Quantum Modeling and Design of Carbon
Nanotube (CNT) Embedded Nanoscale
MOSFETs
Akin Akturk, Gary Pennington and Neil Goldsman
Department of Electrical and Computer Engineering
University 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:
k  kx x 
2
,
d
(    n,
, n )
(1)
Here x 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]:
 3

  
2   
E  3 1  4 cos 
ak x  cos 
  4 cos 

 n 
 n 
 2

(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:

2
k x2
  E  En  1   n  E  En 
n*
2m

(3)
Here En is the conduction band minimum,  n is the parabolicity factor, and mn* 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:

   
En  min  3 1  2cos 

 n 

(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:
 En


n*
1eV 
m
 0.0965 
mo
  
cos 

 n 
(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:
1
1
1


 (n, F ) 1 (n, F ) 2 (n, F )
(6)
Where 1 (n, F ) and 2 (n, F ) 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:
1 (n, F ) 
 o ( n)
F
1
Fc (n)
(7)
Here o (n) 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:


o (n)  40n 2 1 
Fc (n) 
 
(cm2/Vs)
,
n2 / 3 
1  64  
1  2  x106 ,
3/ 2 
n 
n 
(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:
2 (n, F ) 
Vmax (n)

F 
F 1  0.01

Fc (n) 

,
(V/cm)
(10)
Here Vmax is the maximum drift velocity of the electrons, shown in Fig. 6. It is formulated
below as:
 

Vmax (n)  1.5n1/ 3 1   x107 ,
 2n 
(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, lowfield 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:
mn*
DOSn 
2
2
(12)
E  E 
n

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:
non  2
1
2

  DOS e
E
n  kT
dE
(13a)
n
E
Or equivalently:
n 
n
o

kTmn*
2 2
2
e

En 
1  t
kT
t
e dt
(13b)
0
The integral in Eqn. 13b can be recognized as the gamma function with an argument
 . The expression in Eqn. 13 gives one-dimensional
equal to ½, and integral is equal to
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:
n 
n
o


1
2.49n


kTmn*
2
2
2
e

En
kT
(14)
III-Quantum Device Simulator:
We develop a quantum device solver [4]-[6] based on the Schrödinger equation and driftdiffusion equations. They are listed below in the order of Schrödinger, Poisson, electron
current-continuity, hole current continuity and lattice heat flow equations.
E ( y)  
d 2 ( y)
 q ( x, y) ( y)
2m* dy 2
2
 2  
q

 p  n  D
n 1
 .J n  GRn
t q
p
1
  .J p  GR p
t
q
C
(15)
(16)
(17)
(18)
T
 . T    J n  J p  
t
(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:

kT  no ( x, y )  
J n  qnn   
ln 
   n kT n
q  noSi  

(20)
Here no(x,y) is the intrinsic carrier concentration at a grid point on our device, and noSi 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:
kT  nQM ( x, y ) 
ln 

q  nCL ( x, y ) 
(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:
nQM ( y ) 
m*kT

2
  ( y)
i
i
2
 EF  Ei  

kT
ln 1  e



(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:
nDD ( x, y)  noQM ( x, y)e
q n 
kT
(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 CNTMOSFETs. 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 along the 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 probability of finding a
carrier.
Electron Concentration (1/cm 3)
10
10
10
19
18
17
0
1
2
3
Vertical (nm)
4
Figure 10: Calculated electron concentration in the vertical direction of the Si-SiO2
interface. Marked line is for CNT-MOSFET with a large diameter CNT (46Å). For
the other two curves, the one that has slightly higher concentration around 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.
En (eV)
( ) β
0.086
0.527
7
2
0.358
1.146
6
3
0.221
1.854
8
1
0.043
0.244
15
2
0.118
0.507
14
3
0.137
0.929
16
1
0.029
0.159
23
2
0.070
0.326
22
3
0.099
0.616
24
1
0.017
0.093
39
2
0.039
0.189
38
3
0.063
0.367
40
n (m=0)
Band
10
1
22
34
58
mn*
mo
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