Semi-analytical Model for Schottky-barrier
Carbon Nanotube and Graphene Nanoribbon Transistors
Xuebei Yang†, Gianluca Fiori‡, Giuseppe Iannaccone‡, and Kartik Mohanram†
†Department of Electrical and Computer Engineering, Rice University, Houston
‡Information Engineering Department, University of Pisa, Pisa
firstname.lastname@example.org email@example.com firstname.lastname@example.org@rice.edu
This paper describes a physics-based semi-analytical model for
Schottky-barrier carbon nanotube (CNT) and graphene nanorib-
bon (GNR)transistors. Themodel includes thetreatment of(i)both
tunneling and thermionic currents, (ii) ambipolar conduction, i.e.,
both electron and hole current components, (iii) ballistic transport,
and (iv) multi-band propagation. Further, it reduces the compu-
tational complexity in the two critical and time-consuming steps,
namely the calculation of the tunneling probability and the self-
consistent evaluation of the the surface potential in the channel.
When validated against NanoTCAD ViDES, a quantum transport
simulation framework based on the non-equilibrium Green’s func-
tion method, it is several orders of magnitude faster without sig-
nificant loss in accuracy. Since the model is physics-based, it is
parameterizable and can be used to study the effect of common
parametric variations in CNT diameter and GNR width, Schottky-
barrier height, and insulator thickness.
Categories and Subject Descriptors: B.7.1 [Integrated circuits]:
Types and Design Styles—Advanced technologies
General Terms: Design, Performance
Keywords: Carbon nanotubes, graphene nanoribbons, Schottky-
Carbon-based materials such as carbon nanotubes (CNTs) and,
morerecently, graphenenanoribbons(GNRs), haveattractedstrong
interest as alternative device technologies for future nanoelectron-
ics applications [1–4]. Devices based on these materials offer high
mobility for ballistic transport, low drain-induced barrier lowering,
high mechanical and thermal stability, and high resistance to elec-
tromigration. Although different families of CNTFETs and GNR-
FETs have been fabricated and studied, the most important distinc-
tion is between Schottky-barrier-type (SB-type) and MOSFET-type
FETs [5–7]. SB-type FETs (SBFETs henceforth) are the most eas-
ily fabricated devices, since they use intrinsic CNT/GNR channels
with metallic drain and source contacts. MOSFET-type devices are
characterized by doped CNT/GNR channels and Ohmic contacts,
and pose more engineering challenges. In SBFETs, a SB is formed
between the channel and source/drain contacts and the gate modu-
neling current, as opposed to the thermionic current in MOSFET-
This research was supported by NSF grant CCF-0916636.
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type devices, that dominates the device transport. SBFETs are am-
bipolar, i.e., they conduct both electrons and holes, showing a su-
perposition of n- and p-type behavior.
Modeling approaches for such novel devices  are based on
(i) computationally intensive, quantum-theory-based non-equilib-
rium Green’s function (NEGF) approaches [9, 10] or (ii) simpler
semi-classical approaches [11–18]. NEGF-based approaches are
highly accurate but extremely time consuming. Further, they pro-
vide limited intuition necessary for circuit design and optimization
with multiple devices. In comparison to NEGF-based approaches,
the simpler semi-classical approaches are computationally very ef-
ficient. Further, they are physics-based and parameterizable, pro-
viding good intuition to designers. It has been shown that for
MOSFET-type devices, a semi-classical description is valid for a
channel length down to about 10nm [12–14,19].
However, when GNRFETs and CNTFETs are SBFETs where
switching behavior stems fromthe gate modulation ofthe tunneling
probability through the source/drain-channel contacts, a quantum-
mechanical description is indispensable . State-of-the-art mod-
els for SBFETs, such as [15–18], are based on the semi-classical
approach augmented with quantum theory to handle tunneling in
SBFETs. The earliest model described in  was more efficient
than NEGF-based approaches, but still infeasible for routine sim-
ulations required for circuit design due to two computationally in-
tensive steps: (i) the calculation of tunneling probability at the SBs
and (ii) the self-consistent evaluation of the surface potential in the
channel. In , the tunneling probability and the surface poten-
tial are obtained using approximate closed-form expressions at the
cost of physical intuition. In [17,18], the models are made com-
putationally efficient by considering SBFETs in only the quantum
capacitance limit to eliminate the need for self-consistent solution.
However, this restricts the use of the model to ultimately scaled
devices free of parasitics.
This paper describes an accurate and computationally efficient
physics-based semi-analytical model for SB-type CNTFETs and
GNRFETs. The model includes the treatment of (i) both tunnel-
ing and thermionic currents, (ii) ambipolar conduction, i.e., both
electron and hole current components, (iii) ballistic transport, and
(iv) multi-band propagation. Further, it reduces the computational
complexity in the two critical and time-consuming steps of the
semi-classical SBFET modeling approach. First, in the calculation
of the tunneling probability through the SB using the Wentzel-Kra-
mers-Brillouin (WKB) approach, we propose linear solutions to
substitute for the computationally intensive integral for tunneling
probability. Second, in the self-consistent loop to evaluate the sur-
face potential in the channel, we simplify the process to calculate
the carrier densities by identifying multiple regions with closed-
form solutions. Since the proposed model is physics-based and
doesnotrelyonfittingparameters, itcan beusedtostudyvariations
in device parameters including CNT diameter and GNR width, in-
L = 20nm
Figure 1: Coaxial CNTFET. (a) and (b) are cross sections along
and perpendicular to the channel direction of the CNTFET.
sulator thickness, and SB height. When compared to the original
semi-classical SBFET model and an NEGF-based quantum trans-
port simulation framework NanoTCAD ViDES , the proposed
model is faster by one and four orders of magnitude, respectively,
without significant loss in accuracy.
The paper is organized as follows. We introduce semi-classical
SBFET modeling in section 2. In section 3, we describe our ap-
proach to reduce the computational complexity of semi-classical
SBFET modeling. Section IV presents simulation results and sec-
tion V is a conclusion.
Although we consider a coaxial gate geometry SB-type CNT-
FET with the structure shown in Figure 1 to describe SBFET mod-
eling, the proposed approach can treat other geometries and chan-
nel materials, as shown for a double-gated GNRFET in Sec. 4.2.
The current through the SBFET is calculated using the widely-used
Landauer formula  and is given by the following expressions:
I = Ielectron+ Ihole
(f(E − EFS) − f(E − EFD))Te(E)dE
(f(EFD− E) − f(EFS− E))Th(E)dE
where f() is the Fermi function, EFSand EFDare the Fermi levels
at the source and the drain, respectively, and Te (Th) is the trans-
mission coefficient for electrons (holes). When all terminal biases
are added, the only parameters that remain unknown are the trans-
mission coefficients at the contacts.
Calculating the transmission coefficients: At each contact the
transmission coefficients for electron current and hole current need
to be calculated, for a total of four coefficients: TSe, TSh, TDe and
TDh. The final transmission coefficients for electrons Te and holes
Thare obtained by combining TSe, TDeand TSh, TDh.
The current is divided into two components: thermionic current,
which flows above the SBs and tunneling current, which flows
Figure 2: Illustration of the classical turning points. Note that
in (b) there is band-to-band tunneling.
Figure 3: Spatial energy band diagram along the transport di-
rection. EFSand EFDare source and drain Fermi levels.
through the SB. Whereas the transmission coefficient is 1 for the
thermionic current, it is given by the tunneling probability for the
tunneling current. The tunneling probability is calculated using the
simple but accurate Wentzel-Kramers-Brillouin (WKB) approach
that has been widely used in literature . Based on the WKB
approach, the tunneling probability T is given by
where zinit and zfinal are the classical turning points, illustrated in
Figure 2, and kzis the parallel momentum related to the E-k rela-
tionship of CNTs. Consider electron tunneling for example. kz is
given by the expression:
kn =|3n − 4|
T = exp
„Eg/2 − (EC(z) − E)
where knis the perpendicular momentum of the nth sub-band, Eg
is the energy band-gap, EC(z) is the bottom of the conduction band
in the z direction, R is the radius of the CNT, ac-c is the carbon-
carbon bond distance , and Vppπ is the carbon-carbon bonding
energy . Neglecting phase coherence, the overall transmission
coefficient is given by :
TSe(Sh)+ TDe(Dh)− TSe(Sh)TDe(Dh)
Modeling the energy band diagram: In order to evaluate the
transmission coefficients for the electron and hole tunneling cur-
rent, the energy profile for the bottom of conduction band EC(z)
and the top of valence band EV(z) need to be modeled, respec-
tively, as illustrated in Figure 3. For a long channel, where the SBs
at the source and drain contacts do not influence each other, the
bottom of the conduction band inside the channel Ebot is given by
Ebot = ϕSB+ Uscf+ qVfb
where ϕSBis the SB height, Uscfis the surface potential, and Vfbis
the flat band voltage. For the coaxial gate geometry, the conduction
band ECnear the two contacts can be modeled as :
EC,left(z) = ϕSB− (ϕSB− Ebot)(1 − e
EC,right(z) = (ϕSB− qVDS) − (ϕSB− Ebot− qVDS)(1 − e
where tinsis the thickness of insulator and L is the channel length.
Similarly, the top of the valence band is shifted down by a value of
Eg, which is the energy gap of the CNT.
Determining the surface potential: The surface potential Uscfstill
remains to be evaluated in order to model the energy band diagram.
It is solved through a self- consistent loop [12,14,15] according to:
Uscf = UL+ UP
where ULis the Laplace potential due to the applied terminal biases
and UPis the potential due to the change in carrier densities. If the
substrate is sufficiently thick, UL = −qVG. Evaluating UPis com-
putationally demanding, however, since it depends on the change
in carrier densities as explained below. Consider electrons for il-
lustration. When the terminal biases are zero, the electron density
in the channel is:
where D(E) is the density of states at the bottom of the conduction
band  and f(E − EF) is the Fermi function. Note the integral
is from the bottom of the conduction band to infinity. When the ter-
minal biases are not zero, the device is not at equilibrium and the
states at the bottom of the conduction band are filled by two differ-
ent Fermi levels. States with positive velocity (N+) and negative
velocity (N−) are filled by electrons according to:
UPis then evaluated as:
D(E)f(E − EF)dE
D(E − Uscf)f+(−)dE
Cins(N++ N−− N0)
where Cinsis the insulator capacitance. Since this UPis the poten-
tial due to the density change in electrons, the effects of holes must
be added to obtain:
The most computationally intensive steps of SBFET modeling is
has to be repeated for each combination of terminal biases. Within
the self-consistent loop, the core computational steps are (i) the
evaluation of carrier densities, including electron density and hole
and drain contacts for both electrons and holes. The next section
describes the key contributions of this paper to reduce the compu-
tational complexity of these steps in SBFET modeling.
3.Reducing computational complexity
In this section, we describe two key simplifications to greatly
reduce the computational complexity in SBFET modeling. These
simplifications do not rely on fitting parameters, and hence pre-
serve the physics-based properties of the semi-classical approach
necessary for parameterized circuit design and optimization.
3.1The transmission coefficients
While evaluating the transmission coefficient for tunneling cur-
rent, the integral of kzin the WKB approach (Equation 1) has to be
solved numerically. Traditionally, kzis calculated at each grid point
in the channel between the two turning points and is computation-
ally expensive. In this paper, we derive linear solutions to replace
this integral as follows. Without loss of generality, consider kz in
Figure 4: kz(z) when (a) EC(zinit) − E < Eg/2 and (b)
EC(zinit) − E > Eg/2
the lowest sub-band for the electron tunneling current.
Note that EC(z)−E monotonically decreases from zinitto zfinaland
EC(zfinal) = E. Considering the following two cases:
„Eg/2 − (EC(z) − E)
„Eg/2 − (EC(z) − E)
− (EC(z) − E)
− (EC(z) − E)
(i) EC(zinit) − E < Eg/2: As shown in Figure 4(a), kz(z) de-
creases monotonically. The integral in Equation 1 is the area be-
tween the x-axis, y-axis, and kz(z), which is roughly the area of
a triangle. Hence, the integral can be simplified to the product of
) and zfinal− zinit.
(ii)EC(zinit)−E > Eg/2: AsshowninFigure4(b), kz(z)increases
first, then decreases monotonically. The integral in Equation 1, or
the area between the x-axis, y-axis, and kz(z), can be regarded as
the area of a triangle and a trapezoid. The triangle and the trapezoid
are separated at zpeak, where zpeaksatisfies EC(zpeak) − E = Eg/2
and kz(zpeak) reaches its maximum. From the model for the energy
E + Eg/2 − Ebot
Hence, the integral can be simplified to
Therefore, instead of computing kz at each grid point in the chan-
nel between the two turning points, the transmission coefficients
can be evaluated using only one or two expressions for kz. This
reduces the computation time significantly, with negligible impact
on accuracy, as presented in Sec. 4.
(zfinal− zpeak) +(kz(zinit) + kz(zpeak))
3.2Electron and hole densities
In the self-consistent loop, the calculation of UPis computation-
ally demanding since the change in both electron density and hole
density must be evaluated numerically at each grid point. In our
work, we propose a transformation that simplifies the calculation
of carrier densities as follows. Without loss of generality, consider
electron density. The density of electrons in positive velocity states
N+and in negative velocity states N−are:
00.050.10.15 0.20.250.30.35 0.4 0.45 0.5
00.1 0.20.3 0.40.50.60.70.8
Figure 5: (a) Ebot > Eexp, (b) Ebot < Eexpand ϕSB− Ebot < VDS, (c) Ebot < Eexp, ϕSB− Ebot > VDSand ϕSB− VDS< Eexp
D(E − Uscf)f+dE
D(E − Uscf)TSfS+ TDfD− TSTDfD
TS+ TD− TSTD
D0(E − Ebot+ Eg/2)(TSfS+ TDfD− TSTDfD)
2p(E − Ebot+ Eg/2)2− (Eg/2)2(TS+ TD− TSTD)
D0(E − Ebot+ Eg/2)(TSfS+ TDfD− TSTDfS)
2p(E − Ebot+ Eg/2)2− (Eg/2)2(TS+ TD− TSTD)
Here, fS and fD are Fermi function of the source and the drain
respectively. The total electron density is equal to N++N−. Since
both fS and fD that include exponential terms appear in both N+
and N−, we rearrange terms to simplify the integrals as follows.
E − Ebot+ Eg/2
p(E − Ebot+ Eg/2)2− (Eg/2)2
E − Ebot+ Eg/2
p(E − Ebot+ Eg/2)2− (Eg/2)2
Note that since N++ N−is equal to N1+ N2, rearranging terms
does not change the final result. By making the substitution
N1and N2can be written as:
The integrals are then simplified as follows, and we illustrate this
by considering N1:
TS+ TD− TSTD
TS+ TD− TSTD
(E − Ebot+ Eg/2)2− (Eg/2)2
(E?)2+ (Eg/2)2− Eg/2 + Ebot,
2TS(E) − TS(E)TD(E)
TS(E) + TD(E) − TS(E)TD(E)fS(E)dE?
2TD(E) − TS(E)TD(E)
TS(E) + TD(E) − TS(E)TD(E)fD(E)dE?
2TS(E) − TS(E)TD(E)
TS(E) + TD(E) − TS(E)TD(E)fS(E)dE?
(E?)2+ (Eg/2)2− Eg/2 + Ebot
In order to simplify the integral, it is necessary to investigate the
shape of P(E?). P(E?) is the product of two terms: the Fermi
function and the transmission coefficient (T-term). At low ener-
gies, the Fermi function is always close to 1 and the T-term dom-
inates the shape of P(E?). When the energy increases, the Fermi
function has an exponential behavior, and changes in the value of
the T-term are slow in comparison to the exponential fall-off in
the value of the Fermi function. Hence, the shape of P(E?) is
dominated by the Fermi function in this regime. We assume that
Fermi function begins to exhibit exponential behavior at the energy
E?= Eexp where fS(pE2
lowing regimes, as illustrated in Figure 5:
exp+ (Eg/2)2− Eg/2 + Ebot) ≈ 0.90.
With this boundary condition, we divide the problem into the fol-
1. Ebot > Eexp
P(E?) is monotonically decreasing in this situation. In order
to simplify the integral, we use the bisection method to find
the point E?= Emwhere P(Em) = 1/2P(0), and approxi-
mate the integral in Equation 2 to 2P(Em)·Em.
2. Ebot < Eexpand ϕSB− Ebot < VDS
Under these conditions, there are two regions separated by
Eexp. In the first region, the Fermi function is almost con-
stant and the T-term dominates the shape of the curve. Upon
closer examination, it is clear that the T-term is equal to
TS(E) because TD(E) = 1. We use a linear function to sim-
plify the integral by calculating TS(Eexp/2), and simplify the
integral in this region to TS(Eexp/2)·Eexp. In the second re-
gion, the Fermi function dominates, and the conditions are
similar to Ebot > Eexpabove. The total value of the integral
in Equation 2 is given by the sum of the two regions.
3. Ebot < Eexpand ϕSB− Ebot > VDS
Under these conditions, there may be two or three regions,
depending on whether ϕSB − VDS is above Eexp or not. If
ϕSB−VDS ≥ Eexp, it is similar to condition (2) above. There
are two regions, with the T-term dominating the first. Al-
though the T-term is not equal to TS(E), it is still possible
to use a linear function to simplify the integral. In the sec-
ond region, the Fermi function dominates and it is equivalent
to case (1) above. If ϕSB− VDS < Eexp, there are three re-
gions. The first interval is from from 0 to the point where
p(E?)2+ (Eg/2)2− Eg/2 + Ebot = ϕSB− VDS. If the en-
the source has SBs. In both regions, we use a linear func-
tion to simplify the integral. Finally, the third region uses the
same conditions as (1) above.
ergy increases to Eexp, it is in the second region where only
Although the self-consistent loop is still needed to evaluate Uscf,
simplifying the calculation of the transmission coefficients and the
carrier densities reduces the computational burden in the two core
loops significantly. Hence, the overall complexity of the semi-
classical SBFET model is reduced without loss in accuracy.
4. Results and discussions
We begin by comparing the results of our proposed semi-ana-
lytical model to the original semi-classical model as well as to
the rigorous quantum atomistic simulator NanoTCAD ViDES .
We consider a CNTFET with coaxial gate geometry and a 20nm
long (13,0) CNT channel. The gate insulator is SiO2 with tins =
1nm. The SB height is assumed to be half of the band-gap. In
Figure 6, we compare the I-V curves obtained from our semi-
analytical model, the original semi-classical model, and NanoT-
CAD ViDES, respectively, for several bias points. Ambipolar char-
acteristics due to both electron and hole conduction are clearly
shown, and the drain voltage exponentially increases the minimum
leakage current. The point of minimum current is at VGS = VDS/2.
As drain voltage increases, SBFETs show linear behavior in the
overall range of gate bias. For example, the drain current and
the channel charge for VDS = 0.8V are linearly proportional to
VGS, whereas those for VDS = 0.4V show exponential behavior in
the sub-threshold region. Note that the difference between the re-
sults of the proposed semi-analytical model and the original semi-
classical model is very small, which demonstrates that our sim-
plification does not result in significant loss of accuracy. In com-
parison, the difference between the results of the proposed model
and NanoTCAD ViDES is slightly higher. This can be attributed
to the difference in the calculation of the transmission coefficients
between the semi-classical and quantum approaches. NanoTCAD
ViDES considers phase coherence of scattering at the two SBs and
the energy profile is more rigorous, which explains the oscillating
nature of the computed transmission coefficient. However, since
this treatment is at the atomic level and extremely computation-
ally intensive, it is not feasible in semi-classical approaches. Ta-
ble 1 presents the average computational time for 80 bias points
for NanoTCAD ViDES, the original semi-classical model, and the
proposed model. Our semi-analytical model is one order of mag-
nitude faster than the original semi-classical model and four orders
of magnitude faster than NanoTCAD ViDES.
Table 1: Average computational time for NanoTCAD ViDES,
original semi-classical SBFET model, and the proposed model.
Semi-classical SBFET model
Proposed semi-analytical model
Parameter variations play an important role in CNTFET elec-
tronics because they significantly influence both the “on” and “off”
current. Typical parameters considered in CNTFET simulation and
design include CNT chirality, insulator thickness, and SB height.
Our model is able to simulate the effect of parameter variations. Al-
though prior work [9,15] has investigated these problems in detail
using rigorous simulation approaches, our semi-analytical model is
able to provide results that are consistent with these more rigorous
approaches in a fraction of the time. In this section, we will present
the simulation results of variations in CNT chirality (and hence,
diameter), insulator thickness, and SB height.
CNT chirality: The diameter of a CNT is determined by the chi-
rality, while the energy band-gap is inversely proportional to the
diameter. We simulated (11,0), (13,0), and (17,0) CNTFETs, and
00.10.20.3 0.4 0.50.60.70.8
0.4V, 0.6V, and 0.8V.
0 0.10.20.3 0.4 0.50.60.70.8
Figure 7: The effect of CNT chirality on the drain current.
VDS= 0.6V , tinsis 1.5nm, and channel length is 20nm.
the results are shown in Figure 7. As the diameter increases, the
band-gap decreases. This allows more electrons and holes in the
conduction and valence bands, respectively, thereby increasing the
current. However, a small band-gap also increases the “off” cur-
rent, so the Ion/Ioffratio is lower.
Insulator thickness: The insulator thickness tinsinfluences the SB
thickness and the gate capacitance. As tins increases, the SBs at
the two channel ends also become thicker and the gate capacitance
decreases. A low gate capacitance means the gate has less control
over the drain current, and a thick SB lowers the tunneling prob-
ability. Therefore, when tins increases, the current decreases. We
simulated a CNTFET with gate insulator thickness at 1nm, 2nm,
and 3nm, and the results are shown in Figure 8.
SB height: The most commonly studied devices are mid-gap SB-
FETs, i.e., ϕSB = Eg/2, and all the simulations so far have only
considered mid-gap transistors. Under this condition, the minimum
current occurs at VGS = VDS/2. However, depending on the contact
work functions, the SB height can differ from Eg/2 . As the SB
height increases, the electron current is lowered and hole current is
larger, so that the right branch of the curve is shifted down, the
left branch shifted up, and the minimum current point moves to the
right. When the SB height decreases, the right branch shifts up, the
left branch shifts down, and the minimum current point shifts to the
left, as shown in Figure 9. Our results for ϕSB = 0, Eg/2, and Eg
are shown in Figure 9.
4.2 GNRFET simulation
So far, this paper has only considered CNTFETs. However, our
model can also be used to simulate GNRFETs. Since GNRs and
00.1 0.2 0.30.40.5 0.60.70.8
Figure 8: The effect of insulator thickness on CNTFET drain
current. VDS = 0.6V , default gate insulator dielectric constant
is 3.9, and channel length is 20nm.
0 0.10.2 0.30.4 0.50.60.70.8
Figure 9: The effect of SB height on CNTFET drain current.
VDS= 0.4V , tinsis 1.5nm, and channel length is 20nm.
CNTs share many properties, only a few changes are needed. The
major changes include: (i) the energy band-gap of a GNR is deter-
mined by its width instead of the chiralities, (ii) GNRFETs usually
do not have coaxial structures, so the expressions for gate capaci-
tance and energy band diagram need to be changed depending on
the structure (double-gate in our case), and (iii) kz  for GNRs
has a linear relationship with energy that makes the integral of kz
in the WKB even simpler. By making these changes, our model is
compatible with GNRFETs, and the major ideas and simplification
approaches mostly remain valid. Figure 10 presents results com-
paring our model and NanoTCAD ViDES for a double-gated GN-
RFET with an N=12 armchair-edge GNR channel and tins = 2nm.
bond relaxation and third nearest neighbor interactions that have
been shown to play an important role in GNRs.
Current models for Schottky-barrier carbon nanotube and gra-
phene nanoribbon FETs are either computationally expensive or
unable to provide physical intuition for device parameter optimiza-
tion. This paper described a semi-analytical model for such FETs
based on physics-based methods to reduce the computational com-
plexity by simplifying the calculation of the transmission coeffi-
cient and the carrier densities. When compared to the quantum-
theory-based simulator NanoTCAD ViDES and the original semi-
classical model, the proposed model is four orders and one order of
magnitudefaster, respectively, without significantlossofaccuracy.
00.10.20.30.40.5 0.60.7 0.8
N=12 AGNR channel
Figure 10: Results for a GNRFET with N=12 armchair-edge
GNR channel and tins= 2nm.
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