.453.3. Results and DiscussionFigure 3.8: Comparison of the gate-bias dependence of fT at VDS = 0.4VFigure 3.9: Current-energy spectrum for the p-i-n CNFET at VGS=0.5V, VDS=0.4V. The dashed lines are the Fermi levels in the doped contacts, and the solidlines are the band edges.463.3. Results and DiscussionFigure 3.10: Current-energy spectrum for the p-i-n CNFET at VGS=0V, VDS =0.4V.n-i-n and p-i-n devices the \oscillation" in vmax is manifest in fT , but is superimposed on asteadily increasing value of fT with chirality. In n-i-n devices, the increasing trend is due toa reduction of the source/channel barrier height with the lower bandgap that is associatedwith an increase in chirality [12]. In the p-i-n case, the lower bandgap leads to a thinnerbarrier for BTBT (see Fig. 3.12). In each case there is an increase in transconductancewith chirality.473.3. Results and DiscussionFigure 3.11: fT dependence on chirality for n-i-n and p-i-n CNFETs. VGS = 0.5V,VDS = 0.4V.483.4. ConclusionFigure 3.12: Energy band diagrams at VGS=0.5V and VDS=0.4V for p-i-n CNFETsmade from tubes of chirality (10,0) (dotted lines) and (22,0) (solid lines). The dashedlines are the Fermi energies in the contacts.3.4 ConclusionFrom this simulation study of doped-contact CNFETs it can be concluded that:  Use of an energy-dependent e ective-mass model gives less optimistic predictions forfT in both p-i-n and n-i-n CNFETs than does the usual, constant-e ective-massmodel.  The high-frequency performance of both n-i-n and p-i-n CNFETs employing zig-zagtubes improves with chirality.  Operation of n-i-n CNFETs at high drain bias may lead to reduced high-frequencyperformance due to charge build-up in the device as a result of BTBT.493.4. ReferencesReferences[1] J. Knoch, et al. \One-Dimensional Nanoelectroinc Devices{Towards the QuantumCapacitance Limit", 66th Device Research Conference, pp. 173-176, Santa Barbara,USA, June 2008.[2] J. Appenzeller, et al. \Comparing Carbon Nanotube Transistors{The Ideal Choice:A Novel Tunneling Device Design", IEEE Transactions on Electron Devices, vol. 52,issue 12, pp. 2568-2576, 2005.[3] J. Appenzeller, Y.-M. Lin, J. Knoch and Ph. Avouris, \Band-to-Band Tunneling inCarbon Nanotube Field-E ect Transistors", Phys. Rev. Lett., vol. 93, no. 19, 196805,Nov. 2004.[4] S.O. Koswatta, D.E. Nikonov and M.S. Lundstrom, \Computational study of carbonnanotube pin-tunnel FETs", IEDM Tech Dig, pp. 525-528, 2005.[5] S. Poli, S. Reggiani, A. Gnudi, E. Gnani, and G. Baccarani, "Computational study ofthe ultimate scaling limits of CNT tunneling devices", IEEE Transactions on ElectronDevices, vol. 55, pp. 313-321, 2008.[6] Kathy Boucart and Adrian M. Ionescu, \Threshold voltage in Tunnel FETs: physicalde nition, extraction, scaling and impact on IC design", 37th European Solid-StateDevice Research Conference, pp. 299-302, Montreux, Switzerland, September 2006.[7] D.L. John, L.C. Castro, P.J.S. Pereira, and D.L. Pulfrey, \A Schr odinger-Poissonsolver for modeling carbon nanotube FETs", Tech. Proc. NSTI Nanotechnology Conf.and Trade Show, vol. 3, 65-68, Boston, 2004.[8] Li Chen and D.L. Pulfrey, \Is there an Opportunity for Carbon Nanotube FETs inVery-High-Frequency Applications?", 66th Device Research Conference, pp. 111-112,Santa Barbara, USA, June 2008.[9] H. Flietner, \The E(k) relation for a two-band scheme of semiconductors and theapplication to the metal-semiconductor contact", Phys. Stat. Sol. (b), vol. 54, 201-208, 1972.503.4. References[10] David Yuk Kei Ko and J. C. Inkson, \Matrix method for tunneling in heterostructures:Resonant tunneling in multilayer systems", Phys. Rev. B, vol. 38, 9945-9951, 1988.[11] D.L. Pulfrey, L.C. Castro, D.L John, and M. Vaidyanathan, "Regional signal-delayanalysis applied to high-frequency carbon nanotube FETs", IEEE Transactions onNanotechnology. vol. 6, 711-717, 2007.[12] D.L. Pulfrey and Li Chen, \Examination of the high-frequency capability of carbonnanotube FETs", Solid-State Electronics, accepted for publication, Feb. 22, 2008.[13] Li Chen, \High-frequency limits of carbon nanotube transistors", M.A.Sc. thesis, TheUniversity of British Columbia, 2008.[14] M P Anantram and F L eonard, \Physics of carbon nanotube electronic devices", 2006Rep. Prog. Phys. 69, 507-561.51Chapter 4Summary, Conclusions andFurther WorkThe semiconductor industry continues to scale metal-oxide-semiconductor transistors(MOSFETs), maintaining the predictions of Moores Law. Many physical restrictionsare becoming more apparent and di cult to surpass with this aggressive scaling, suchas drain induced barrier lowering (DIBL) and gate oxide leakage current. Carbonnanotube  eld-e ect transistors (CNFETs) are being considered as one of the futureelectronic technologies because of their excellent performance. The variable bandgap, depending on the chirality and CNT diameter, gives rise to nanotube propertiesranging from metallic to semi-metallic to semiconducting. The work presented in thisthesis displays their potential and limitations for high-frequency performance. Fromthe simulation study of the e ect of an energy-dependent e ective mass model, we nd the estimates for the band-structure-determined velocity are much superior tothose using a constant e ective-mass model, particularly at high energies; this supe-riority shows itself as the drain-source bias is increased, and indicates that estimatesof fT using a constant e ective mass can be in large error.In Chapter 2, new results are added to a recent critique of the high-frequency per-formance of carbon nanotube  eld-e ect transistors. From a materials standpoint,carbon nanotubes are potentially good for high-frequency because of high band-structure-limited velocity, which reduces the channel charging time. The image-charge e ects, which can lead to the signal velocity being higher than the propa-gation velocity (band-structure-limited velocity) in  eld regions of bipolar transis-52Chapter 4. Summary, Conclusions and Further Worktors, are not signi cant in nanoscale FETs. The comparison between doped n-i-nand SB CNFETs shows the higher transconductance occurs in doped-contact CN-FETs due to the reduced quantum-mechanical re ection of electrons at the injectingsource/intrinsic-nanotube interface. From a device practical standpoint, however,CNFETs may be limited by parasitics associated with gate-metal thickness and con-tact resistance.The constant-e ective-mass model (CEM) used here gives optimistic predictions forfT especially at high drain-source bias. When electrons are injected into the drainreservoir with high energy, the propagation velocity should be lower than the valuededucted from the parabolic band structure from CEM. This case has been explicitlyexplained in Chapter 3. Its impact on both n-i-n and p-i-n doped-contact CNFETsis explored and compared. Band-to-band tunneling (BTBT), which occurs in tradi-tional devices such as Zener diodes, can apparently be exploited to build nano-scalep-i-n transistors for low-power applications. This e ect shows its ambipolar naturein p-i-n transistors and also reduces fT in n-i-n CNFETs due to charge build-up inthe device as a result of BTBT. For zig-zag tubes, the propagation velocity has amaximum value (for the  rst sub-band) of vb;max = 9:1   105 m/s for tubes of chi-rality (3i+1,0), where i is an integer, and a peak value that increases towards vb;maxas i increases for tubes of chirality (3i+2,0). When this oscillation in peak velocityis combined with the improvement in gm for both n-i-n and p-i-n CNFETs, fT alsooscillates with the chiral index increase of zig-zag tubes for both devices.In conclusion, this study of the high-frequency performance of carbon nanotube FETshas shown that:1. A constant-e ective mass model can lead to large overestimates of fT. This er-ror can be greatly reduced by using the energy-dependent e ective-mass modeldeveloped in this thesis.2. Incorporation of an energy-dependent e ective-mass model into our Schr odinger-Poisson solver allows band-to-band tunneling to be investigated. This transport53Chapter 4. Summary, Conclusions and Further Workmechanism is the main conduction process in p-i-n CNFETs, but it also existsin n-i-n CNFETs, where it can reduce fT due to charge build-up in the channel.3. Inclusion of parasitics, particularly gate-source and gate-drain capacitance dueto having practical values of gate-metal thickness, can seriously reduce fT. Thereduction may be so severe that CNFETs will not realize the advantage theyhave over other transistors due to their superior intrinsic properties.During the course of implementing BTBT into our SP solver, the question of whatvalue to use for the energy level that distinguishes electrons from holes arose. Issuesconnected with this are summarized in Appendix C, which will hopefully be of useto someone wishing to study the e ect in more detail.54Appendix AMatrix Method forHeterostructure TransportThis appendix describes solving the schr odinger equation for the carrier transport in theheterostructure.A.1 Scatter MatrixIn  gure A.1, for a  nite N+1 layer structure, a and b are the coe cients of the \forward"and the \backward" states, respectively. As explained in [1], the forward states are de- ned as waves which propagate or exponentially decay in the positive-z direction, and thebackward states behave similarly, but in the negative-z direction. The relationship for thetransfer matrix from layer 0 to N is:"a0b0#= T(0;N)"aNbN#: (A.1)The equivalent scatter matrix S(0;N) would be:"aNb0#= S(0;N)"a0bN#: (A.2)55A.1. Scatter MatrixFigure A.1: Tunneling in multilayer systemThe matching conditions for the (n+1)th interface can be expressed below [2]:"anbn#= K(n + 1)"an+1bn+1#: (A.3)The details of the construction of the K interfacial matrix are given in the literature [2],and will be demonstrated for our speci c case in the next section.For simplicity, the S matrices and the K represent S(0;n) and K(n + 1), respectively.Assume a semi-in nite boundary condition and a wave that propagates from left to right,then set a0 = 1; bN = 0. Combined with the deduction in [1], the coe cients are given by:8<:an = [1   S12(0;n)S21(n;N)] 1S11(0;n)bn = [1   S21(n;N)S12(0;n)] 1S21(n;N)S11(0;n): (A.4)56A.1. Scatter MatrixIn the above equations, S11(0;n), S12(0;n) could be solved through iteration starting fromthe  rst layer [1]:8<:S11(0;n + 1) = S11(K11   S12K21) 1S12(0;n + 1) = S11(0;n + 1)(S12K22   K12)S 111: (A.5)The remaining problem is to  nd an expression for S21(n;N) expression as follows.First we de ne matrix K0:"bn+1an+1#= K0(n + 1)"bnan#; (A.6)and matrix I = K 1 .It's easy to prove Iii = K0jj, Iij = K0ji and Sii(n;N) = Sjj(N;n); where i; j = 1;2 andi 6= j.Use the relations above and equations (A.5)S21(n;N) = S12(N;n)= S11(N;n)[S12(N;n + 1)K022   K012)]S(N;n + 1)= [S21(n + 1;N)K022   K012][K011   S21(n + 1;N)K021] 1= [S21(n + 1;N)I11   I21][I22   S21(n + 1;N)I12] 1: (A.7)Therefore coe cients in (A.4) can be solved by equations (A.5) and (A.7). The approachhereinabove only requires initial boundary conditions a0 and bN, hence the errors in thecoe cients do not propagate from one layer to the next.57A.2. Interface MatrixA.2 Interface MatrixIn  gure A.1, de ne an, bn as the wavefunction coe cients of the beginning point of layern; a0n, b0n are coe cients at the end point of layer n. Assuming the coordinate of interfacebetween layer n and n + 1 is zero, the wavefunctions within layers n and n + 1 can beexpressed as: (z) =8<:a0neiknz + b0neiknz z < 0an+1eikn+1z + bn+1eikn+1z z > 0;(A.8)where kn is the wavevector in layer n. From quantum mechanic of boundary conditions:8<:a0n + b0n = an+1 + bn+1(a0nkn   b0nkn)m 1n = (an+1kn+1   bn+1kn+1)m 1n+1;(A.9)where mn is the e ective mass in layer n; an and a0n are in the same layer with intervallength d, therefore a0n = aneiknd, b0n = bne iknd. Written in matrix form:"an+1bn+1#= 0:5"1 +   1    1     1 +  #"eiknd 00 e iknd#"anbn#(A.10)= I"anbn#; (A.11)where   = knmn+1kn+1mn and I = K 1.The advantage of using (A.10) is that the probability in each layer can be expressed as the\sum of coe cients": j nj2 = jan + bnj2.58A.2. ReferencesReferences[1] David Yuk Kei Ko and J. C. Inkson, \Matrix method for tunneling in heterostructures:Resonant tunneling in multilayer systems", Phys. Rev. B, 38(14), 9945-9951 (1988).[2] A.C. Marsh and J. C. Inkson, \Scattering matrix theory of transport in heterostruc-tures", Semicond. Sci. Technol. 1, 285-290 (1986).59Appendix BZigzag CNT Band VelocityIn this appendix, the maximum  rst sub-band-determined velocity for zigzag carbon nan-otubes is derived from the Tight-Binding approximation.B.1 Maximum Band VelocityFrom [1], for zigzag tube (n;0), the E-k relationship for conduction band is:E =  [1 + 4 cos(3ak2 ) cos( pn ) + 4cos2( pn )]0:5: (B.1)Here, k is the longitudinal wavevector and k 2 [0;  3a], a = 0:142 nm is the carbon-carbonbond length, p is an integer from 1 to 2n, and   is the overlap parameter.1Equation (B.1) yields the band velocity:v = 1~ dEdk =   2~3acos( pn ) sin(3ak2 )E ; (B.2)1Here,  =2.8 eV.60B.1. Maximum Band Velocitywhere ~ is Dirac constant.2 De ne Ea and ka as the corresponding band energy and wavevector, respectively, to achieve the maximum velocity vmax, where the derivative of v withrespect to k is zero:dvdk   k=ka= 0:This renders 3a2 cos(3aka2 )Ea   sin(3aka2 )dEdk   k=ka= 0.Utilize the de nition of velocity v = 1~ dEdk , vmax is given by:vmax = 3a2~ cot(3aka2 )Ea; (B.3)Ea =   sin(3aka2 )[ 2 cos( pn )cos(3aka2 ) ]0:5: (B.4)Eliminate Ea, vmax can be expressed as:vmax = 3a2~ r 2ccos(3aka2 ): (B.5)Where c = cos( pn ) and the c; k relation is written as:cos(3aka2 ) =  4c2   1   j4c2   1j4c : (B.6)2~ = 1:0546   10 34 J   s.61B.2. The First Sub-band vmaxB.2 The First Sub-band vmaxThe lowest energy for each sub-band occurs when k = 0. Hence from (B.1):E =     1 + 2 cos( pn )   : (B.7)Obviously, for the  rst sub-band, p is an integer to satisfy: cos( pn )  !  0:5. With theconstraint p = 1;2;:::2n, then p ! 2n=3 or 4n=3. Considering p  ! 4n=3 as a degenerateband, this also indicates the degeneracy of the zigzag CNT  rst sub-band is always 2.c is negative for the  rst sub-band, from (B.6), cos(3aka2 ) =8<: 1=(2c);  1 < c 6  0:5 2c;  0:5   c < 0.Substituting into (B.5):vmax = 3a2~ 8<:1;  1 < c 6  0:5 2c;  0:5   c < 0:(B.8)The equation above indicates that the  rst band maximum velocity for semiconductorzigzag tube is 9:0609   105 m=s. Through further investigation, this maximum velocity isdivided into two categories for di erent zigzag tubes. One is (3i + 1;0), the \ rst class" oftube with i as an integer. For these tubes, vmax is constant. The other one is the (3i+2;0)tube for which the vmax is smaller than the previous class and will go close to the \ rstclass" vmax when i increases. Details of the proof are in the next section.B.3 Choice of Parameter pFrom (B.7), the  rst sub-band half-band gap E1 satis es:62B.3. Choice of Parameter pE1  =8<:1 + 2 cos( pn ); cos( pn ) >  0:5 1   2 cos( pn ); cos( pn ) <  0:5:(B.9)Lemma1: For (3i + 1;0) and (3i + 2;0) tubes, p for the  rst sub-band satis escos( pn ) <  0:5 and cos( pn ) >  0:5, respectively.Proof: For (3i + 1;0) tube, as mentioned before, cos( pn )  !  0:5 to attain minimumE1. It's quickly to pick p either 2i or 2i + 1 and yield the following expression: pn  2 3 =4y =8<: 24x; p = 2i4x; p = 2i + 1;where 4x =  3 13i+1:Substitute  pn = 4y + 2 3 into (B.9), And the statement that cos( pn ) <  0:5 to attainminimum E1 is equal to the inequality below:1 + cos(2 3   24x) + cos(2 3 + 4x) > 0:Starting from the left hand side and using the inequality 3 cos(2 3  24x) > cos(2 3  4x):LHS > 1 + cos(2 3   4x) + cos(2 3 + 4x) dd= 1 + 2 cos(2 3 ) cos(4x)= 1   cos(4x) > 0:This proves the statment for a (3i + 1;0) tube.3From 4x expression  313i+1, when i = 0, 4x = 3 , thus 4x   3 and both2 3   4x ,2 3   24xstay in the same range where the cosine function is monotonically decreasing.63B.4. ConclusionFor (3i + 2;0) tube, similarly, it's equivalent to prove the inequality below:1 + cos(2 3   4x) + cos(2 3 + 24x) < 0;with 4x =  3 13i+2 and p = 2i + 1 or 2i + 2. Using \Sum to product" to change the lefthand side:LHS = 1 + 2cos(2 3 + 4x2 ) cos(34x2 )Obviously, 4x6  6 . Hence LHS is a monotonic decreasing function with respect to 4x,and LHS  ! 0 when M x  ! 0. Thus the result follows.B.4 ConclusionFrom the discussion above, vmax and the energy Ea to achieve vmax in the  rst sub-band can be expressed as:For (3i + 1;0) tube, p = 2i + 1; vmax = 3a2~ ; Ea =  q4 cos2(2i+13i+1 )   1;For (3i + 2;0) tube, p = 2i + 1; vmax =  3a~   cos(2i+13i+2 ), Ea =  q1   4 cos2(2i+13i+2 ).64B.4. ReferencesReferences[1] M P Anantram and F L eonard, \Physics of carbon nanotube electronic devices", Rep.Prog. Phys. 69, 507-561 (2006).65Appendix CIssues of Band-to-band Tunnelingin CNFETs (BTBT)In this appendix, the problem in modeling BTBT has been provided and is the topic forfuture work.C.1 Modeling without Band to Band TunnelingThe wavevector k = k(E;z) is given by:k2 =8<:2mb~2 (E   E0   4b); for electrons2mb~2 ( E + E0   4b); for holes;(C.1)where E0 is the midgap energy, 4b is the energy distance between the bottom of sub-bandb and E0, and mb is the e ective mass for sub-band b.From the speci ed band structure, we can compute k(E;z) for electrons and holes sepa-rately.We use the scatter matrix method in [1] to obtain the carrier wavefunction  (E;z) prop-agating from either source or drain.This is done by the function scattmat in our Schr odinger-Poisson (SP) solver:66C.1. Modeling without Band to Band Tunneling[AS,BS,AD,BD]=scattmat(kS,kD,dz)The input of this function are kS, kD the wavevectors from source and drain, respectively,and dz the length of each piece-wise rectangular barrier step. The outputs are coe cientsof the wavefunctions from the source and drain. Both Ai and ki (i=Source or Drain) havethe same dimensions.Take the source-originating charge for example, the carrier is an electron or a hole depend-ing on whether the carrier energy level is larger than the source conduction band or smallerthan the source valence band, respectively. The local density of states (ldos: Gs(E;z)) fromthe source is expressed by [2]:Gs(E;z) = Dbgs(E) s(E;z)  s(E;z); (C.2)and s(E;z)  s(E;z) = jAS(E;z) + BS(E;z)j2; (C.3)where Db is the degeneracy of the sub-band b, gs(E) is the 1D non-local density of statesin the neutral region of the source contact.Therefore:Qs(E;z) =8<: qGs(E;z)fs(E) for electrons;qGs(E;z)[1   fs(E)] for holes.(C.4)Similar expressions are used for charges originating in the drain contact.67C.2. Modeling with Band to Band TunnelingC.2 Modeling with Band to Band TunnelingThe wavevector k = k(E;z) is given by:k2 = 2mb~2 (jE   E0j   4b): (C.5)Except for the wavevector formula, the ldos is computed in the same way as for the non-interband tunneling case.The ldos Gs(E;z) near the source Fermi level injected from source is shown in Fig. C.1Figure C.1: p-i-n CNFET band structure, VGS=0.6V, VDS=0.7V. The gate length is16 nm. The dotted horizontal line illustrates carriers interband tunnelling at energy0.02 eV, the dashed line between the conduction and valence band is E0 and the uctuant solid line is ldos.68C.2. Modeling with Band to Band TunnelingAnother di erence that arises with interband tunneling in the SP solver is how to calculatecharge density. Take the source-originating charge Qs for example:Qs(E;z) = qGs(E;z)[u(E;z)   fs(E)] (C.6)The parameter u is used to di erentiate between electrons and holes and the division levelis set at E0:u(z;E) =(0; E > E0 (electron)1; E < E0 (hole): (C.7)However the setting of the division line at the middle level of the band gap is still arbitrary.Further work needs to be done to arrive at a more rigorous de nition for E0. In the followingwe show how the placement of the dividing energy level can a ect the charge calculations.Figure C.2 is the simulation results using the same p-i-n CNFET made from (13,0) nan-otubes described in [3]: the gate length is 16 nm, the oxide thickness is 3.2 nm, the ox-ide relative permittivity is 16 (HfO2), the source and drain contact doping densities are1:5   109 m 1and the source and drain length 50 nm. The bias voltage VGS = 0:6 V andVDS = 0:4 V. The left  gure is setting the division level at E0. When the level is shiftedto the valence band edge, there will be more charges with negative sign (electrons). Thisis shown by the \dip" in the right  gure.69C.3. n-i-n CNFET: with and without BTBTFigure C.2: The source- and drain-originating components of the spatial chargedistribution. The total charge (solid line), QS (dashed line), QD (dotted line). Theleft  gure is the result from equation (C.7) and the right is generated by moving thedivision line to the valence band edge.C.3 n-i-n CNFET: with and without BTBTThe spatial change of the charge is shown in  gure C.3 and C.4. The left and right  guresare for no BTBT and for BTBT, respectively. In  gure C.3, the device used here is thesame to the reference [2] with LS = LD = 30nm, LG = 7nm. The interband tunnellingonly makes up a small portion of the current in this case. However, for a n-i-n CNFET withthe same the con guration as in [4], with LS = LD = 50nm, LG = 16nm, the interbandtunnelling signi cantly changes the charge distribution and this e ect is demonstrated in gure C.4.70C.3. n-i-n CNFET: with and without BTBTFigure C.3: Change in the source- and drain-originating components of the spatialcharge for @VGS = +5 mV at VGS = VDS = 0:7 V. (11,0) CNFET, without (left) andwith (right) interband tunnelling simulation.Figure C.4: Change in the source- and drain-originating components of the spatialcharge for @VGS = +5 mV at VGS = 0:4 V; VDS = 0:6 V. (22,0) CNFET, without(left) and with (right) interband tunnelling simulation.71C.3. ReferencesReferences[1] David Yuk Kei Ko and J. C. Inkson, \Matrix method for tunneling in heterostructures:Resonant tunneling in multilayer systems," Phys. Rev. B, 38(14), 9945-9951, 1988.[2] D.L. Pulfrey, L.C. Castro, D.L John, and M. Vaidyanathan, "Regional signal-delayanalysis applied to high-frequency carbon nanotube FETs", IEEE Trans. Nanotechnol-ogy. vol. 6, 711-717, 2007.[3] S. Poli, S. Reggiani, A. Gnudi, E. Gnani, and G. Baccarani, "Computational study ofthe ultimate scaling limits of CNT tunneling devices", IEEE Transactions on ElectronDevices, vol. 55, pp. 313-321, 2008.[4] Li Chen and D.L. Pulfrey, \Comparison of p-i-n and n-i-n carbon nanotube FETsregarding high-frequency performance", Solid-State Electronics, submitted.72

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