ABSTRACT:
This paper presents a geometry-physics based compact model for
MOSFET-like CNTFET. This model is developed in terms of basic
geometry parameters like chiral index, doping concentration, no. of
tubes, pitch etc. The non-idealities modeled in the design are the
quantum tunneling and the acoustical/optical phonon scattering in
the channel region and the screening effect by parallel CNTs. The
I-V characteristics have been obtained over the entire region of
operation for large signal (digital) application purposes. The
CNTFET parameters have been optimized for an operating frequency of
100MHz for channel length of 32nm with objective function of Iratio
to be maximized. The optimization algorithm in the design is
Particle Swarm Optimization which is evolutionary based
optimization technique. The optimal Nd and (n, m) are obtained with
desired pitch and number of tubes.
1. INTRODUCTION:
As the scaling of conventional MOS devices is never ending, the
problems associated with scaling of silicon based devices are
highly visible. CNTFETS are undoubtedly the ideal successors of
conventional MOS devices. Modelling of CNTFET require accurate
treatment of quantum tunnelling and the non-idealities of the
device. Several approaches have been made for accurate modeling of
the device. The use of Schrdinger- Poisson based solver was one of
such approaches but it lacks quantum treatment of the device.[2]
NEGF treatment though solves the problem of quantum modelling but
the numerical modelling tasks become more and more complicated.
Wigner formalism of scattering effect along with Greens function
provided more accurate model of the FET.[5] the complication in
deriving the model are getting more and more intense.[7] . The need
of accurate model for CNTFET is still an issue.This paper presents
a simpler way to model the CNTFET using its energy profile. The E-k
curve of CNT derived from quantum physics is used as the base for
the model.[8] The energy level change is then used to compute the
device transport equation. [9] the capacitance extraction is done
using device geometry and electrostatics physics. [10] The acoustic
/ optical phonon effects are modeled in the design [12]. The model
is designed in terms of primary design parameters and the I-V
characteristics is obtained. The device parameters are optimized
for a specific frequency of operation. Commonly used optimization
algorithms are gradient-based methods but it has sevaeral
limitations like initial guess requirement, singularities in
objective functions, etc. [14-15]. Genetic algorithm does not
suffer from the problems and can be used for parameter
optimization. [16]. Compared to GA, the PSO is reported to be more
efficient has proved to be more efficient in parameter optimization
[17-18]. The paper uses PSO algorithm to optimize the parameters of
the FET for operating frequency of 100 MHz with objective of high
mobility and low capacitance value for channel length of 32nm.
2. CNT INTERCONNECTS:
2.1 CARBON NANOTUBES
CNT is a tubular form of carbon with diameter as small as 1nm.
The length of the tube ranges from a few nm to m. A CNT is
characterized by its Chiral Vector: Ch = n 1 + m 2, - Chiral Angle
with respect to the zigzag axis , where 1 and 2 are lattice basis
vectors of graphene sheet and the chiral angle represents the angle
that the axis of rolling up sheet forms with one of the basis
vectors. On the basis of chirality CNT can be classified as, zig
zag ( = 0), Chiral (0 < < 30), or armchair (with = 30) Figure
1 shows the 3-D structure of CNT for chiral index (13, 0) with N =
1000 electrons. The diameter of the tube can be achieved by
following equation
Figure 1: 3D Plot of CNT of chiral index (13, 0) in MATLAB
The band structure of the carbon nanotube can be obtained by
zone-folding the band structure of the graphene sheet.[8-10].The
E-k plot (Figure 2)developed using this theory shows that
conduction band and valence band consists of several subband
regions with ith subband minima occurring at energy level Ei,min =
i*Eg/2, [12] where Eg is energy gap of the tube and is given by
The E-k relation at subband minima[21] can be described by the
equation
Figure 2: Momentum versus Energy
2.2 Carrier concentration in CNT Channel
The carrier concentration of the CNT channel is the product of
FermiDirac distribution * Density of state, f(E) *D(E), over the
carrier energy,i.e., ni=,
The density of states is found to be maximum only at subband
minima of the CNT channel and is extremely low over other subband
region (Figure 3). The nanotube density of states [13] at subband
minima sbbd is given by
Here H(E) is heaviside unit function and E0 is the mid-gap
energy of the intrinsic region
Figure 3: Density of states versus Energy
Carrier concentration for Pth subband in CNTFET is given by,
where b is the carboncarbon distance equal to 0.142 nm, and is
the carboncarbon bonding energy estimated at 3.033 eV,The partial
solutions of the above equation in different cases[15] has been
shown in figure 4. The first case include low bias application
across the channel V < sbbd[p], while the second case considers
V > sbbd[p] and then carrier concentration equation is modelled
for both situation (a)at low energy, and (b) at higher energy
level. Finally, a complete solution is obtained using smoothing
function(fsmo) as shown in figure 5. The final carrier
concentration expression is
Figure 4: Linear channel Density versus voltage bias Figure 5:
Linear charge density versus temperature
3. Modelling the CNTFET
3.1 Band Diagram of CNTFET
The band diagram of a material plays an important role in
deciding the behaviour of any device. The band diagram can be
designed by the help of the knowledge about Fermi level. The Fermi
Dirac distribution lets you know about the energy at which the
probability of finding a electron in conduction band becomes equal
to the probability of finding a hole in valence band. The Fermi
level for both the n-type as well as p-type semiconductor vary
vastly in band diagram. The Fermi dirac function (figure 6) is
written as follow:
The equation above clearly tells us that the Fermi dirac
distribution depends on temperature & mobility the below
diagram shows how temperature affects it. The Fermi level of n-type
& p-type(figure 7) can also be modelled keeping into
consideration of the fact that the Fermi level for n-type
semiconductor moves towards the conduction band & known as
donor energy level, whereas the Fermi level for p-type
semiconductor moves towards the valence band and is known as the
acceptor energy level.
Figure 6: Fermi Dirac distribution for different Temperature
Figure 7: Fermi level for N and P Type Si
The Band diagram of the CNTFET has been achieved here we can
change the structure of the Band Diagram by the application of gate
& drain voltages. Here an interesting fact we are able to
witness regarding the band structure when we are increasing the
gate voltage & keeping the same drain voltage it decreases the
conduction band structure of channel.
3.2 Circuit model of the CNTFET
Figure 8: CNTFET geometryThe quasi-1D CNTFET can be modelled by
the model shown above consisting of a semiconducting source, a
semiconducting drain, a metallic gate, a high K dielectric the
channel is made up of CNT (Carbon nanotube). The movement of
electrons from source to drain is being done by means of Ballistic
Transport by which electrons where not facing any energy losses
from the walls of the channel. As we are dealing in the nanoscale
region the bands of the device becomes discrete in nature. The
bands become discretised and are termed as sub bands. The behaviour
of these sub bands will decide the amount of current flowing
through our device. The bands are broken into sub band minima and
sub band maxima. The total current flowing through the device can
be calculated by simply adding all the sub bands together.
Figure 9: Large Signal Model of CNFET
Figure 10: Effect of gate voltage on CNFET subband in terms of
Qcnt and Cins for Current Calculation
The current flowing through the device is dependent on the
energy band of the device. This in turn is modulated by the gate
voltage of the device. Since the current is constant throughout the
channel it can be calculated at the top of the energy barrier which
can also be considered as the beginning of the channel. Here for
calculating the total amount of current we have to make use of the
large signal equivalent of it.(Fig 9).A flat band voltage Vfb to
consider the difference between the gate metal work function &
the nanotube electron affinity. The movements of electron has been
modelled here as follow. The electrons coming from the source fill
up the +k states of the Fermi level and the electron coming from
the drain fill up the k states of the Fermi level these two make up
for the QCNT. (Fig 10). In order to model the current flowing
through the device we need to first out Qcnt in terms of sub band,
Fermi level & Vcnt. Then we will be able to find out Id. Then
at last we will find out the gate voltage corresponding to Vcnt.To
find out the first part the total charge of each sub band can be
thought as the each charge contributed by source and drain.The
total current for each sub band can be calculated as
To start the calculation for I we need to know about s. Here s
can be calculated as below: where Cins is known as
The quantum capacitance is also necessary for calculating the
exact current flowing through our device which is given as
Using all these equations we can model the CNTFET with all its
necessary parameters in MATLAB with accuracy.
5. Optimization
Using the optimization technique we can choose the best result
for the parameters available to us for modelling the CNTFET. There
are various optimization techniques available to optimize our
devices in nanoscale region, out of them all here we have opted for
ANT colony and PARTICLE SWARM optimization technique which falls
under Generic optimization technique. The generic optimization is
inspired by evolutionary biology. Generic Optimization simply
implemented as a computer simulation. It is highly advisable
because it solves problems with multiple solutions and is easily
transferrable to any model. There are two types of genetic
optimization models with each having its own advantage.The ANT
colony optimization is the first approach. This optimization
technique depends on the pheromone which discharged by the ANTs in
there colony. The whole idea of an ANT colony on how they can find
the shortest path for their food can be readily implemented on real
life encounters. The ANT which tracks the path finds this out by
the density of the pheromone which remains while it retracts from
the same path. Since the pheromone tends to evaporate with time,
the ANT chooses that path in which the pheromone density is high.
It has the advantage of distributed computing, it is robust and it
can adapt to the system in which it is applied in real time. The
second approach is the PARTICLE SWARM approach which is inspired by
bird flocking. It is an advanced form of genetic algorithm. It is
easier to implement than Genetic approach. It depends on the
ability of the birds to find food at a certain area. Each bird
returns a particular value which shows the distance of that bird
from the food. The value which is best for the group is taken first
and a second best value is considered. The algorithm return backs
two values in which the first one is known as best value and the
second one is known as the second best value. Depending upon these
two values the algorithm can find out the shortest path necessary
for finding the food for the birds. It has numerous advantages like
in ANT colony the entire population moves as a group to the
destination but in PARTICLE SWARM the population moves towards the
best solution. 6. Results and Discussion
The energy band diagram of CNTFET for chiral index (13,0) have
been obtained (figure 11) with fermi level set at 0eV. Figure 11
shows only first subband minima plot of CNT in CNTFET and the
variation of these bands with application of drain and source
voltage has been studied which shows that the drain voltage
application lowers the Fermi level of drain side by the same
amount.
Figure 11: Energy Band Diagram Variation at different S-D
bias(Vds)
. Figure 12: Change in Energy Band Diagram with gate voltage
variation
The effect of application of Gate to Drain Voltage on Enegry
band Diagram of device has been studied (figure 12) which shows
that as we keep on increasing our gate to source voltage the Fermi
level of the channel gets reduced by some amount (i.e. Vcnt) .The
mobility of the CNTFET devive has been found out and the result
shows that the mobility maxima occurs at subband minima of the CNT
channel because of high carrier density presence at that
states.(figure 13). The quantum capacitance of the device is
calculated from the carrier concentration and the graph is plotted
against applied potential.(figure 14).
Fig. 13 mobility verses energy
Fig.14 Quantum capacitance for the first sub band minima as
function of VThe insulator capacitance of the device has been
obtained in COMSOL multiphysics using its geometry and the results
for different no. of tubes and with various pitch distance has been
recorded in the table which shows that Insulator capacitance Cins
increases as the number of tubes increases in our device and as we
are going on increasing the pitch for our device the insulator
capacitance of the device decreases gradually.
Fig. 15 COMSOL Image of CNTFET with three tubes.
Table 1: Variation of Cins With number of tubes nt at different
Pitch =10 nm. Nt (Number of Tubes) Cins(pF/m) ( Insulator
Capacitance)
1 138
2 342
3 670
4 968
Table 2: Variation of Cins with Pitch at nt = 3
Pitch (n,m) Cins (pF/m)
6 734
8 699
10 670
Finally the current model has been obtained and the results show
that the I-V characteristics of this device are similar to
conventional MOSFET devices. Now optimisation of the parameters has
been done with specification Lt = 32 nm , nt = 3 , pitch = 8 nm.
The optimization with aim to maximize the current ratio for our
device with optimizing the parameters present in our device has
been performed and the result shows that Ioff Minima is observed
for (n,m) = (16,0) The optimized Nd obtained is 1e20(cm-3). So we
conclude that Modelling of the device based on device physics
predicts its characteristics to a good extent and Parameter
extraction using optimization algorithm which converges the result
within the desired range.
Figure 17: Drain current verses the gate to source voltage with
constant VDS
Figure 18: drain current verses drain to source voltage at
constant VGS
CONCLUSION:The conclusion of this entire work is that a good and
robust model of CNTFET has been achieved with optimized parameters.
The parameter which has been considered has been optimized in order
to get maximum performance of the device. The main objective of
this optimization process was to achieve a good ION by IOFF ratio
which has been achieved as the IOFF has been minimized. As a
compact model, good agreement with the fabricated device on both dc
and ac characteristics can be demonstrated. Further improvements to
this implemented model, in order to improve the accuracy and
improve run time can be done by using a more accurate band
structure model or by the considerations of more parameters which
have been ignored in the models. One such parameter is the
diffusion at the source/drain junctions are ignored.Regarding
future work after optimizing these parameters we can model this
device using its parameter. So we can end up specifying a design
library especially dedicated to our device and design
standards.
