# Carbon nanotube and graphene device modeling and simulation

T ABLE OF CONTENTS page A CKNOWLEDGMENTS.................................4 LIST OF TABLES.....................................7 LIST OF FIGURES....................................8 ABSTRACT........................................13 CHAPTER 1 INTRODUCTION..................................16 1.1 Overview....................................16 1.2 Carbon Nanotube (CNT) vs.Graphene Nanoribbon (GNR)........16 1.3 Nanoelectronics Simulation...........................18 1.3.1 Review of NEGF Formalism......................18 1.3.2 Selfconsistent Simulation Scheme...................19 2 ANALYSIS OF STRAIN EFFECTS IN BALLISTIC CARBON NANOTUBE FETS.........................................22 2.1 Introduction...................................22 2.2 Approach....................................24 2.3 Results......................................27 2.3.1 Transfer Characteristics.........................27 2.3.2 Minimum Current,I min .........................29 2.3.3 On Current,I on .............................30 2.3.4 Intrinsic Delay..............................31 2.4 Conclusions and Discussions..........................33 3 A COMPUTATIONAL STUDY OF VERTICAL PARTIAL GATE CARBON NANOTUBE FETS.................................40 3.1 Introduction...................................40 3.2 Approach....................................42 3.2.1 Device Structure.............................42 3.2.2 Quantum Transport...........................42 3.2.3 Electrostatics..............................44 3.3 Results......................................44 3.3.1 Transfer Characteristics.........................44 3.3.2 Device Operation............................45 3.3.3 Output Characteristics.........................45 3.3.4 Eﬀect of Gate Length..........................46 3.3.5 Eﬀect of Air Pore............................47 3.3.6 Eﬀect of CNT Diameter........................48 5

3.4 Conclusion....................................48 4 EFFECTS OF SCATTERING ON RF PERFORMANCE OF CARBON NANOTUBE FETS.........................................55 4.1 Introduction...................................55 4.2 Approach....................................56 4.3 Results......................................57 4.3.1 On Current...............................57 4.3.2 Charge and Velocity...........................58 4.3.3 Intrinsic Cutoﬀ Frequency and Delay.................59 4.4 Discussions and Conclusions..........................60 5 A COMPUTATIONAL STUDY OF CARBON-NANOTUBE-ENABLED VERTICAL ORGANIC FIELD-EFFECT TRANSISTORS...................64 5.1 Introduction...................................64 5.2 Approach....................................65 5.3 Results......................................68 5.4 Discussions and Conclusions..........................71 6 EFFECT OF NON-IDEALITIES IN GRAPHENE NANORIBBON FETS...77 6.1 Introduction...................................77 6.2 Approach....................................78 6.3 Results......................................79 6.3.1 Ideal Structure..............................79 6.3.2 Atomistic Vacancy...........................79 6.3.3 Edge Roughness.............................80 6.3.4 Ionized Impurity.............................81 6.4 Conclusion....................................82 7 FRAMEWORK OF DEVICE SIMULATIONS FOR NON-UNIFORM GRAPHENE NANORIBBONS...................................89 7.1 Introduction...................................89 7.2 DFT-TOB Model................................89 7.3 Doping Eﬀect in Graphene Nanoribbon by Edge-Chemistry Engineering..91 7.4 DFT-NEGF Method..............................92 7.5 Conclusion....................................93 8 CONCLUSION AND FUTURE WORK......................100 8.1 Conclusion....................................100 8.2 Suggestion for Future Works..........................102 REFERENCES.......................................103 BIOGRAPHICAL SKETCH................................111 6

LIST OF TABLES T able page 1-1 Carbon nanotube (CNT) vs.graphene nanoribbon (GNR)............21 7

LIST OF FIGURES Figure page 1-1 Lattice structure of graphene............................20 1-2 Band structure of graphene,which has a linear E-k relation near the Fermi point.20 1-3 Two special kinds of GNRs.A) N = 10 armchair-edge GNR.B) N = 7 zigzag-edge GNR..........................................20 1-4 Band structure of semiconducting CNT and GNR.A) (13,0) CNT.B) N = 12 armchair-edge GNR..................................21 1-5 Density of states of (13,0) CNT (solid line) and N = 12 A-GNR (dashed line)..21 2-1 Type of applied strains.A) Tensile uniaxial strain.B) Compressive uniaxial strain.C) Torsional strain...............................34 2-2 Real to mode space approach for strained CNT.A) Part of 2-D (n,0) zigzag nanotube lattice in real space.B) Uncoupled 1-D mode space lattice.......34 2-3 Transfer characteristics with diﬀerent strains.A) Uniaxial strain on (16,0) CNT. B) Uniaxial strain on (17,0) CNT.C) Torsional strain on (16,0) CNT.D) Torsional strain on (17,0) CNT.................................35 2-4 Band gap vs.strain.A) Uniaxial strain.B) Torsional strain............35 2-5 Minimum leakage current vs.strain.A) Uniaxial strain.B) Torsional strain...36 2-6 On current vs.strain.A) Uniaxial strain.B) Torsional strain...........36 2-7 On state of 2% strained (16,0) CNTFET compared to the unstrained device. A) Conduction band proﬁle along the channel position.B) Energy-resolved current spectrum........................................37 2-8 Intrisic delay vs.I on /I off under diﬀerent strain.A) Uniaxial strain on (16,0) CNT.B) Uniaxial strain on (17,0) CNT.C) Torsional strain on (16,0) CNT. D) Torsional strain on (17,0) CNT..........................38 2-9 Conduction band proﬁle along the channel position for 2% strained (17,0) CNTFET at a high gate overdrive................................39 2-10 First and second lowest subband of (16,0) CNTs under unstrained (solid line) and 2% strained (dashed line) condition......................39 3-1 Vertical partial-gate CNTFET.A) Cross section of the modeled geometry.B) Device parameters of the nominal device.......................50 8

3-2 Suppression of ambipolar conduction.A) I D -V G characteristics at V D = −0.4 V.B) Band proﬁle along the channel position at V G = −0.9 V.C) Band proﬁle along the channel position at V G = 0.6 V......................51 3-3 Device operations of vertical partial-gate CNTFET.A) P-type operation with V G = V D = −V DD .B) N-type operation with V G = V D = +V DD ..........52 3-4 Output characteristics of partial-gate CNTFET.A) I D -V D curve with diﬀerent gate voltages.B) Band proﬁle along the channel position at V D = −0.4 V and −0.1 V with V G = −0.5 V...............................52 3-5 Eﬀect of gate length in partial-gate CNTFET.A) I D -V G curves with diﬀerent gate lengths.B) Band proﬁle along the channel position for diﬀerent gate lengths.53 3-6 Eﬀect of air pore in partial-gate CNTFET.A) I D -V G curves with and without air pore.B) Electric ﬁeld contour near the source contact with (left) and without (right) air pore.....................................54 3-7 Eﬀect of CNT diameter in partial-gate CNTFET..................54 4-1 Eﬀect of phonon scattering on the current with diﬀerent gate voltages or diﬀerent channel lengths.A) I D vs.V G at V D = 0.5 V at the ballistic limit (the dashed line) and in the presence of phonon scattering (the solid line).The channel length is 100 nm.B) On current vs.channel length at ballistic limit (dashed line) and in the presence of phonon scattering (solid line).The inset sketches the ﬁrst subband proﬁle at the on state............................62 4-2 Eﬀect of phonon scattering on average carrier velocity and electron density.A) Average carrier velocity vs.channel position at V G = V D = 0.5 V at the ballistic limit (dashed lines) and in the presence of phonon scattering (solid lines).B) Electron density vs.channel position at V G = V D = 0.5 V at the ballistic limit (dashed lines) and in the presence of phonon scattering (solid lines)........62 4-3 Eﬀect of phonon scattering on cutoﬀ frequency and intrinsic delay.A) Cutoﬀ frequency vs.channel length at on state.The circles are numerically computed f T at the ballistic limit and the dashed line is a ﬁtting curve of f T = 110 GHz · µm/L ch .The crosses are numerically computed f T in the presence of phonon scattering and the solid line is a ﬁtting curve of f T = 40 GHz · µm/L ch .B) Intrinsic delay vs.channel length at the ballistic limit (the circles) and in the presence of phonon scattering (the crosses).The dashed line is a linear ﬁtting of the ballistic result by τ = L ch ×1.71 ps/µm...................63 5-1 A) Schematic structure of a vertical organic FET.Gate,source,active layer, and drain are vertically stacked up.Porous source electrode of percolating carbon nanotube network allows gate electric ﬁeld to penetrate to the channel region. B) Cross-section of the simulated device structure.................73 9

5-2 A) Cross-sectional potential energy contour at V G = 0 and V D = −5 V.The inset is potential energy at V G = 10 V and V D = −5 V around the CNT (shown by white circle).B) Vacuum energy level shifted by an ionization energy of the active layer along the channel at x = 0 under V G = 0 and 10 V (V D = −5 V). This shows HOMO level (E H ) in the active layer region (y > 2 nm) and the original electron potential energy (E m ) of CNT without gate bias in the source electrode region (0 < y < 2 nm).The inset is a zoom-in plot near the source contact.........................................73 5-3 A) Transfer characteristics for V D = −2.5 and −5 V in a along scale (linear plots are shown in the inset).B) Vacuum energy level shifted by an ionization energy of the active layer along the channel at x = 0 under V G = 0 V.......74 5-4 A) Output characteristics for V D = 0,5,and 10 V.No current saturation is observed within the simulated bias range (upto V D = −30 V).B) Hole density in the channel for V D = −2 and −10 V...........................74 5-5 Channel length scaling:I D -V G curves for diﬀerent channel length in A) a linear and B) a log scale...................................75 5-6 Eﬀect of channel mobility:A) I D -V G curves with diﬀerent mobility of the active layer (the unit is given in cm 2 /V· s).B) Transconductance vs.channel mobility plot.The inset shows S vs.mobility.........................75 5-7 Eﬀect of channel dielectric:A) I D -V G curves for diﬀerent permittivity of active layer.B) Vacuum energy level shifted by an ionization energy of the active layer along the channel at x = 0 under V G = 10 V.....................76 5-8 Eﬀect of gate oxide:A) I D -V G curves for gate dielectric constant of 4 and 16 in a log scale and a linear scale (inset).B) S and g m (inset) are plotted as a function of inverse eﬀective gate oxide thickness:t ox is scaled down from 200 to 50 nm,and ox is varied from 4 to 20.........................76 6-1 Simulated device structure..............................83 6-2 Transfer characteristics A) before and B) after work-function engineering.....83 6-3 Transfer characteristics in the presence of a single lattice vacancy along the transport direction in a log scale (left axis) and in a linear scale (right axis).........84 6-4 A) Conduction band proﬁle along the channel position in the presence of a lattice vacancy at the ON state.B) Energy-resolved current spectrum in the presence of a vacancy near the source.............................84 6-5 A) Transfer characteristics in the presence of a single lattice vacancy along the channel width direction.B) Energy-resolved current spectrum...........85 6-6 Atomistic conﬁguration of a simulated GNR channel in the presence of edge roughness........................................85 10

6-7 A) Transfer characteristics and B) LDOS at the OFF state (V G = 0 V and V D = V DD ) with the GNR channel shown in Fig.6-6.The solid line shows the band proﬁle of the ideal GNRFET.............................85 6-8 Histogram of I on in the presence of edge roughness of GNR by adding or removing carbon atoms with probability P = 0.05.One-hundred samples are randomly generated and simulated.The mean is 6.36 µA,the median is 6.31 µA,and the standard deviation is 2 µA..............................86 6-9 Transfer characteristics in the presence of a positive ionized impurity along the transport direction in a log scale (left axis) and in a linear scale (right axis)...86 6-10 A) Conduction band proﬁle along the channel position in the presence of a positive ionized impurity at the ON state.B) Energy-resolved current spectrum in the presence of a positive ionized impurity near the source...............87 6-11 Histogram of I on in the presence of an ionized impurity with +0.4q at 1.84 ˚ A away from the GNR surface..............................87 6-12 A) Transfer characteristics in the presence of a charge impurity with −q at 0.5 nm away from the GNR surface.B) Conduction band proﬁle along the transport position at the ON state................................88 7-1 A) Quasi-1D graphene nanoribbon (GNR) structure.B) GNR subbands,which comes from the quantum conﬁnement along the width direction..........94 7-2 Eﬀect of edge atoms:a part of GNR and its band structure with A) H,B) OH, and C) F termination.................................95 7-3 Top-of-the-barrier ballistic transistor model.....................95 7-4 Output characteristics of OH-terminated N = 12 armchair-edge GNRFET....96 7-5 P-type doping eﬀect:A) O atom is attached on the edge of H-terminated N = 21 A-GNR.B) It’s band structure and density-of-states (DOS) and projected DOS (PDOS) on C atoms...............................96 7-6 Stability of O-attached A-GNR.Bonding between O and C makes it have lower energy..........................................97 7-7 N-type doping eﬀect:A) N replaces C atom in H-terminated N = 21 A-GNR. B) It’s band structure and density-of-states (DOS) and projected DOS (PDOS) on C atoms.......................................97 7-8 Stability of N-substitutional A-GNR.Bonds between N and C make it have lower energy..........................................98 11

7-9 GNR with one impurity atom.Blocks for the input of ab-initio calculation are shown.α 0 and α 1 correspond to on-site blocks of uniform GNR and one with impurity,respectively,and β 00 is for coupling of uniform GNR and β 01 and β 10 are couplings between the uniform GNR and one with impurity..........98 7-10 A) Transmission and B) local density-of-states (LDOS) for the structure shown in Fig.7-9.......................................99 12

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy CARBON NANOTUBE AND GRAPHENE DEVICE MODELING AND SIMULATION By Young Ki Yoon December 2008 Chair:Jing Guo Major:Electrical and Computer Engineering The performance of the semiconductors has been improved and the price has gone down for decades.It has been continuously scaled down in size year by year,and now it encounters the fundamental scaling limit.We,therefore,should prepare a new era beyond the conventional semiconductor technologies.One of the most promising devices is possible by carbon nanotube (CNT) or graphene nanoribbon (GNR) in terms of its excellent charge transport properties.Their fundamental material properties and device physics are totally diﬀerent to those of the conventional devices.In this nano-regime,more sophisticated device modeling and simulation are really needed to elucidate nano-device operation and to save our resources from errors.The numerical simulation works in this dissertation will provide novel view points on the emerging devices. In this dissertation,CNT and GNR devices are numerically studied.The ﬁrst part of this work is on CNT devices,and a common structure of CNT device has CNT channel,metal source and drain contacts,and gate electrode.We investigate the strain, geometry,and scattering eﬀects on the device performance of CNT ﬁeld-eﬀect transistors (FETs).It is shown that even a small amount of strain can result in a large eﬀect on the performance of CNTFETs due to the variation of the bandgap and band-structure-limited velocity.A type of strain which produces a larger bandgap results in increased Schottky barrier (SB) height and decreased band-structure-limited velocity,and hence a smaller minimum leakage current,smaller on current,larger maximum achievable I on /I off ,and 13

larger intrinsic delay.We also examine geometry eﬀect of partial gate CNTFETs.In the growth process of vertical CNT,underlap between the gate and the bottom electrode is advantageous for transistor operation because it suppresses ambipolar conduction of SBFETs.Both n-type and p-type transistor operations with balanced performance metrics can be achieved on a single partial gate FET by using proper bias schemes.The eﬀect of phonon scattering on the intrinsic delay and cut-oﬀ frequency of Schottky barrier CNTFETs is also examined.Carriers are mostly scattered by optical and zone boundary phonons beyond the beginning of the channel.The scattering has a small direct eﬀect on the DC on current of the CNTFET,but it results in signiﬁcant decrease of intrinsic cut-oﬀ frequency and increase of intrinsic delay.Semiconducting CNT is useful for the channel in CNTFETs,whereas metallic CNT can be used as an electrode.If a porous CNT ﬁlm is used as a source electrode,vertical thin-ﬁlm transistors (TFTs) can be constructed. Vertical organic FET (OFET) shows clear transistor switching behavior allowing orders of magnitude modulation of the source-drain current even in the presence of electrostatic screening by the source electrode.The channel length should be carefully engineered due to the trade-oﬀ between device characteristics in the subthreshold and above-threshold regions. The second subject is device simulations of GNRFETs.Even though GNR is also graphene-based quasi-1D nanostructure like CNT,the diﬀerences in shape,boundary condition,and existence of edges and dangling bonds make it operate in a diﬀerent way. Atomistic 3D simulation study of the performance of GNR SBFETs is presented.The impacts of non-idealities on device performance have been investigated.The edges of GNR,which do not exist in CNT,can be advantages or disadvantages.If an appropriate control by diﬀerent edge atoms is possible,it would be deﬁnitely positive.Totally new electronic band structure is obtained by diﬀerent edge-termination atoms.In addition, only a fraction of impurity atom can also much aﬀect on the material properties of GNR.In order to perform device simulations of non-uniform GNR devices,multiscale 14

sim ulation scheme can be used in non-equilibrium Green’s function (NEGF) formalism and density-functional method. 15

CHAPTER 1 INTRODUCTION 1.1 Overview Carbon-based nanostructure has attracted strong interests during the last decade. Since carbon nanotubes (CNTs) were discovered by Iijima in 1991 [1],a number of active researches have been achieved to explore the fundamental material properties of CNTs and its possible applications [2].Recently,in addition,graphene and graphene nanoribbon (GNR) have been strong topics of researches since the graphene monolayer was experimentally obtained for the ﬁrst time in 2004 [3].CNTs and GNRs are strong candidates for future nanoelectronic applications due to their excellent electric characteristics as follows.The graphene-based nanostructures have extremely high carrier mobilities [3,4].The band structures are symmetric and direct,which is useful for optoelectronic applications.They can be metallic or semiconducting depending on the structures.There is no dangling bonds on the surface,and they are amenable to the deposition of high-κ gate insulator.Signiﬁcant eﬀorts have been devoted to the applications of CNTs and GNRs for the last decade.The ﬁrst type of CNT and GNR ﬁeld-eﬀect transistors (FETs) were demonstrated in 1998 and 2007,respectively [5–8].Even though a great deal of eﬀort has been made to understand the device physics of nanodevices based on CNTs or GNRs, many of transistor operations are still unclear under various conditions. In this study,device simulations are performed to understand the fundamental device operation of CNTFETs or GNRFETs and to elucidate the key factor controlling the device performance.Our numerical simulation will also help experimentalists save their intensive eﬀorts in the fabrication by sharing the understanding on nanodevice operations. 1.2 Carbon Nanotube (CNT) vs.Graphene Nanoribbon (GNR) Carbon-based materials are attracting strong interests for potential nanoelectronic applications due to their excellent electrical properties.Among them,quasi-one-dimensional (1D) nanostructures such as carbon nanotubes (CNTs) or graphene nanoribbons (GNRs) 16

are remarkable candidates for future nanoelectronics,which can be considered as 1D nanostructures derived from two-dimensional (2D) graphene.Therefore,it would be very helpful to review the properties of graphene brieﬂy prior to paying attention to 1D nanostructures. Graphene is an atomistically thin 2D monolayer of carbon atoms as shown in Fig. 1-1.Band structure of graphene can be calculated by a tight-binding (TB) model in a p z orbital basis set per carbon atom by considering its nearest neighbors only [9].First, Hamiltonian matrix can be written as H(

k) = 0 t

1 +e −i

k·a 1 +e −i

k·a 2

t

1 +e i

k·a 1 +e i

k·a 2

0 ,(1–1) where t ≈ −3 eV is tight-binding parameter between carbon atoms and a 1 and a 2 are basis vectors of graphene.Then,the band structure of graphene can be obtained as E(

k) = ±

t(1 +e i

k·a 1 +e i

k·a 2 )

,(1–2) which can be simpliﬁed by Taylor expansion near the Fermi point as E(

k) = ±v F

k −

k F

,(1–3) where v F ≈ c/300 is Fermi velocity.The band structure of graphene near the Fermi point is shown in Fig.1-2,where there is no band gap between the conduction band and the valence band.Pure graphene itself,therefore,is not suitable for nanoelectronic applications.For a graphene-based material,quantum conﬁnement can give rise to a band gap,which will be shown in the following two cases:CNT and GNR. CNT and GNR are quasi-1D nanostructures derived from 2D graphene.CNT is conceptually a rolled-up graphene,and a chiral vector c = na 1 +ma 2 describes its unique structure along the circumferential direction of the tube.The periodic boundary condition of CNT is characterized by

k · c = 2πq,where q is a quantum number.Due to the 17

b oundary condition,

k is quantized and,CNT is metallic if (n −m) is a multiple of 3,and it is semiconducting otherwise. GNR is a narrow strip of graphene,which can be obtained by state-of-the-art patterning technique like e-beam lithography.The structure of GNR can be deﬁned with a vector c = n a 1 2 +m

a 2 2 along the width direction of the GNR.Figure 1-3 shows two special cases of GNRs:(i) armchair-edge GNR (A-GNR) and (ii) zigzag-edge GNR (Z-GNR). According to the simple tight-binding method,Z-GNR is always metallic,and A-GNR can be metallic or semiconducting depending on structure (it is metallic if n+1 is a multiple of 3,and semiconducting otherwise).In comparison to CNT,GNR has inﬁnite-wall boundary condition.Table 1-1 is a brief summary of the comparison of CNT and GNR. Band structures of semiconducting CNT and GNR are compared in Fig.1-4,which shows the same band gap for both (13,0) CNT and N = 12 A-GNR under the simple tight-binding method.However,CNT and GNR with the same band gap show roughly a factor of 2 diﬀerent density-of-states (DOS) as compared in Fig.1-5,which is caused by the band degeneracy of CNT.Even though the simple tight-binding method predicts the same band gap for the above semiconducting CNT and GNR,the band gap of N A-GNR has a deviation from E g of (n + 1,0) CNT due to the edge bond relaxation [10]. Furthermore,the band gap of Z-GNR is also slightly opened and it has a band gap owing to the spin-polarized edge states [10]. 1.3 Nanoelectronics Simulation 1.3.1 Review of NEGF Formalism The non-equilibrium Green’s function (NEGF) formalism,which is a useful bottom-up approach for nanoscale device simulations in terms of atomistic description and quantum eﬀects,solves Schr¨odinger equation under non-equilibrium bias conditions [11,12].The validity of NEGF formalism has already been shown from the previous studies in various areas [13–17].In this section,brief summary of NEGF method is given. 18

The retarded Green’s function of the channel material is given by G r =

(E +i0 + )I −H−Σ 1 −Σ 2

−1 ,(1–4) where H is the Hamiltonian matrix of the channel,Σ 1 and Σ 2 are self-energies of the source and drain contacts,respectively. The charge density can be computed as ρ i (x) = (−q)

+∞ −∞ dE · sgn[E −E N (x)] {D 1 (E,x)f (sgn[E −E N (x)](E −E F1 )) + D 2 (E,x)f (sgn[E −E N (x)](E −E F2 ))},(1–5) where sgn(E) is the sign function,E F1,F2 is the source (drain) Fermi level,and D 1,2 (E,x) is the local density-of-states due to the source (drain) contact,which is computed by the NEGF method.The charge neutrality level,E N (x),is at the middle of band gap because the conduction band and the valence band of the CNT or GNR are symmetric. The source-drain current is calculated as I D = q h

∞ −∞ T race

Γ 1 G r Γ 2 G r+

(f 1 −f 2 )dE (1–6) per spin per valley,where Γ 1,2 = i

Σ 1,2 −Σ + 1,2

are broadening functions of the source (drain) contacts,f 1,2 are equilibrium Fermi functions of source (drain) contacts [11]. 1.3.2 Selfconsistent Simulation Scheme For a reliable device simulation,self-consistent scheme is crucial because of the following aspects.(i) To solve the transport equation for a charge density,the electrostatic potential should be included in the diagonal components of the Hamiltonian matrix. (ii) Charges in the device,in turn,aﬀect the electric ﬁeld in the device.The iteration between the transport equation and the Poisson equation,therefore,is necessary until the self-consistency is achieved.Once self-consistency is satisﬁed,correct charge density and souce-drian current can be obtained by NEGF formalism using the self-consistent potential. 19

a 1 a 2 Figure 1-1.Lattice structure of graphene Figure 1-2.Band structure of graphene,which has a linear E-k relation near the Fermi point. A B Figure 1-3.Two special kinds of GNRs.A) N = 10 armchair-edge GNR.B) N = 7 zigzag-edge GNR. 20

kt/kt max 1 0.5 0 0.5 1 E (eV) 1 0 5 0 5 1 0 A kt/kt max 1 0.5 0 0.5 1 E (eV) 1 0 5 0 5 1 0 B Figure 1-4.Band structure of semiconducting CNT and GNR.A) (13,0) CNT.B) N = 12 armchair-edge GNR. −2 0 2 0 1 2 x 10 8 E [eV] DOS(E) [eV.cm] − 1 Figure 1-5.Density of states of (13,0) CNT (solid line) and N = 12 A-GNR (dashed line). Table 1-1:Carbon nanotube (CNT) vs.graphene nanoribbon (GNR) CNT GNR Deﬁnition Rolled-up graphene Patterned graphene Shape Tube Ribbon Vector c = na 1 +ma 2 c = n a 1 2 +m a 2 2 Special cases m= 0:Zigzag CNT m= 0:Armchair-edge GNR m= n:Armchair CNT (metallic) m= n:Zigzag-edge GNR Boundary condition Periodic b.c.Particle in a box 21

CHAPTER 2 ANALYSIS OF STRAIN EFFECTS IN BALLISTIC CARBON NANOTUBE FETS The eﬀect of uniaxial and torsional strain on the performance of ballistic carbon nanotube (CNT) Schottky barrier (SB) ﬁeld-eﬀect transistors (FETs) is examined by self-consistently solving the Poisson equation and the Schr¨odinger equation using the non-equilibrium Green’s function (NEGF) formalism.A mode space approach can be used to reduce the computational cost of atomistic simulations for strained CNTs by orders of magnitude.It is shown that even a small amount of uniaxial (< 2%) or torsional (< 5 o ) strain can result in a large eﬀect on the performance of the CNTFETs due to the variation of the band gap and band-structure-limited velocity.Semiconducting CNT channels with diﬀerent chiralities are inﬂuenced in drastically diﬀerent ways by a certain applied strain,which is determined by a (n-m) mod 3 rule.In general,a type of strain which produces a larger band gap results in increased Schottky barrier height and decreased band-structure-limited velocity,and hence a smaller minimum leakage current,smaller on-current,larger maximum achievable I on /I off ,and larger intrinsic delay.The other type of strain that reduces the band gap results in the opposite eﬀect on the device performance metrics of CNTFETs. 2.1 Introduction Strain engineering has been extensively explored for boosting the performance of nanoscale silicon ﬁeld-eﬀect transistors (FETs) [18,19].Strain also plays an important role in the electrical properties of carbon nanotubes (CNTs),and has been a subject of strong research interest.The pioneering experimental works [20–23] showed that the conductance of a CNT can be varied by orders of magnitude by applying a local strain using a STM tip,and the phenomenon was subsequently investigated by theoretical calculations [24].With the recent advance on CNT devices [25,26],uniaxial or torsional strain can be applied throughout the whole CNT channel as a potential approach to improve the device performance.A CNT device under uniaxial strain has been 22

in vestigated by stretching a ﬂexible substrate,on which the CNT device was fabricated [27].The results can be important for the applications such as ﬂexible electronics [28] and nanoscale pressure sensors.A CNT electromechanical device under torsional strain has been examined by pressing a pedal that is attached to a suspended CNT channel [29]. In addition to the controlled ways of applied strains to the CNT channel,strain can also be unintentionally induced in the device fabrication process.Previous theoretical studies have been focusing on local deformation of CNTs or the eﬀects of strain at the material level [9,24,30–32],and the strain eﬀects on the device performance are unclear.With the signiﬁcant advance on experimental techniques for applying uniaxial and torsional strains, it is important to explore how strain aﬀects the performance of CNTFETs and whether it is possible to use strain for improving the performance of CNTFETs. In this work,the eﬀects of uniaxial and torsional strains on the characteristics of ballistic CNT Schottky barrier (SB) FETs are examined using the non-equilibrium Green’s function (NEGF) formalism.A mode space approach previously developed for unstrained CNTs [16] can be extended to a CNT channel under uniform uniaxial or torsional strain, which reduces the computational cost of the atomistic quantum simulation by orders of magnitude.We show that even a small uniaxial (< 2%) or torsional (< 5 o ) strain can result in a large eﬀect on the device performance metrics,such as the on-current,the minimum leakage current,and the intrinsic delay,due to the variation of the band gap and band-structure-limited velocity.Semiconducting CNT channels with diﬀerent CNT chiralities respond in drastically diﬀerent ways to the same type of strain.For example, the tensile strain on a (17,0) CNT causes a larger on-current and a smaller intrinsic delay,but the same strain on a (16,0) CNT results in the opposite eﬀect.The results indicate the important eﬀects of uniaxial and torsional strain on the characteristics of CNTFETs and the possibility to improve the performance of CNTFETs by applying a carefully designed type of strain.At the same time,careful trade-oﬀ between diﬀerent device performance metrics must be taken care of. 23

2.2 Approach We simulated a coaxially gated CNT SBFET at room temperature (T=300 K).The nominal device has a 3 nm HfO 2 gate oxide with a dielectric constant of 16.For the channel material,40 nm-long strained (16,0) or (17,0) CNT is used.Two diﬀerent types of strains are applied on the channel CNT.Figures 2-1A and 2-1B show uniaxial tensile and compressive strain,respectively,and the uniaxial strain is described by the percentage of the length change.Figure 2-1C shows torsional strain and γ represents the torsional strain by degree.Within the range of strain we considered (< 2%),CNTs show a linear response to the uniaxial strain [33],and hence strain energy per carbon atom is assumed to be equal for all atoms.Severe torsion may occur ﬂattening on tubes [34],which, however,is neglected in this study because our range of torsional strain (< 5 o ) is much smaller than the predicted critical angle (70 o and 60 o for 40 nm-long (16,0) and (17,0) CNT,respectively) that changes the atomistic conﬁguration.A power supply voltage of 0.4 V is used.The metal source/drain is directly attached to the CNT channel,and the Schottky barrier height between the source/drain and the channel is Φ Bn = E g /2,where E g is the band gap of channel CNT.The metal contact Fermi level lies in the middle of the CNT band gap.The gate leakage current is neglected for simplicity. The DC characteristics of ballistic CNTFETs are simulated by solving the Schr¨odinger equation using the non-equilibrium Green’s function (NEGF) formalism self-consistently with the Poisson equation.A tight binding (TB) Hamiltonian with a p z orbital basis set is used to describe an atomistic physical observation of the channel.The atomistic treatment is computationally expensive in real space,but signiﬁcant saving of computational cost can be achieved by the mode space approach [16]. Figure 2-2A shows a part of (n,0) CNT lattice in real space.t 1 ,t 2 ,and t 3 are binding parameters between the nearest neighbor carbon atoms;t i = t 0 (r 0 /r i ) 2 [35],where subscription i=1,2,and 3,t 0 is binding parameter between carbon atoms of unstrained CNT,r 0 is bonding length between carbon atoms of unstrained CNT,and r i is bonding 24