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12-1-2003
Electrostatics of nanowire transistors
Jing Guo
Jing Wang
E. Polizzi
Supriyo Daa
Birck Nanotechnology Center and Purdue University, daa@purdue.edu
Mark S. Lundstrom
School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, lundstro@purdue.edu
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Guo, Jing; Wang, Jing; Polizzi, E.; Daa, Supriyo; and Lundstrom, Mark S., "Electrostatics of nanowire transistors" (2003). Other
Nanotechnology Publications. Paper 14.
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IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 329
Electrostatics of Nanowire Transistors
Jing Guo, Student Member, IEEE, Jing Wang, Student Member, IEEE, Eric Polizzi, Supriyo Datta, Fellow, IEEE, and
Mark Lundstrom, Fellow, IEEE
Abstract—The electrostatics of nanowire transistors are
studied by solving the Poisson equation self-consistently with
the equilibrium carrier statistics of the nanowire. For a one-di-
mensional, intrinsic nanowire channel, charge transfer from the
metal contacts is important. We examine how the charge transfer
depends on the insulator and the metal/semiconductor Schottky
barrier height. We also show that charge density on the nanowire
is a sensitive function of the contact geometry. For a nanowire
transistor with large gate underlaps, charge transferred from bulk
electrodes can effectively “dope” the intrinsic, ungated region and
allow the transistor to operate. Reducing the gate oxide thickness
and the source/drain contact size decreases the length by which
the source/drain electric field penetrates into the channel, thereby,
improving the transistor characteristics.
Index Terms—Electrostatic analysis, nanotecnology, transistors.
I. INTRODUCTION
W
ITH the scaling limit of conventional silicon transistors
in sight, there is rapidly growing interest in nanowire
transistors with one-dimensional (1-D) channels, such as
carbon nanotube transistors [1], [2] and silicon nanowire tran-
sistors [3]. Due to the 1-D channel geometry, the electrostatics
of nanowire devices can be quite different from bulk silicon
devices. Previous studies of carbon nanotube p/n junctions and
metal/semiconductor junctions demonstrated unique properties
of nanotube junctions [4]–[6]. For example, the charge transfer
into the nanowire channel from the metal contacts (or heavily
doped semiconductor contacts) can be significant [4], [5].
In this paper, we extend previous studies by looking at the
dependence of the charge transfer on the metal/semiconductor
Schottky barrier height, the insulator dielectric constant, and the
metal contact geometry. We show that if an intrinsic nanowire
is attached to bulk metal contacts at two ends, large charge
transfer can be achieved if the Schottky barrier is low and the
insulator dielectric constant is high. If, however, the intrinsic
nanowire is attached to 1-D metal contacts, the charge den-
sity on the nanowire depends critically on the electrostatic en-
vironment rather than the properties of the metal contacts. Re-
ducing the gate oxide thickness and the contact size decreases
the distance overwhich the source/drain field penetrates into the
nanowire channel and can, therefore, help to suppress the short
channel effects and improve the transistor performance.
Manuscript received September 9, 2003. September 10, 2003. This work was
supported in part by the National Science Foundation (NSF) under Grant EEC-
0085516, in part by the NSF Network for Computational Nanotechnology, and
by the MARCO Focused Research Center on Materials, Structure, and Devices,
which is funded at MIT, in part by MARCO under Contract 2001-MT-887 and
DARPA under Grant MDA972-01-1-0035. This paper was presented in part at
the IEEE Silicon Nanoelectronics Workshop, Kyoto, Japan, June 2003.
The authors are with the School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 47907 USA.
Digital Object Identifier 10.1109/TNANO.2003.820518
Fig. 1. Modeled, coaxially gated carbon nanotube transistor. The intrinsic
nanotube channel has a diameter of 1.4 nm and the flat band voltage of the
gate is zero. The cylindrical coordinates for solving the Poisson equation is
also shown.
II. APPROACH
We simulated the coaxially gated carbon nanotube transistor
shown in Fig. 1. Although the calculations are for carbon nan-
otube transistors, the general conclusion should apply to other
nanowire transistors with 1-D channels. The equilibrium band
profile and charge density were obtained by solving the Poisson
equation in cylindrical coordinates self-consistently with the
equilibrium carrier statistics of the carbon nanotube. The charge
density per unit length on the nanotube,
, is calculated
by integrating the “universal” nanotube density-of-states (DOS)
[7],
, over all energies
(1)
where
is the electron charge, is the sign function,
and
is the Fermi energy level minus
the middle gap energy of the nanotube,
. Since the
source/drain electrodes are grounded, the Fermi level is set
to zero,
. The nanotube middle gap energy is com-
puted from the electrostatic potential at the nanotube shell,
, where is the nanotube radius.
The electrostatic potential
satisfies the Poisson equation
(2)
where
is the charge density, is the dielectric constant. The
following boundary conditions were used:
at the left metal contact;
at the right metal contact;
at the gate cylinder (the flat band voltage is as-
sumed to be zero)
Also,
is the nanotube bandgap, is the Schottky barrier
height for electrons between the source/drain and the nanotube,
and
is the gate voltage.
We numerically solved the Poisson equation by two methods:
1) the finite difference method and 2) the method of moments
1536-125X/03$17.00 © 2003 IEEE
330 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003
Fig. 2. Schematic plots for (a) a bulk Si structure where the cross-sectional area is assumed to be large and (b) a carbon nanotube channel between bulk metal
electrodes. The Schottky barrier heights for electrons are zero. (c) The conduction band edge and (d) the electron density in the units of doping fraction. Results
for the bulk Si structure are shown as dashed lines and for the nanotube as solid lines.
[8]. In order to improve the convergence when iteratively
solving (1) and (2), the Netwton–Ralphson method (with
details in [9]) was used. The results obtained by the finite
difference method and by the method of moments agree well.
III. R
ESULTS
We first compare the charge transfer from bulk contacts to
the 1-D carbon nanotube to the charge transfer to a bulk silicon
channel. We simulated two cases: 1) an intrinsic bulk Si channel
sandwiched between two metal contacts as shown in Fig. 2(a)
and 2) an intrinsic carbon nanotube channel between metal con-
tacts as shown in Fig. 2(b). In both cases, the Schottky bar-
rier heights between the metal contacts and the semiconductor
channel are zero, which aligns the metal Fermi level of to the
conduction band edge of the semiconductor. Electrons are trans-
ferred from metal contacts into the intrinsic channel due to the
work function difference between the metal and the semicon-
ductor. Fig. 2(c) plots the conduction bands, and Fig. 2(d) plots
the charge densities in the unit of electron per atom for the bulk
Si and nanotube channel. Compared to the bulk Si channel, the
barrier in the nanotube is much lower, and the charge density is
much higher. Although the nanotube is 3
m long, the charge
density at the center of the tube is still as high as 10
e/atom,
about five orders of magnitude higher than that of the bulk Si
in terms of electron fraction. As the result, the carbon nanotube
channel is more conductive.
The charge transfer to the tube is significant because the
charge on tube doesn’t effectively screen the potential pro-
duced by the bulk contacts. Compared to the bulk channel,
the charge element on the nanotube only changes potential
locally. For example, in the bulk channel, the charge element
is a two-dimensional sheet charge, which produces a constant
field. The charge dipole formed by charge sheet in bulk Si and
metal contacts shifts the potential far away. In contrast, for the
nanowire channel, the charge element is a point charge, which
produces a potential decaying with distance
and has little
effect far away (the potential of a point charge dipole decays
even faster as
) [4]. As the result, for the 1-D channel,
the potential produced by the bulk contacts is not screened
by the charge on the nanotube near the metal/semiconductor
interface. The bulk contacts tend to put the conduction band
edge near the Fermi level over the whole 3-
m-long tube if the
metal/CNT barrier height is zero.
We next estimate the charge density in the channel. The es-
timation provides a simple way to understand how the charge
density of the tube varies with the contact and insulator prop-
erties. For the device structure shown in Fig. 2(b), if the metal
contacts are grounded, and the metal/semiconductor work func-
tion difference is
, where is the
nanotube (metal) work function, the electron density is
(3)
where
is the electron potential energy produced by charge
in the channel, and
is the average density-of-states for the
GUO et al.: ELECTROSTATICS OF NANOWIRE TRANSISTORS 331
Fig. 3. Electron density (the dashed line) and hole density (the solid line) at the
center of the 3-
m-longCNT[inFig.2(b)]versus the Schottky barrierheightfor
electrons,
, and that for holes,
. The left axis shows the charge density in
the unit of number of electrons (holes) per unit length and the right axis shows
the same quantity in the unit of charge fraction.
energy between the nanotube middle gap energy and the Fermi
level. The charge element in the 1-D channel only shifts the
potential locally, we approximately relate the potential
to
the electron density at the same position
as follows:
(4)
where
is the electrostatic capacitance per unit length be-
tween the nanotube and the bulk contacts. The electron density
due to the charge transfer from the bulkcontacts can be obtained
from (3) and (4) as
(5)
where the quantum capacitance [10], [11] is defined as
, which is proportional to the average DOS of the nanotube.
Equation (5) can be interpreted in a simple way. The bulk elec-
trodes modulate the charge density of the nanotube through an
insulator capacitor,
, which is in series with the quantum
capacitance of the nanotube.
We now examine how the charge transfer varies with the
Schottky barrier height and the insulator dielectric constant.
Fig. 3, which plots the charge density at the center of the
tube as shown in Fig. 2(b) versus the barrier height, shows
that when the barrier height decreases, the charge density first
increases. Fig. 4, which plots the charge density at the center of
the tube versus the insulator dielectric constant, shows that the
charge density increases as the dielectric constant increases.
The dependence of the charge density on the barrier height
and the dielectric constant can be easily understood based on
(5). Lowering the barrier height increases the metal/CNT work
function difference,
, and increasing the insulator dielectric
constant increases
, both of which increase the electron
density,
(or hole density if the metal/semiconductor
barrier height is lower for holes).
The importance of charge transfer into the carbon nanotube
channel by one- dimensional metal contacts has been previously
discussed in [4]. We, however, reached the same conclusion that
charge transfer into the 1-D channel is significant for a different
contact geometry (the bulk contacts). We also explored the 1-D
Fig. 4. Electron density at the center of the 3-
m-long tube [in Fig. 2(b)]
versus the insulator dielectric constant. The Schottky barrier height for
electrons,
, is zero.
contacts. In this case, the results are quite different from bulk
contacts. The charge density of the nanotube channel is critically
determined by the electrostatic environment (i.e., the potential
and location of nearby bulk contacts) rather than the metal-con-
tact properties, as will be discussed in detail next.
Fig. 5 illustrates the important role of the contact geometry.
We simulated: 1) a CNT between grounded bulk contacts as
shown in Fig. 5(a) and 2) a CNT between grounded wire con-
tacts as shown in Fig. 5(b). In both cases, the tube length is 3
m and a grounded, coaxial gate cylinder is far away with a
radius of 30
m. The S/D contacts have zero Schottky barrier
heights for electrons thus tend to dope the tube n-type, while the
gate has a high work function and zero barrier height for holes
thus tends to modulate the tube to p-type. For the bulk contact
case, the whole tube is doped to n-type by bulk contacts and
the charge density on the tube is independent of the voltage on
the gate cylinder. In contrast, for the wire contacts, the tube is
lightly modulated to p-type and the charge density on the tube
is very sensitive to the potential on the gate, although it is far
away. The results shown in Fig. 5 can be explained as follows.
For the bulk contacts, because the gate cylinder is far away,
the bulk contacts at the ends collect all field lines and image
all charge on the tube, as shown in Fig. 5(a). For the wire con-
tacts, however, the potential produced by the charge on the 1-D
wire decays rapidly with distance, thus several nanometer away
from the metal/semiconductor interface, the wire contacts have
littleeffects.On the other hand, the capacitance between the gate
cylinder and the tube decays slowly (logarithmically) with the
tube radius, thus several nanometer away from the metal/semi-
conductor interface, the charge on the tube images on the gate
rather than the wire contacts nearby. As a result, the charge den-
sity is determined by the potential on the gate. The charge den-
sity on the nanotube channel is essentially determined by the
electrostatic environment.
One consequence of the significant charge transfer is that
nanowire transistors with large gate underlap can still operate.
Fig. 6(a) shows a coaxially gated CNTFET with a 500-nm gate
underlap and the bulk electrodes. Fig. 6(b) plots the conduction
band profile at
and 0.3 V. In the OFF-state (
V), a large barrier is created in the channel and the transistor
is turned off. In the
ON-state ( V), the barrier under
the gate is pushed down. Because the low dimensional charge
332 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003
Fig. 5. Contact geometry. A 3-
m-long CNT between (a) the bulk contacts and (b) the 1-D wire contacts. The tube diameter is 1.4 nm. and Schottky barrier
heights for electrons are zero. A coaxial gate far away with a 30-
m radius is grounded. The workfunction of the gate metal equals to the semiconductor affinity
plus the band gap, so that the gate tends to dope the CNT to p-type. (c) The band profile for (a). (d) The band profile for (b).
Fig. 6. (a) Coaxially gated CNTFET with bulk electrodes and a large gate
underlap. (b) The conduction band profile at
V
=0
V and 0.3 V. The
metal/CNT barrier height for electrons is 50 meV, the ZrO
gate oxide
thickness is 8 nm, the tube diameter is 1.4 nm, the gate length is 2
m, and the
gate underlap is 500 nm.
on the ungated nanotube does not effectively screen the poten-
tial produced by the gate and S/D electrodes, the potential at
the ungated region is close to the Laplace potential produced
by the source and gate electrodes. The conduction band edge is
approximately linear in the ungated region. If the Schottky bar-
rier height between S/D and the channel is
50 meV, the bar-
rier height at the ungated region at the
ON-state is low enough
to deliver an
ON-current of A. This mechanism provides
a possible explanation for the operation of the n-type CNTFET
in a recent experiment by Javey et al. [12], in which an n-type
CNTFET with large, intrinsic gate underlaps still had a good
ON–OFF ratio.
Fig. 7. (a) A coaxially gated CNTFET with a 20-nm-long, intrinsic channel.
The source/drain radius,
R
, is equal to the oxide thickness. The metal/CNT
barrier height for electrons is zero, the tube diameter is 1.4 nm and the dielectric
constant of the gate insulator is
"
=25
(b) the equilibrium conduction band
edge at
V
=0
for the gate oxide thickness
t
=
2, 8, and 20 nm.
One concern about the nanowire transistors with low
meta/CNT Schottky barriers is that due to the significant
charge transfer, it might be difficult to turn off the transistor.
To examine this concern, we simulated the coaxially gated
CNTFET as shown in Fig. 7(a) with different gate oxide thick-
ness. Fig. 7(b), which plots the equilibrium band profile, shows
that when the gate oxide thickness is the same as the channel
length, the source/drain field penetrates into the channel the
channel and the transistor cannot be turned off. When the gate
oxide is thin, however, the gate still has very good control over
the channel and the transistor is well turned off. By solving
![Fig. 4. Electron density at the center of the 3- m-long tube [in Fig. 2(b)] versus the insulator dielectric constant. The Schottky barrier height for electrons, , is zero.](/figures/fig-4-electron-density-at-the-center-of-the-3-m-long-tube-in-2pkogouh.webp)
![Fig. 3. Electron density (the dashed line) and hole density (the solid line) at the center of the 3- m-long CNT [in Fig. 2(b)] versus the Schottky barrier height for electrons, , and that for holes, . The left axis shows the charge density in the unit of number of electrons (holes) per unit length and the right axis shows the same quantity in the unit of charge fraction.](/figures/fig-3-electron-density-the-dashed-line-and-hole-density-the-vojfnskb.webp)
