Quantum Modeling and Design ofCarbon Nanotube (CNT) Embedded

Nanoscale MOSFETs

Akin Akturk, Gary Pennington and Neil Goldsman

Department of Electrical and Computer EngineeringUniversity of Maryland, College Park, MD 20742, USA

akturka@glue.umd.edu, garyp@glue.umd.edu,neil@eng.umd.edu

Abstract

We propose a novel MOSFET design that embodies single wall

zig-zag semiconducting Carbon Nanotubes (CNT) in the

channel, as shown in Fig. 1. Investigations show that CNTs

have high low-field mobilities, which can be as great as

2x105 cm2/Vs. Thus we predict that MOSFET performance can be

improved by embedding CNTs in the channel. To investigate

the performance of the new CNT-MOSFET device, we develop a

methodology that connects CNT modeling to MOSFET

simulations. Our calculations indicate that by forming high

mobility regions in the channel, MOSFET performance can be

boosted. The gain in the drive current is as much as 30-40%

over standard MOSFETs. Our predictions are based on

calculations obtained from our CNT Monte Carlo (MC)

simulator [1]-[3] and quantum based device solver [4]-[6].

Keywords: Single-Wall Zig-Zag Carbon Nanotube, CNT Mobility

Model, CNT Embedded MOSFET Design, Simulation.

I-Introduction:

As we approach the end of the semiconductor roadmap,

investigators are exploring device technologies to push

electronics into the 21st century. Carbon nanotubes (CNTs)

are being explored as a structure that may play a leading

role in future electronic systems. CNTs are planar graphite

sheets (graphene) that are seamlessly wrapped into tubes, as

shown in Fig. 2. CNTs possess favorable electrical

characteristics and can be fabricated in dimensions as small

as 4-8Å in diameter. The electrical characteristics of a CNT

vary with its diameter and the wrapping angle of the

graphene [7]. Both the diameter and the wrapping angle can

be described by the tube’s fundamental indices (n,m). Theory

indicates that CNTs can be metallic or semiconducting

depending on the fundamental tube indices (n,m), and bandgap

depends on the diameter. Analysis shows semiconducting CNTs

have very high low-field mobilities, with peak drift

electron velocities that can be as much as five times higher

than that of silicon. It has also been shown that tubes can

be doped by donors and acceptors [7,8]. Experiments and

calculations also indicate that CNTs could facilitate

devices with large transconductances and high drive currents

[4,5], [7]-[15]. Experiments have demonstrated the viability

of CNT-based FETs [12,13], and CNT-SOI type MOSFETs [14,15].

Preliminary research has been done to model, design CNT

embedded bulk MOSFETs [4,5],

We propose a novel CNT-MOSFET device that is shown in

Fig. 1. Our calculations indicate that the CNT-MOSFET can

have improved device performance over the conventional

MOSFET [4,5]. To investigate the potential attributes of the

new design, we developed a methodology for modeling

nanoscale CNT-MOSFETs. It includes determination of the

electrical characteristics of single wall zig-zag CNTs, and

the merging of the CNT results into our quantum device

solver. To electrically characterize the CNT, we developed a

Monte Carlo (MC) simulator for CNTs. We first calculate

electron and phonon dispersion relations for single wall

zig-zag CNTs with different tube indices n. We then derive

the selection rules and scattering matrix elements, using

Fermi’s Golden Rule. Using MC results, we derive analytical

models for CNT parameters such as mobility and density of

states. Once we obtain CNT parameters, we import them to our

quantum device solver. Our device solver is based on the

semiconductor equations, modified to account for quantum

effects, and the Schrödinger equation. We solve these

coupled equations on a discretized mesh of our CNT-MOSFET

device shown in Fig. 1. The solution gives results which

include: CNT-MOSFET current voltage curves, electron

concentration profile, in the bulk MOSFET, and in the CNT

enhanced channel.

We summarize our algorithm in Fig. 3.

II-CNT Simulator:

As mentioned above, we first employ a Monte Carlo (MC)

simulator [1]-[3] to characterize fundamental transport

properties of CNTs, and then we incorporate these properties

into our device simulator. These properties include electron

drift velocity versus electric field curves as well as CNT

mobility for zig-zag single wall CNTs, which is the most

studied topology. To obtain these properties, we begin with

the physical CNT system, where electrons are confined around

the circumference, and move relatively freely along the tube

in the direction of the longitudinal axis. Therefore, one

can write the appropriate plane wave solutions that satisfy

periodicity for the given CNT circumference. However along

the tube, electrons are not confined. Thus the wavevector

can be written as follows:

, ( ) (1)

Here is parallel to the tube axis, and is the unit

vector along the circumference. Discrete values around the

circumference, β, are bounded by the fundamental tube index

n, to take advantage of the symmetry lines in the CNT

Brillouin zone. Note that n count as one unique zone

boundary. A zig-zag single wall CNT and its corresponding

Brillouin zone can be found in Figs. 2 and 4, respectively.

We consider one-dimesional semiclassical charge transport

along the tube.

As external field is applied, electrons are accelerated

along the tube up until they scatter with phonons. Here we

consider scattering by optical, and acoustic inter and intra

subband, and intervalley. Both scattering mechanisms are

treated within the deformation potential approximation using

the Fermi’s Golden Rule. In this work, we use 3eV and 9eV

for the nearest-neighbor π-hopping integral and the

deformation potential, respectively. For more details of our

MC simulations we refer the reader to [1,2].

A. CNT Bandstructure:

To account for the CNT related quantum effects in the CNT-

MOSFET channel, we need to determine the band-structure of

the CNTs. Due to confinement, the bandstructure splits into

a system of subbands when graphene is wrapped into a CNT.

Each of the subbands has a characteristic effective mass,

mobility and band energy minima.

We determine the energy levels of CNTs by applying zone-

folding methods to graphene. From the two-dimensional

graphene band diagram, we cut one-dimesional slices, whose

numbers and locations are set by the fundamental tube

indices (n,m).

We use zone-folding of the graphene conduction band to

calculate the electronic bandstructure of CNT. The following

formula gives the energy dispersion for a zig-zag CNT[7]:

(eV) (2)

Where a (=2.46Å) is the lattice constant of two dimensional

graphite. It has been shown that this analytical formula

agrees with experimental results [7].

To extract the pertinent information that can be easily

integrated into our device simulator, we transform Eqn. (2)

into a quadratic form which is consistent with traditional

semiconductor analysis:

However we use a parabolic energy bandstructure, instead of

the one in Eqn. 2. Because it is too complicated for use in

an MC code and the parabolic dispersion curves well

approximate the tight-binding bandstructure for energies of

our interest, which is up to 2eV above the conduction band

minimum. In Fig. 5, we show how well the parabolic

bandstructure fits the results of the tight-binding model.

The parabolic energy dispersion relation for different

subbands can be written as follows:

(3)

Here is the conduction band minimum, is the

parabolicity factor, and is the effective mass in

subband β for a tube with index n. Conduction band minimum,

effective mass and parabolicity factor can be calculated

using Eqn. 2 for different subbands. For example, conduction

band minimum for each subband can be found by setting kx to

zero in Eqn. 2, and choosing the appropriate signs for the

expression below such that E is minimum:

(eV) (4)

Likewise, effective mass is the curvature of the energy

dispersion relation given in Eqn. 2 around kx=0. We obtain

the following formula for the effective masses of each

subband:

(5)

Our calculations indicate that only the lowest three

subbands are effective in the electron transport

calculations along the tube. Thus we only deal with those

subbands, which correspond to βo, 2(n-βo), and 2(2βo-n),

where βo is equal to 2n/3 rounded to the nearest integer.

They are written in an order starting from the subband that

has the lowest energy minimum and increasing. Note that

these subbands are not around the gamma point, but rather

correspond to higher frequencies. In Table 1, we give a list

of conduction band minimums and effective masses for the

first three subbands of different tubes.

B. CNT Mobility Model:

We derived a mobility model based on our MC simulation

results of drift velocity versus electric field curves. They

are plotted in Fig. 6 for CNTs with tube indices 10 to 58.

They are equivalently ranging in diameter from 8Å to 46Å.

Simulation indicates that electron drift velocity first

increases linearly with the applied field, reaches a

maximum, and then rolls off, showing a negative differential

mobility (NDM). We find that peak electron velocities are as

much as four times higher than what they are in silicon.

Electrons reach velocities as high as 5x107cm/s in large

diameter CNTs (n=58), while the maximum velocity, which is

approximately 3x107cm/s for n=10, drops for smaller diameter

tubes. However this is still larger than the corresponding

values in other semiconductors. Calculated results also show

that the critical field, where we have the peak drift

velocity, increases from 1kV/cm to 10kV/cm as we enlarge the

tube diameter from 8Å to 46Å.

Figure 6 shows three main characteristics of the CNTs.

First, CNTs attain larger drift velocities than other

semiconductors. Investigations show that is due to decreased

scattering rates, which is a result of the one-dimensional

system we have. Second, electrons in large diameter tubes

have higher velocities than the ones in small diameter tubes

for a given applied field, unless the applied field is too

large. This leads to higher low-field mobilities for larger

diameter tubes. Analysis show that is due to increased

density of states of the large diameter tubes. Third, all

CNTs show NDM. It is similar to GaAs in that respect, where

conduction band velocity of the first subband is larger than

that of the second.

We develop an analytical mobility model considering the two

lowest subbands, which dominate the conduction. By that

means, we embed the effects of NDM in our model. We then

express the final mobility using Mathiessen’s rule, as

follows:

(6)

Where and refer to the mobilities in the

first and second subbands, respectively. They are functions

of fundamental tube index, n, and the electric field, F. The

mobility of the first subband is:

(7)

Here is the low-field mobility, and Fc(n) is the

critical electric field. The critical electric field

corresponds to the peak electron drift velocity. We have the

following expressions for the low-field mobility and the

critical field:

, (cm2/Vs) (8)

, (V/cm) (9)

Here ρ=1-gcd(n+1,3) (= 0, -2), where gcd(n+1,3) is the

greatest common divisor of n+1 and 3.

We can write the mobility of the second subband as follows:

, (V/cm) (10)

Here Vmax is the maximum drift velocity of the electrons,

shown in Fig. 6. It is formulated below as:

, (cm/s) (11)

In Fig. 7, we show our calculated mobility versus field

curves, using Eqns. 6-11. CNTs can sustain large mobilities,

larger than all known semiconductor devices. For n=58, low-

field mobility is as high as 2x105cm2/Vs, while for n=10 it

is approximately 4x103cm2/Vs. So they can be used to form

high mobility regions.

C. CNT Intrinsic Carrier Concentration:

To embed a CNT in our device solver, we need a value for the

intrinsic carrier concentration of the CNT. Thus we develop

a methodology to obtain the intrinsic concentrations of

different tubes. We start from the parabolic energy

dispersion relation in Eqn. 3. However we ignore the

parabolicity factor to simplify calculations. The density of

states for each subband is:

(12)

We also assume that we have a Maxwellian type distribution.

Thus we get the following expression for the intrinsic

carrier concentration, where Fermi level is set to zero:

(13a)

Or equivalently:

(13b)

The integral in Eqn. 13b can be recognized as the gamma

function with an argument equal to ½, and integral is equal

to . The expression in Eqn. 13 gives one-dimensional

carrier concentration, whereas we need the concentration per

volume. Thus we further assume that the vertical cross

section of the tube can be fit in a square, whose side is

one diameter long. Finally, we arrive at the following

formula for the intrinsic carrier concentration, which is

only a function of fundamental tube index n:

(14)

III-Quantum Device Simulator:

We develop a quantum device solver [4]-[6] based on the

Schrödinger equation and drift-diffusion equations. They are

listed below in the order of Schrödinger, Poisson, electron

current-continuity, hole current continuity and lattice heat

flow equations.

(15)

(16)

(17)

(18)

(19)

Here the newly introduced variables , ψ, n-p, Jn-p, T, D,

GRn-p, C and κ are electrostatic potential, wave function,

electron-hole concentrations, electron-hole current

densities, lattice temperature, net dopant concentration,

electron-hole net generation-recombination rates, heat

capacity and thermal diffusion constant, respectively. In

Eqn. 15, x is parallel to the Si-SiO2 interface in MOSFET,

and y is normal to x.

Solution of the CNT-MOSFET system requires proper handling

of two dynamics. First one is the quantum well formed at the

Si-SiO2 interface that causes band splitting thus lowering

of the carrier concentration. Second one is the

heterojunction formed at the CNT-silicon interface that

reflects the different bandstructures of the CNT and

silicon.

We first solve our system classically. This includes the

coupled solution of Eqns. 16-19. At this stage, we also

resolve the effects of heterostructure, through the use of a

revised Scharfetter-Gummel discretization scheme. The

details can be found in [4]. We keep the Scharfetter-Gummel

discretization intact, but modify electron current density

as follows:

(20)

Here no(x,y) is the intrinsic carrier concentration at a

grid point on our device, and is the intrinsic carrier

concentration of silicon.

Investigations show that carrier confinement at the Si-SiO2

interface can significantly reduce carrier concentration

adjacent to the interface [15]. Thus we also solve for the

quantum effects using the classical solution as our initial

guess. To inherent quantum effects in our device model, we

solve the Schrödinger equation, Eqn. 15, in addition to

Eqns. 16-19. We solve the one-dimensional Schrödinger

equation along the normal direction of Si-SiO2 interface,

starting from the interface and going down the substrate

about 200Å. We apply the same procedure to different mesh

points along x in the channel, as well. The physical system

and the quantum well can be found in Fig. 8.

We treat the quantum induced effects like the formation of

position dependent heterostructures in the quantum well.

Thus we sum quantum effects in the carrier concentration

term, where quantum confinement is reflected as bandgap

broadening. This methodology introduces an extra term within

the parentheses of the gradient in Eqn. 20, as follows:

(21)

Here subscripts refer to quantum (QM) and classical (CL)

solutions. We calculate the quantum carrier concentration

using the solution of the Schrödinger equation, which gives

energy eigenvalues and eigenfunctions. We then combine the

resulting subband structure with the occupation densities,

which can be seen in Fig. 9. Thus we get the following

formula for the quantum carrier concentration, where we have

a sum over different subbands:

(22)

In summary, we first solve the coupled Poisson, electron

current continuity, hole current continuity and heat flow

equations, classically. During the simulation, we also

include the effects of the heterostructure formed by the CNT

and silicon. We achieve this by using an effective

potential, argument of the gradient in Eqn. 20, instead of

the classical electrostatic potential, , in the drift-

diffusion equations. After we have the coupled solution of

the device equations, we use it as an initial guess for the

quantum system. We iteratively solve Schrödinger, Poisson,

quantum electron and hole current continuity, and heat flow

equations until coupled solution of the Schrödinger equation

and quantum electron current continuity equation give the

same electron concentration. Carrier concentration using the

solution of the Schrödinger equation is calculated as shown

in Eqn. 22. Whereas we get the carrier concentration within

the modified drift-diffusion (DD) formalism as follows:

(23)

Here noQM(x,y) is the intrinsic carrier concentration at a

grid point on our device, that includes the effects of the

heterostructure, as well as the quantum.

Once we have the coupled solution of the system, our quantum

device solver outputs current-voltage curves, and profiles

of quantum electron concentration, electrostatic potential,

hole concentration and such. In Fig. 10, we show our

calculated electron concentration in the vertical direction

of the MOSFET channel, starting from the Si-SiO2 interface.

We applied 0.7V to the gate terminal, and grounded others.

CNT-MOSFETs with large diameter tubes have higher

concentrations in the channel. Thus they are likely to

sustain large transconductances but may be at the expense of

weaker isolation in the “off” state of the MOSFET. We also

obtain the current-voltage characteristics of the 0.1μm CNT-

MOSFETs in the linear and saturation regions. We show the

drain current density versus applied drain voltage curves in

Fig. 11. There are three configurations for comparison.

First one is the conventional MOSFET without any CNTs in the

channel. The other two are the CNT-MOSFETs with small (8Å)

and large (46Å) diameter CNTs in the MOSFET channel. We find

that CNT-MOSFETs utilizing smaller diameter tubes attain

higher drive currents than the ones having larger diameter

tubes, followed by the conventional MOSFET. The ordering

within CNT-MOSFETs can be attributed to the height of the

barrier formed at the CNT-silicon heterojunction. However

they both have improved drive current characteristics than

the conventional MOSFET due to larger channel electron

concentrations shown in Fig. 10 and larger mobility values

on CNTs. In Fig. 12, we plot the subthreshold

characteristics of these three devices. Large diameter tubes

have higher leakage currents than others, making them

undesirable for low power applications. However small

diameter tubes give rise to improved performance in

subthreshold, linear and saturation regions, making them

favorable for low power and high frequency applications.

IV-Conclusion:

We propose a novel device structure that combines the MOSFET

technology with CNT nanostructures. We report on our

predictions that the CNT-MOSFET device is likely to have

better performance than the conventional MOSFET. To analyze

the new design, we develop a methodology for

modeling and simulating CNTs and CNT-MOSFETs. We first

employ MC techniques to electrically characterize single

wall zig-zag CNTs. For MC calculations, we obtain the energy

dispersion relation and the phonon spectrums of CNTs by

applying zone-folding methods to graphite. We derive

selection rules and scattering rates from Fermi’s Golden

rule. We also include collision broadening effects. We then

derive analytical models for CNT parameters like mobility.

We import these results to our quantum device solver and

determine the current-voltage characteristics of CNT-

MOSFETs. We predict that the drive current of CNT-MOSFETs is

higher than the conventional MOSFETs, with the smaller

diameter tube has the highest. Likewise in the subthreshold

region, narrow diameter tube has the best performance. Thus

we predict that CNT-MOSFETs employing lower diameter tubes

are likely to have good device performance and can be used

in low power and high frequency devices.

Figure 1: Simulated CNT-MOSFET device.

Figure 2: Simulated zig-zag single wall CNT structure.

Figure 3: Coupled algorithm flowchart.

Figure 4: CNT Brillouin zone on graphene k-space. x is alongthe tube axis, and θ refers to the circumference. High symmetry directions of ΓMK are also shown.

Figure 5: Bandstructure for a zig-zag CNT with n=10. Solid and dashed lines refer to parabolic and tight-binding bandstructures, respectively. Inset is for n=59.

Figure 6: Electron drift velocity versus electric field curves for zig-zag CNTs with different tube indices.

Figure 7: Calculated mobility versus electric field curves for zig-zag CNTs with different tube indices.

a)

b) c)

Figure 8: a) CNT-MOSFET device system. Schrödinger equation is solved along the dashed line starting from the Si-SiO2 interface for a 200Å long line. b) Quantum well at Si-SiO2 interface. c) Quantum well with nanotube.

a)

b)

Figure 9: a) Quantum well at the Si-SiO2 interface causes band splitting. b) We calculate the quantum carrier concentration by summing charges in different subbands, weighted by the density of states, D(E), and the probabilityof finding a carrier.

0 1 2 3 4

1017

1018

1019

Vertical (nm )

Electron Co

ncentration (1/cm

3 )

Figure 10: Calculated electron concentration in the verticaldirection of the Si-SiO2 interface. Marked line is for CNT-MOSFET with a large diameter CNT (46Å). For the other twocurves, the one that has slightly higher concentrationaround 0.5nm is for CNT-MOSFET with a small diameter tube(8Å). The remaining one is for the conventional MOSFET.

Figure 11: Calculated current-voltage curves. CNT-MOSFETs show 30-80% increase in drain current. We applied gate biases of 0.4V, 0.7V, and 1.0V.

Figure 12: Calculated subthreshold current-voltage curves. Narrow diameter tubes show improved subthreshold performance. Drain bias is 0.1V.

Table 1: Band structure parameters for zig-zag CNTs.

n (m=0) Band (eV) ( ) β

10 1 0.086 0.527 72 0.358 1.146 63 0.221 1.854 8

22 1 0.043 0.244 152 0.118 0.507 143 0.137 0.929 16

34 1 0.029 0.159 232 0.070 0.326 223 0.099 0.616 24

58 1 0.017 0.093 392 0.039 0.189 38

3 0.063 0.367 40

V. References

[1] G. Pennington, N. Goldsman, “Semiclassical Transport and

Phonon Scattering on Electrons in Semiconducting Carbon

Nanotubes,” Phys. Rev. B, vol. 86, pp. 45426-37, 2003.

[2] G. Pennington, N. Goldsman, “Monte Carlo Study of

Electron Transport in a Carbon Nanotube,” IEICE Trans. Electron.,

vol. E86-C, pp. 372-8, 2003.

[3] G. Pennington, A. Akturk, and N. Goldsman, “Drift

Velocity and Mobility Model for Carbon Nanotubes,” IEEE Elect.

Dev. Lett, “submitted for publication”.

[4] A. Akturk, G. Pennington, and N. Goldsman, “Modeling the

Enhancement of Nanoscale MOSFETs by Embedding Carbon

Nanotubes in the Channel,” Third IEEE Conf. on Nanotech., pp. 24-

7, 2003.

[5] Akturk, G. Pennington, and N. Goldsman,

“Characterisation of Nanoscale Carbon Nanotube (CNT)

Embedded CMOS Inverters,” ISCAS 2004, “submitted for

publication”.

[6] A. Akturk, N. Goldsman, G. Metze, “Increased CMOS

Inverter Switching Speed with Asymmetrical Doping,” Solid-State

Electronics, vol. 17, no. 2, pp. 185-92, 2003.

[7] R. Saito, M. S. Dresselhaus, and G. Dresselhaus,

Physical Properties of Carbon Nanotubes, Imperial College

Press, London, 1998.

[8] R. S. Lee, H. J. Kim, J. E. Fischer, A. Thess, R. E.

Smalley, “Conductivity Enhancement in Single-Walled Carbon

Nanotube Bundles Doped with K and Br,” Nature, vol. 388, pp.

255-7, 1997.

[9] L. Grigorian, G. U. Sumanasekera, A. L. Loper, S. Fang,

J. L. Allen, and P. C. Eklund, “Transport Properties of

Alkali-Metal-Doped Single-Wall Carbon Nanotubes,” Phys. Rev.

B, vol. 58, pp. R4195-8, 1998

[10] Z. Yao, C. L. Kane, and C. Dekker, “High-Field

Electrical Transport in Single-Wall Carbon Nanotubes,” Phys.

Rev. Lett, vol. 84, pp. 2941-4, 2000.

[11] J. Hone, M. Whitney, C. Piskoti, and A. Zettl, “Thermal

Conductivity of Single-Walled Carbon Nanotubes,” Phys. Rev. B,

vol. 59, pp. R2514-6, 1999.

[12] A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker,

“Logic Circuits with Carbon Nanotube Transistors,” Science,

vol. 294, pp. 1317-20, 2001.

[13] R. Martel, V. Derycke, J. Appenzeller, S. Wind, Ph.

Avouris, “Carbon Nanotube Field-Effect Transistors and Logic

Circuits,” IEEE Design Automation Conf., pp. 94-8, 2002.

[14] S. J. Tans, A. R. M. Verschueuren, and C. Dekker, “Room

Temperature Transistor Based on a Single Carbon Nanotube,”

Nature, vol. 393, pp. 49-52, 1998.

[15] S. J. Wind, J. Appenzeller, R. Martel, V. Derycke, Ph.

Avouris, “Vertical Scaling of Carbon Nanotube Field-Effect

Transistors using Top Gate Electrodes,” Applied Physics Letter,

vol. 80, no. 20, pp. 3817-9, 2002.

[16] C. K. Huang, and N. Goldsman, “2-D Self-Consistent

Solution of Schrödinger Equation, Boltzmann Transport

Equation, Poisson and Current-Continuity Equation for

MOSFET,” SISPAD 2001, pp. 148-51, Spring-Verlag, 2001.