Simulation Study of Carbon Nanotube Field Emission Device –
from Ab-initio Field Emission Calculation to Macroscopic Electron
Trajectory Simulation
Yung-Chiang Lan and Li-Gan Tien
National Center for High-performance Computing
7 R&D Rd. VI, Science Park, Hsinchu, Taiwan 300, Taiwan
Abstract: The workfunction of capped single-walled zigzag (10, 0) carbon nanotube has been calculated by the
plane-wave pseudopotential, density functional theory method. The calculated workfunction under no external
applied field is 5.16 eV, which is in a good agreement with the experiment measurement. The relation between
the workfunction and the external electric field is also established, which is then used in the macroscopic field
emission model for the electron trajectory simulation. The electron trajectory simulation of the field emission
display with planar-gate structure is performed. The simulation results exhibit that, in this type structure, the
gate voltage has a strong effect on display’s resolution. The good resolution is achieved when the gate voltage is
adjusted to converge the electron beams on anode plate. Due to lack of field effect in y-direction, the electrons
will diverge in this direction. So the spot size of the current density on anode plate and the resolution of the
display will be affected by this factor.
Key-Words: carbon nanotube, workfunction, plane-wave pseudopotential, density functional theory, field
emission display, planar-gate structure
used in the F-N equation to model the field emission
and the classical electrodynamics is used to study the
electron movement.
For the development of FEDs, the triode
structures are more attractive for their lower driving
voltage and higher light efficiency. Recently, a new
CNTs-based FED structures, i.e. planar-gate [12]
structures, is proposed and exhibits good
characteristics. In this type structure, the lateral gate
electrode generates transverse electric field to pull
out electrons from the neighboring emitters. But how
the current density distributed on the anode plate and
how the display’s resolution affected by the bias
conditions of the emitter and the gate are not clear
yet. In this paper, the simulation studies, based on
the above-mentioned multiscale methods, of such
triode structure CNT-FEDs were performed to
investigate these problems.
The paper is organized as follows. Section 2
describes the simulation models and methods.
Section 3 contains the simulation results and
discussions. Finally, the conclusions are given in
section 4.
1 Introduction
The discovery of carbon nanotubes (CNTs) has
drawn a lot of attention due to their unique physical
properties and various potential applications [1-5].
Due to their high aspect ratios and small radii of
curvature, CNTs exhibit excellent field emission
characteristics. A high field emission current density
of 10 mA/cm2 and low turn-on electric field of 0.75 ~
0.8 V /m have been reported [6][7], which are very
advantageous for flat panel display applications.
For field emission display (FED) design, the
emission characteristics of CNTs and the electron
dynamics under the electric field are very crucial,
which will affect the brightness and resolution
explicitly. But it is very difficult to study these two
processes at the same time for their divergence in the
spatial dimension. The CNT’s field emission are
often treated by way of quantum mechanics [8][9].
The moving electrons, however, is viewed as
classical charged particles and treated via classical
mechanics generally [10]. The connection between
the nano-scale field emission characteristics and the
macroscopic field emission model used in the
electron dynamics is the so-called Fowler-Nordheim
(F-N) equation and the equivalent workfunction [11].
Therefore, the study of CNTFED system can be
divided into two steps. In the fist step, the equivalent
workfunction is obtained from the experiments or the
quantum mechanical calculations such as the density
function theory. Then the equivalent workfunction is
2 Simulation Methods
2.1 Quantum-mechanical Calculations of
Carbon Nanotube
In this study, the quantum-mechanical
calculations of CNT’s effective workfunction are
1
performed using the CASTEP code [13][14], which
is a plane-wave, pseudopotential program based on
density functional theory (DFT). The generalized
gradient approximation (GGA) [15] is adopted in the
exchange-correlation potential with the Perdew
-Wang Scheme (PW91) [16][17]. And the nuclei and
core electrons are represented by ultrasoft
pseudopotentials.
A capped single-walled zigzag (10, 0) carbon
nanotube, as seen in Fig. 1, is considered in this study.
The CNT is modeled by 5 layers of carbon ring with
one mouth capped by a half of a C60 molecule and the
dangling bounds at the other end are saturated by
hydrogen atoms to avoid the boundary effects, given
a final structure consisting 120 carbon atoms and 10
hydrogen atoms. The nanotube is placed in the center
of a rectangular unit cell with the tube axis along the
electromagnetic fields are advanced in time at each
time step. In each time step, the charged particles are
moved according to the Lorentz equation using the
fields advanced in each time step. The weighted
charge density and current density at the grids are
subsequently calculated.
The obtained charge
density and current density are subsequently used as
sources in the Maxwell equations for advancing the
electromagnetic fields.
In the field emission process, electron emission
is modeled by the Fowler-Nordheim (F-N) equation
as
J
(1)
where A  1.5415  106 , E is the normal
component of the electric field at the emitter surface,
 is the work function of the emitter,

c axis. The cell dimensions are a  b  20  and

c  26  . A constant external electric field is
applied to the tube along the direction of the c axis.
The magnitude of the potential linearly decreases

A E2
B( y)3 2
exp
(
),
 t2
E
B  6.8308  109 , t 2 is taken as approximately 1.1,
and ( y)  0.95  y2 with y  3.79  105  E1 2  in
SI units. The MAGIC code provides an F-N equation
module to simulate the field emission. Initially, the
electrostatic field along the emitter surface is
determined for a given geometry and applied
voltages. The emission charge is determined by Eq.
(1) according to the local electric field. The
simulation proceeds by pushing the emitted
electrons, weighting the current and charge densities
to the grids, updating the electromagnetic fields, and
calculating the emission charges. These processes
are repeated for each time step until the specified
number of time steps is reached. The space-charge
effects are automatically included in such a
simulation algorithm.
The configuration of planar-gate type FED
investigated in this study is shown in Fig. 3. The
Cartesian x-y-z coordinate system is adopted in this
simulation study. The simulation domain and the
geometry parameters are shown in Fig. 4. Due to the
extremely small dimension of the carbon nanotubes,
it is impractical to precisely model every nanotube in
the whole display structure. Instead, we model the
emitter as a thin film electrode and adjust the work
function of the artificial emitter, which is deduced
from the quantum-mechanical calculation mentioned
in section 2.1. In our study, the voltage difference
between cathode and anode is set to 2000 V. And the
voltage differences between cathode and gate are set
to 30 V, 90 V, 150 V, and 210 V.

from 13  of c axis to  13  , so that the nanotube
in the center feels the constant electric field with
preset voltage from the direction of the c axis only.
This external potential is added to the Hamiltonian in
the unit cell when the total energy is calculated. The
fast-Fourier-transform (FFT) grid is set to
100  100  128 and a cutoff energy of 300 eV is used
to expand the valence electronic wavefunctions in
plane waves. Besides the valence bands, 12 extra
bands are added to represent the conduction bands.
For each external potential case, the workfunction is
calculated after CNT’s geometry is optimized. And
the calculations are executed on IBM P690 at NCHC
with 16 CPUs and 48 hours for a typical case.
The workfunction of a metal surface is defined
as the energy needed to take an electron from the
Fermi level  to the vacuum level, i.e.      ,
where  is the electrostatic potential energy in the
vacuum region. But due to the unique geometry of
single-walled carbon nanotube, the surface of CNT is
represented by the tip and wall. Therefore, the
vacuum level is defined, in this study, as the
maximum average potential energy along the tube
axis and near the tip of the tube, as shown in Fig. 2.
2.2 Simulation of Electron Trajectory
A particle-in-cell computer simulation code
MAGIC [18] was used for the trajectory simulation.
MAGIC is a three-dimensional, finite-difference,
time domain code for self-consistent simulation of
the electromagnetic fields and charged particles. The
2
Therefore the current density distributions look like
strips on anode plates.
Figure 9 shows the current density distributions
along x-direction at y = 0.0011 m and on anode plate
for planar-gate structure device. It can be found
clearly that, for the planar-gate structure, the current
density profiles converge to single peaks gradually
when the gate voltage is increased from 30 V to 150
V. But when the gate voltage is increased more (from
150 V to 210 V), the single current density profile
becomes wider.
3 Results and Discussions
The averaged potential energy distributions of
the carbon nanotube along the tube axis have been
shown in Fig. 2, with 0 V and 2 V external voltages.
The Fermi levels for each case are also shown in Fig.
2 for comparison. Figure 2 also exhibits that the
averaged potential energy oscillates in tube’s wall
region.
The valleys of the potential profile
correspond to the places of the carbon nuclei. The
variation of calculated workfunctions with respect to
the external voltages is shown in figure 5. The
calculated workfuction with no applied external
voltage is 5.16 eV, which shows a good agreement
with the experiment measurement with a value of
5.08 eV by Ye et al. [19]. From Fig. 5, a linear
relation between the workfunction and the external
electric field can be established, as
 (eV) = 5.053 - 0.512  E (V/2.6 nm) .
4 Conclusion
In summary, we have studied the workfunction
of capped single-walled zigzag (10, 0) carbon
nanotube based on the plane-wave pseudopotential,
density functional theory calculations.
The
calculated workfunction of CNT under no external
applied field is 5.16 eV, which is in a good agreement
with the experiment measurement. A linear relation
between the workfunction and the external electric
field is established, as described by Eq. 2, which is
then used in the macroscopic field emission model.
The electron trajectory simulation results
exhibit that, for planar-gate structure, the gate voltage
has a strong effect on the display’s resolution. The
good resolution is achieved only when the gate
voltage is adjusted to converge the electron beams on
anode plate. Due to lack of field effect in y-direction,
the electrons will diverge in this direction. So the
spot size of the current density on anode plate and the
resolution of the display will be affected by this
factor.
(2)
This relation will be used in the macroscopic field
emission model when the electron trajectory
simulations are performed.
Figure 6 shows the simulation results of the 3-D
electron trajectories of the planar-gate CNT-FED
device at the emitter voltage of 0 V, the anode
voltages of 2000 V, and the gate voltage of 90 V, 150
V, and 210 V, respectively. The geometry and
dimension of the simulated triode-type CNT-FED
device have been depicted in section 2.2. The
corresponding current density distributions on anode
plates and the equal-potential contours on x-z plane
are shown in Fig. 7 and Fig. 8, respectively. As
shown in Figs. 6 - 8, the gate electrode attracts the
electrons emitting from its neighboring (left- and
right-side) CNT emitters in x-direction. For the
90-V-gate-voltage case (Fig. 6(a), Fig. 7(a) and Fig.
8(a)), the attracting force from gate is not enough
strong that the electrons from left- and right-side
emitters can’t converge on anode plate (in
x-direction). Therefore the resolution of the display
is not good under this bias condition. When the gate
voltage is 150 V, as shown in Fig. 6(b), Fig. 6(b) and
Fig. 8(b), the electrons from left- and right-side
emitters can converge on anode plate (in x-direction).
And the resolution of the display will become better.
When the gate voltage is increased to 210 V, as
shown in Fig. 6(c), Fig. 7(c) and Fig. 8(c), the gate’s
attractive force becomes too strong that the line
widths along x direction are larger. In this case, the
display’s resolution will becomes poor again. It can
also be found that, from Fig. 6 and Fig. 7, due to lack
of electric fields to attract electrons in y-direction, the
electrons will distribute divergently in this direction.
Acknowledgments
The authors would like to acknowledge
National Center for High-performance Computing
for providing the computing facilities. This work is
supported partially by ERSO/ITRI.
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Fig. 2 Averaged electrostatic potential
distribution along the carbon nanotube
axis for 0 V and 2 V external voltages.
Fig. 3 Schematic of the planar-gate CNT-FED
structure.
Fig. 4 Schematic of simulation domain and
geometrical parameters used in the
electron trajectory simulation. The units
are in m.
Fig. 1 Structure model of capped single-walled
zigzag (10, 0) carbon nanotube.
Fig. 5 Variation of calculated workfunctions
with the external applied voltage.
4
(a)
(b)
(c)
Fig. 6 Three-dimensional electron trajectories of the planar-gate structure with emitter voltage of 0 V,
anode voltage of 2000 V, and gate voltage of (a) 90 V, (b) 150 V, and (c) 210 V, respectively.
(a)
(b)
(c)
Fig. 7 Current density distributions on anode plate of the planar-gate structure with emitter voltage of 0
V, anode voltage of 2000 V, and gate voltage of (a) 90 V, (b) 150 V, and (c) 210 V, respectively.
(c)
(a)
(b)
Fig. 8 Equalpotential contours on x-z plane of the planar-gate structure with emitter voltage of 0 V,
anode voltage of 2000 V, and gate voltage of (a) 90 V, (b) 150 V, and (c) 210 V, respectively.
Fig. 9 Current density distributions along
x-direction at y = 0.0011 m and on
anode plate for planar-gate structure
with different emitter-to-gate voltage.
5