Investigation of chirality dependence of carbon nanotube-based ring oscillator

Yaser Mohammadi Banadaki, Safura sharifi, Ashok Srivastava*

Division of Electrical and Computer Engineering

Louisiana State University

Baton Rouge, LA 70803, U.S.A. *ashok@ece.lsu.edu

ABSTRACT

Carbon nanotube (CNT)-based integrated circuits including CNT field effect transistors (FET) and CNT

interconnects are a promising emerging technology. In this paper, we have designed and simulated a single-

walled CNT-based ring oscillator (RO) to investigate the chirality dependence of oscillation frequency using

a numerical quantum-based model of CNT FET with 5 nm gate length and a compact diffusive model for

CNT interconnects. We have obtained oscillation frequency of 25 GHz assuming contact resistance of

24𝐾𝛺, which reduces by ~%8.5 and ~%10 for 1 nm increase in CNT chiral diameter of zigzag CNT FET

and 1𝜇𝑚 metallic CNT interconnects between inverter stages, respectively.

Keywords: Single-walled carbon nanotube ring oscillator; Field effect transistors; Interconnects.

INTRODUCTION

The International Technology Roadmap for

Semiconductors (ITRS) [1] has mentioned future

technological and intrinsic device difficulties of

scaling Si MOSFETs in CMOS technologies

enforcing the semiconductor industry to explore the

use of novel materials and devices. Some of the

most promising alternatives for emerging VLSI

integrated circuits have been categorized under

“Carbon-based Nanoelectronics” such as CNT FET

[2-5] or the graphene-nanoribbon FET [6-7].

Basically, a single-walled CNT is a rolled-up sheet

of graphene discovered in 1991 by Iijima [8]. A

SWCNT offers high mechanical strength, high

carrier density, high thermal and electrical

conductivity with near ballistic transport. CNT FET

has similar current-voltage characteristics as that of

Si MOSFETs while the complementary design can

be fabricated using identical transistors leading to

less fabrication processes for designing CNT-based

integrated circuits [9]. Carbon nanotubes can be

either metallic or semiconducting depending on

rolling direction. In a CNT-based circuit, the

semiconducting and conducting CNTs are used for

transistors and for interconnects, respectively.

Recently, the experimental reports [10, 11] reveal

that the speed of CNT growth has the chirality

dependence developing the ability to selectively

grow CNT with specific chirality. This has

significantly influenced on the characteristics of

CNT device and interconnects urging to be

investigated in designing CNT-based integrated

circuits [12, 13] such as the ring oscillator, which

has been the purpose of this paper.

A ring oscillator is the ultimate test circuitry for

new materials and a key building block in radio

frequency, digital electronic and optical

communication systems for the frequency

translation of information signals, channel selection

and time reference [14, 15]. Javey et al have

fabricated the first complementary SWCNT-based

3-stages RO with the CNT length and diameter of 3

µm and 2 nm obtaining the oscillation frequency of

∼220 Hz [16]. IBM has announced the fabrication

of high performance CNT-based 5-stages RO with

the CNT length and diameter of 18 µm and 2nm,

respectively, which results in oscillation frequency

of 52 MHz [17]. A seven-stage CNT-based RO has

also been proposed and simulated for the CNT FET

with the gate length of 200 nm and the oscillation

frequency of 3.75 MHz [9]. As Moore’s law

requires shrinking the channel length of transistors

below 10nm within the next decade, an important

next step would be predicting and exploring the

characteristics of CNT-based integrated circuits

with regard to physical parameters of CNT such as

its chirality. Thus, a simulation approach needs to

consider the quantum confinement of the tube

circumferential direction and quantum tunneling

along the channel [18]. The non-equilibrium

Green’s function (NEGF) formalism [18-22] is a

strength simulation method which fully treats the

quantum phenomena such as the tube confinement,

quantum tunneling and contact effect leading to a

very accurate extraction of chirality dependence of

few-nanometer CNT channels. In this case, the

length of semiconducting CNT is usually less than

its mean-free-path which allows for ballistic

simulation of carriers transport in CNT FETs. For

interconnects in VLSI design, Cu has much less

mean-free path than a metallic CNT and suffers

from electron surface scattering, grain-boundary

scattering, electromigartion and void formation [23-

27]. Thus, the application of metallic CNTs for

interconnects along with CNT FETs can increase

the performance of VLSI circuits.

In this paper, we have investigated the chirality

dependence of CNT-based ring oscillator using a

numerical ballistic model for CNT FETs and a

compact diffusive model for CNT interconnects.

The arrangement of the paper is as follows; in

Section 2, the gate-all-around CNT FET structure

has been modeled by solving the Poisson and

Schrödinger equations within the NEGF formalism.

The current-voltage characteristics and

corresponding voltage transfer characteristics of the

CNT FET-based inverter have been obtained as a

function of the chirality of few nanometer CNT

channels. In section 3, the semi-classical model of

CNT interconnect has been introduced for a metallic

single-walled CNTs based on our previously

developed one-dimensional fluid model of

electronic transport considering scattering

mechanisms in long conducting CNT [24]. The

equivalent transmission model of a metallic

SWCNT isolated above a ground plane consists of

circuit elements distributed along its length

including magnetic and kinetic inductance, quantum

and electrostatic capacitances, and a non-linear

resistance resulting from acoustic and optical

phonon scatterings. Section 4 is devoted to the

simulation of CNT-based RO circuits, where

semiconducting and metallic CNTs have been used

as transistors and interconnects between inverters,

respectively. The obtained data from the device-

level simulation of CNT FETs and the model of

CNT interconnect have been introduced into SPICE

circuit simulator. The multi-stage CNT-based RO

has been simulated to explore the chirality

dependence of the oscillation frequency as a

fundamental physical parameter of CNT FETs and

CNT interconnects in designing a CNT-based RO

circuits.

Figure 1. (a) The 3-D schematic view of gate-all-around

CNTFETs. LD, LG and LS are the lengths of drain, gate and

source where the spaces between the regions are due to

visibility purpose and assumed zero in the simulations. tox and

D are the insulator thickness and the diameter of

semiconducting CNT corresponding to chirality n=3k+1 (k is

positive integer), respectively. (b) The atomic configuration in

real space and corresponding conceptual configuration in

mode space approaches. The x arrows are the direction along

the CNT FET device, c and kc are the circumferential

directions in real space and reciprocal space, respectively.

Coupling parameters between the nearest neighbor carbon

rings in mode space are shown with t and b2q, respectively.

CNT FET

Figure 1(a) shows a three-dimensional (3-D)

schematic view of the gate-all-around CNTFETs,

where the semiconductor zigzag nanotube has been

used in the channel with chirality n=3k+1 (k is

positive integer) [13]. The source and drain regions

are heavily doped and the metallic coaxial gate

manipulates the carrier transport in CNT channel.

The proposed CNT FET can be modeled

accurately using NEGF formulism, which can treat

quantum-mechanical confinement, reflection, and

tunneling [20]. The CNT FET was simulated by

converging an iterative procedure, where the

Poisson and Schrödinger equations are solved self-

consistently using NEGF formulism [19-22].

The gate-all-around CNT transistor results in

invariant potential and charge density around the

nanotube. The electrostatic potential 𝑈𝑖 along the

CNT can be determined by 2-D Poisson equation as

follows,

∇2Ui(r, x) = −q

ε[p(xi) − n(xi) + ND

+ − NA−] (1)

where ND+and NA

− are the doping profile of ionized

donor and acceptor, respectively and p(xi) and n(xi)

are hole and electron density distribution which are

updated using the calculated NEGF formulism in

iterative procedure. The retarded Green’s function

is given by,

𝐺(𝐸) = [(𝐸 + 𝑖η+)𝐼 − 𝐻 − ΣS − ΣD]−1 (2)

where E is the energy, η+ is an infinitesimal positive

value, I is the identity matrix and H is the

Hamiltonian matrix for an isolated CNT channel. ΣS

and ΣD are the self-energy matrices of the source

and drain, which describe how the ballistic channel

couples to the contacts. The mean-free-path of

nanotube is around hundreds of nanometers at room

temperature which allows near ballistic carriers

transport. In addition the carrier transport can be

describe in terms of the circumferential modes

known as “Mode space approach” [18, 19] since the

potential variation around coaxial-gated nanotube is

small. In other word, the periodic boundary

conditions kcC = 2πq , where C is the

circumference of the nanotube and q is the angular

quantum number, mathematically allows to

decouple the two dimensional nanotube lattice of a

(n,0) zigzag nanotube into n-uncoupled one-

dimensional mode space lattices as shown in

Fig.1(b). This reduces the size of the Hamiltonian

matrix and the corresponding computational cost.

By constructing two different types of rings along

the tube, the Hamiltonian matrix for the qth

subband

can be calculated as follows,

𝐻 =

[

𝑈1

𝑏2𝑞

𝑏2𝑞

𝑈2

𝑡

0

𝑡

𝑈3

𝑏2𝑞

⋱

0

𝑡

𝑈𝑁−1

𝑏2𝑞

𝑏2𝑞

𝑈𝑁 ]

𝑁×𝑁

(3)

where b2q = 2tcos(πq/n) and t are coupling

parameters between the nearest neighbor carbon

rings, and N is the total number of carbon rings

along CNT device. The diagonal elements Ui’s are

the electrostatic potential at the ith

carbon ring which

obtained from Eq. (1).

Corresponding to boundary conditions of the

Schrödinger equation, the source self-energy matrix

has only non-zero value in the (1, 1) element given

by,

Σ𝑆(1,1) =

(E−U1)2+t2+b2q2 ±√[(E−U1)2+t2+b2q

2 ]2−4(E−U1)2t2

2(E−U1) (4)

By replacing U1 with UN, the drain self-energy

matrix with non-zero value in the (N, N) element

can be calculated using an equation similar to

Eq.(4). The level broadening of source and drain

can be obtained using the self-energies of the

corresponding contacts as follows,

ΓD(S)(E) = i[ΣD(S)(E) − ΣD(S)+ (E)] (5)

where ΣD(S)+ is the Hermitean conjugate of ΣD(S)

.

The electron and hole correlation functions

𝐺 𝑛(𝑝)(𝐸) can be calculated by,

Gn(E) = G(E)Σin(E)G+(E) (6a)

Gp(E) = G(E)Σout(E)G+(E) (6b)

where Σ𝐷(𝑆)𝑖𝑛 = ΓD(S)fD(S) and Σ𝐷(𝑆)

𝑜𝑢𝑡 = ΓD(S)(1 −

fD(S)) are in(out) scattering functions for drain

(source) contact couplings, respectively, which are

the level broadening weighted by Fermi distribution

function fD(S) . The induced electron and hole

distributions on carbon ith

ring can be calculated

using the following equations,

n(xi) =4

∆x∫

Gj,jn (E)

2π

+∞

Em(j)dE (7a)

p(xi) =4

∆x∫

Gj,jp

(E)

2π

Em(j)

−∞dE (7b)

where Em is the midgap energy and the factor of four

is due to the two spin states and the twofold valley

degeneracy. The calculated p(xi) and n(xi) have been

applied to Poisson equation to calculate 𝑈𝑗𝑛𝑒𝑤 and

the corresponding Hamiltonian in the self-consistent

iterative loop. Once the algorithm converges, the

transmission coefficient T(E) and the corresponding

current can be calculated,

T(E) = trace(ΓS(E)G(E)ΓDG+(E)) (8)

I =4e

h∫ T(E)[f(E − EFS) − f(E − EFD)]

+∞

−∞dE (9)

where h is the Plank constant, f is the Fermi-Dirac

distribution and EFS and EFD are the Fermi energy

levels in the source and drain regions. The details of

the quantum transport calculation of CNT-based

field effect transistor can be followed in [18-20].

The drain-source currents of the proposed CNT

FETs have been obtained using NEGF model for

different chirality (n=3k+1) from 7 to 28. Figure

2(a) and 2(b) show the chirality dependence of

drain current as a function of drain voltage and gate

voltage, respectively. Increasing the chirality leads

to the increase in drain-source current due to the

contribution of more quantum channels

corresponding to the angular quantum number q in

equation kcC = 2πq . By intersecting the current-

voltage characteristics of n- and p-type CNT FETs,

the voltage transfer characteristic (VTC) of

complementary CNT FET-based inverter can be

obtained as a function of chirality as shown in Fig.

3(a). The VTC curves of low-chirality CNT FETs

have sharp transition from ON-state to OFF-state

while the non-linearity in VTC curves decreases by

increasing chirality of CNT FETs. The current of

CNT FET-based inverter as a function of input

voltage corresponding to VTC curve has been

shown in Fig. 3(b). Increasing the chirality from 7

to 28 leads to four orders of magnitude increase in

the maximum current of inverter from ~5.7 ×10−9𝐴 to ~4.2 × 10−5𝐴.

Figure 2. The chirality dependence of CNT FET

characteristics. (a) the IDS-VDS and (b) the IDS-VGS

characteristics as a function of CNTFET chirality from n=7 to

25. The device geometries of the CNT FETs are tox = 2nm,

LG = 10nm and LD = LS = 10nm , where the dielectric

constant is assumed εox=16.

Figure 3. (a) Voltage transfer characteristic of CNTFET-based

inverter using NEGF model for different chirality (n,0), (b) the

current of CNTFET-based inverter as a function of input

voltage for different chirality (n,0) and (c) Schematic of

CNTFET-based inverter. The device geometries and

parameters of the CNT FETs are kept as mentioned in Fig. 2.

CNT INTERCONNECT

A metallic CNT has less problem with

electromigartion and void formation and exhibits the

high thermal and electrical conductivity as well as

mechanical properties [23, 24]. Electrons in a Cu

interconnect not only have much less mean-free path

than a CNT interconnect, but also suffer from

electron surface scattering and grain-boundary

scattering leading to increasing its resistivity. These

performances introduce a metallic CNT interconnect

as a promising alternative to current Cu interconnect

in VLSI technology [1].

Srivastava et al. [25] have been developed the

semi-classical model for a conducting metallic

SWCNT using one-dimensional fluid model. The

equivalent circuit model of a metallic SWCNT

isolated above a ground plane consists of circuit

elements distributed along its length as shown in fig.

4(a). The inductance (L) consists of the series of

magnetic inductance (LM) and kinetic inductance

(LK). The magnitude of magnetic inductance is

logarithmically sensitive to the ratio of distance

between nanotube and the ground ( tins ) and

nanotube diameter (Din) given by,

LM =μ

2πln(

tins

Din) (10)

The kinetic inductance is dominant inductance

with several orders of magnitudes and thus plays a

vital role in high frequency applications, which is

contrary to macroscopic long thin wires with

relatively larger magnetic inductance. The kinetic

inductance is given by,

Lk ≈πℏ

e2vF (11)

where ℏ is reduced Plank constant.

Figure 4. (a) Transmission line model of a SWCNT

interconnect. The inductance (L) consists of the series of

magnetic (LM) and kinetic (LK) inductances. CE and CQare the

electrostatic and quantum capacitance. Rc is the contact

resistance, which is assumed equal to zero in figures 5-7 while

its dominant effect is investigated in figure 8. R(V, Din) is the

resistance of CNT interconnect which depends on voltage

across the interconnect and CNT diameter, (b) Equivalent

capacitance of a CNT interconnect with the length of Lin =1 μm as a function of chirality (n,n) and (c) Resistivity of a

CNT interconnect with the length of Lin = 1 μm as a function

of voltages and its chirality (n,n).

The capacitance also consists of quantum

capacitance of the nanotube and electrostatic

capacitance between a wire and a ground plane. The

quantum capacitance is due to the Pauli Exclusion

Principle in quantum electron gas of a metallic

SWCNT interconnect given by,

CQ ≈e2

πℏvF≈ 100 aF/μm (12)

Similar to the magnetic inductance, the

electrostatic capacitance depends on the insulator

thickness and nanotube diameter while its

contribution in calculating equivalent capacitance is

important and given by,

CE ≈2πε

ln(tinDin

) (13)

The increase of equivalent capacitance as a

function of chirality is shown in Fig 4(b). The

resistance in the equivalent transmission model has

the voltage (V) and diameter (Din ) dependence,

which can be determined by the effective mean free

path of electron in a metallic SWCNT interconnect

(λeff) considering acoustic and optical phonon

scattering mechanisms. The spontaneous scattering

lengths for emitting an optical phonon (λop) and

acoustic phonon (λac) have diameter dependence,

which can be estimated as follows [5],

λop = 56.4 Din (14)

λac =400.46×103Din

T (15)

In order to calculate the effective mean free path,

three scattering lengths needs to be calculated for

optical phonon scattering, which are optical phonon

absorption (λop,abs), optical phonon emission for the

absorbed energy ( λop,emsabs ), and optical phonon

emission for the electric field across the SWCNT

length (λop,emsfield ) as follows [24],

λop,abs = λopNop(300)+1

Nop(T) (16)

λop,emsabs = λop,abs +

Nop(300)+1

Nop(T)+1λop (17)

λop,emsfield =

ℏωop−KBT

qV

Lin

+Nop(300)+1

Nop(T)+1λop (18)

where Nop = 1 [1 + exp (ℏωop KBT⁄ )]⁄ is the

optical phonon occupation in Bose-Einstein

statistics, KB is the Boltzman constant, ℏωop is

optical phonon energy and Lin is the CNT

interconnect length. The effective mean free path

and the corresponding resistance can now be

calculated by Matthiessen’s rule as follows,

1

λeff=

1

λac+

1

λop,emsabs +

1

λop,emsfield +

1

λop,abs (19)

R(V, Din) =R0

4

Lin

λeff (20)

where R0 ≈ πℏ 2e2⁄ ≈ 12.9 kΩ is quantum

resistance. Figure 4(c) shows the voltage and

diameter (interconnect chirality) dependence of the

resistance in the equivalent transmission model for a

CNT interconnect with the length of Lin = 1 μm.

The calculated resistance of SWCNT interconnect is

highly sensitive to the value of voltage across the

tube, in which the dependence increases by

decreasing chirality. The increase in chiral value

leads to the increase in the number of energy bands

in energy-wavevector diagram of the metallic CNT

enhancing the electron current capability. By

increasing chiral value, the equivalent capacitance

increases while the resistance decreases depending

on voltage across the tube, which results in a RC

delay and consequently variation in oscillation

frequency of CNT-based RO.

CNT-BASED RING OSCILLATOR

In a ring oscillator, the input of each inverter is the

output of the previous inverter, closing the ring by

returning the output of the last inverter to the first

stage. The cascade combination of inverters results

in sequentially change of logic state and the

corresponding oscillation depending on propagation

delay (𝑡𝑝) in voltage switching between successive

inverter stages. Figure 5(a) shows the schematic of

the multi-stage RO, where semiconducting and

metallic CNTs have been used as transistors and

interconnects between inverters, respectively. The

developed analytical or numerical model of a nano-

meter device and interconnect can be incorporated

in SPICE using Verilog-AMS [28] or SPICE

elements [29, 30]. Here, we have incorporated the

device-level data of CNT FET-based VTC curve

and the non-linear resistance of CNT interconnects

using GTABLE and ETABLE elements in SPICE

as shown in Fig. 5(b). The former is a four

terminals element for initially modulating the

current-controlled by voltage, which has been

modified for non-linear resistance of CNT

interconnects by connecting the input and output

terminals in pair creating a two-terminal element

[29].

Figure 6(a) shows the output waveforms and its

Fourier transform of 3-stages CNT-based RO for

CNTFET (7, 0) and CNT interconnect (27, 0). The

circuit has a large signal oscillation at about 220

GHz and the odd components are dominant in the

frequency spectrum of the signal. The frequency of

a RO is given by f = 1 (2tpN)⁄ , where N is the

number of stages. The proposed RO, N=3, results in

an average propagation delay of 0.75 pico-second

for each stage due to its intrinsic CNT FET and

CNT interconnect assuming the perfect electrode

contacts (Rc = 0) in the model. The short channel

length of CNT FET (LG = 5 𝑛𝑚) and high intrinsic

mobility of CNT results in low propagation delay of

inverter stages [16]. Figure 6(b) shows the output

waveforms of the proposed RO for the chiralities of

CNT FET equal to n=7 and 28. The square wave

has been obtained for chirality n=7 while changing

to the sinusoidal wave for n=28.

Figure 5. (a) Circuit schematic of multi-stage CNT-based ring

oscillator and (b) SPICE schematic of CNT interconnect and

CNT FET-based inverter showing the correct connection of

GTABLE and ETABLE elements.

Figure 6. (a) Output waveforms and its Fourier transform of 3-

stage CNT-based RO for CNTFET (7,0) with 𝜀𝑜𝑥 = 4 and

CNT interconnect (27,0) and (b) Output waveforms of 3-stage

CNT-based RO for chiralities of CNTFET equal to 7 and 28

for 𝜀𝑜𝑥 = 25. The dielectric thickness of CNTFET (tox) and

the length of interconnect (Lin) are assumed 2nm and 100nm,

respectively. The lengths of drain (LD), gate (LG) and source

(LS) regions are 5nm in all the simulations.

The oscillation frequency decreases by

increasing CNT FET chirality as shown in Fig 7(a).

The oscillation frequency of 3-stage RO has been

obtained f = ~250 GHz reducing to ~ 215 GHz for

the change of chirality, n, from 7 to 28. The

diameter of CNT can be related to chirality by

𝑑 = √3𝑎𝑐𝑐𝑛 𝜋⁄ , where 𝑎𝑐𝑐 = 0.142 𝑛𝑚 is inter-

atomic distance between two nearest-neighbor

carbon atoms. Thus, by changing the diameter of

semiconducting CNT about 1.64 nm, the oscillation

frequency is reduced by ~ %14.6 leading to

straightforward estimation of frequency reduction

by ~ % 8.5 per 1 nm increase in diameter of zigzag

CNT. The oscillation frequency is determined by

the propagation time delay of CNT-based inverter

stages and by the number of stages used in the ring.

The latter creates the multi-phase outputs as each

stage provides a phase shift of 𝜋 𝑚⁄ , where m is

number of stages. Figure 7(b) and 7(c) show the

frequency of CNT-based RO as a function of

chirality for ring oscillators with m = 5 and m = 9,

respectively. For all three types of ring oscillators,

the oscillation frequency is decreased by increasing

the CNT chirality. However, the trends are different

as the change of frequency by chirality becomes

more linear for ring oscillator with higher number

of stages.

Figure 7. Frequency of CNT-based RO as a function of

chirality for ring oscillators with (a) three, (b) five and (c) nine

stages. The simulation parameters are LD = LG = LS = 5nm,

𝑡𝑜𝑥 = 2𝑛𝑚, 𝜀𝑜𝑥 = 16, Lin = 100 n𝑚 and nin = 27.

Figure 8. (a) Frequency of 3-stage CNT-based ring oscillator as

a function of chirality for different dielectric thicknesses when

the dielectric constant of CNTFETs are 𝜀𝑜𝑥 = 4 and (b)

Frequency of 3-stage CNT-based ring oscillator as a function

of chirality for different dielectric constants when the dielectric

thickness of CNTFETs are 𝑡𝑜𝑥 = 2𝑛𝑚 . ( Lin = 100 n𝑚

and nin = 27).

Figure 9. Frequency of 3-stage CNT-based RO as a function

of chirality for ring oscillators for a conducting CNT

interconnect with different chiralities when the contact

resistance is Rc = 24KΩ. The simulation parameters are LD =LG = LS = 5nm,𝑡𝑜𝑥 = 2𝑛𝑚, 𝜀𝑜𝑥 = 16, Lin = 1μ𝑚 and nin =27.

Figure 8(a) shows the oscillation frequency of 3-

stage CNT-based ring oscillator as a function of

CNT FET chirality for different dielectric

thicknesses (tox ). The other circuit parameters as

well as the chirality of CNT interconnect have been

kept constant in the simulation. For tox = 1nm, the

oscillation frequency decreases from 245 GHz to

217 GHz by increasing CNT FET chirality form n =

7 to 22 (blue curve). For CNT FET chirality more

than n=25, the oscillation of CNT-based RO is

eliminated, which results from reducing non-

linearity of VTC curves of CNT-based inverters. By

increasing the dielectric thickness, the non-

oscillation chiralities have been limited to lower

values n=19, 16, and 13 corresponding to the

dielectric thickness of tox = 1.5nm, 2nm

and 2.5 nm, respectively. The oscillation frequency

is more sensitive to CNT FET chirality for higher

dielectric thickness resulting in higher reduction of

frequency from n=7 to n=10 at tox = 2.5nm (black

curve). Also, at the same chirality n=7, the

oscillation frequency of 245 GHz at tox =1nm decreases to ~232 GHz at tox = 2.5 nm due to

the reduction of gate controllability on the

modulation of CNT FET channel by increasing the

thickness of gate oxide. Figure 8(b) shows the

frequency of 3-stage CNT-based RO as a function of

chirality for the dielectric constants of εox = 10, 16

and 25. For εox = 10 , the oscillation frequency

decreases from 250 GHz to 215 GHz by increasing

CNT FET chirality form n = 7 to 28 (blue curve).

Increasing the dielectric constant results in a shift of

the whole curve to higher oscillation frequencies.

Until now, we have assumed a perfect electrode

contacts (Rc = 0) in the model shown in Fig. 4(a) to

provide a proof-of-concept for the chirality

dependence of oscillation frequency regardless of

total contact resistance (2Rc ) of a single-walled

CNT which has been reported different values

between 10 kΩ to 100 kΩ for Pt electrodes [26].

Figure 9 shows the frequency of 3-stage CNT-based

RO as a function of CNT FET chirality for

conducting single CNT interconnects with different

chiralities, in which we have simulated the

proposed RO assuming Rc = 24KΩ . Taking into

account contact resistances, results in an order of

magnitude reduction in the oscillation frequency.

However, the trend of frequency reduction as a

result of increasing the CNT FET chirality remains

the same as in the case of perfect electrode contacts

in the previous simulations. The figure shows the

effect of different chiralities of a conducting

SWCNT interconnect on the oscillation frequency.

Increasing the chirality of SWCNT interconnect has

the same influence on oscillation frequency as the

increase of the CNT FET chirality. The oscillation

frequency reduces from 25 GHz to 22 GHz for

increasing the CNT FET chirality from n=7 to 28

with the constant chirality of CNT

interconnect nin = 27 and with the length of

Lin = 1μ𝑚 between inverter stages. In the same

way, the oscillation frequency reduces from 25 GHz

to 20.3 GHz for increasing the chirality of SWCNT

interconnects from nin = 6 to 30 with the constant

chirality of CNT FET n=7. The reduction of

oscillation frequency has been estimated by ~10%

per 1 nm increase in diameter of 1μ𝑚 metallic CNT

interconnects.

CONCLUSION

We have designed and simulated a carbon nanotube-

based ring oscillator, where semiconducting and

metallic CNTs have been used as transistors and

interconnects between inverters, respectively. We

have investigated the chirality dependence of the

proposed ring oscillator using a numerical ballistic

model based on NEGF formalism for CNT FETs

with the gate length of 5 nm and a compact diffusive

model for CNT interconnects. We have found that

the oscillation frequency of all 3-, 5-, and 7-stages

CNT-based ring oscillator decreases by increasing

the chirality of both (n, 0) CNT FETs and (n, n)

CNT interconnects while the trends becomes more

linear by increasing the number of stages. The

frequency reduction has been estimated by ~8.5%

per 1 nm increase in diameter of zigzag CNT and

~10% per 1 nm increase in diameter of 1μm

metallic CNT interconnects. We have found that the

oscillation frequency is reduced by increasing

dielectric thickness and decreasing dielectric

constant leading to the lack of oscillation for high-

chirality CNTFETs. Assuming the contact resistance

of Rc = 24KΩ , the oscillation frequency of ~25

GHz has been obtained for 5 nm (7, 0) zigzag CNT

FETs and 1μm (6, 6) metallic CNT interconnects.

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