arXiv:0704.1682v1  [cond-mat.mtrl-sci]  13 Apr 2007
Modeling the Field Emission Current Fluctuation in Carbon Nanotube
Thin Films
N. Sinha*, D. Roy Mahapatra**, J.T.W. Yeow* and R. Melnik***
* Department of Systems Design Engineering, University of Waterloo,Waterloo, ON, Canada
** Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India
*** Mathematical Modeling and Computational Sciences,Wilfrid Laurier University, Waterloo, ON, Canada
jyeow@engmail.uwaterloo.ca
ABSTRACT
Owing to their distinct properties, carbon nanotubes
(CNTs) have emerged as promising candidate for field
emission devices. It has been found experimentally that
the results related to the field emission performance
show variability. The design of an efficient field emit-
ting device requires the analysis of the variabilities with
a systematic and multiphysics based modeling approach.
In this paper, we develop a model of randomly oriented
CNTs in a thin film by coupling the field emission phe-
nomena, the electron-phonon transport and the mechan-
ics of single isolated CNT. A computational scheme is
developed by which the states of CNTs are updated in
time incremental manner. The device current is calcu-
lated by using Fowler-Nordheim equation for field emis-
sion to study the performance at the device scale.
Keywords: carbon nanotube, field emission, electro-
dynamics, current density.
1 INTRODUCTION
Field emission from carbon nanotubes (CNTs) was
first reported in 1995 [1],[2]. With advancement in syn-
thesis techniques, application of CNTs in field emis-
sion devices, such as field emission displays, gas dis-
charge tubes and X-ray tube sources has been success-
fully demonstrated [3], [4]. Field emission performance
of a single isolated CNT is found to be remarkable due to
its structural integrity, geometry, chemical stability and
high thermal conductivity. One can use a single CNT
to produce an electron beam in a single electron beam
device. However, in many applications (such as X-ray
imaging systems), a continuous or patterned film is re-
quired to produce several independent electron beams.
However, the situation in these cases becomes complex
due to coupling among (i) the ballistic electron- phonon
transport at moderate to high temperature range, (ii)
field emission from each of the CNT tip and (iii) electro-
dynamic forces causing mechanical srain and deforming
CNTs (and thus changing the electron density and dy-
namic conductivity). In such cases, the individual CNTs
are not always inclined normal to the substrate surface
(as shown in Fig. 1, where CNT tips are oriented in a
random manner). This is the most common situation,
which can evolve from an initially ordered state of uni-
formly distributed and vertically oriented CNTs. Such
evolution process must be analyzed accurately from the
view point of long-term performance of the device. The
interest of the authors’ towards such an analysis and de-
sign studies stem from the problem of precision biomed-
ical X-ray generation.
In this paper, we focus on a diode configuration,
where the cathode contains a CNT thin film grown on a
metallic substrate and the anode is a copper plate acting
as emission current collector. Here, the most important
requirement is to have a stable field emission current
without compromising the lifetime of the device. As
the CNTs in the film deplete with time, which is due
to burning and fragmentation that result in a decreas-
ing number of emitting sites, one observes fluctuation
in the output current. Small spikes in the current have
also been observed experimentally [5], which can gener-
ally be attributed to the change in the gap between the
elongated CNT tip and the anode, and also possibly a
dynamic contact of pulled up tip with the anode under
high voltage. As evident from the reported studies [5],
it is important to include various coupled phenomena in
a numerical model, which can then be employed to un-
derstand the effects of various material parameters and
also geometric parameters (e.g. CNT geometry as well
as thin film patterns) on the collective field emission
performance of the thin film device. Another aspect of
interest in this paper is the effect of the angle of orien-
tation of the CNT tips on the collective performance.
A physics based modeling approach has been developed
here to analyze the device-level performance of a CNT
thin film.
2 MODEL FORMULATION
The physics of field emission from metallic surfaces
is fairly well understood. The current density (J) due to
field emission from a metallic surface is usually obtained
by using the Fowler-Nordheim (FN) equation [6]
CΦ3/2
, (1)
where E is the electric field, Φ is the work function of
the cathode material, and B and C are constants. How-
http://arxiv.org/abs/0704.1682v1
Figure 1: SEM image showing randomly oriented tips
of CNTs in a thin film.
E(x,y)
Cathode
Anode
Figure 2: CNT array configuration.
ever, in the case of a CNT thin film acting as cathode,
the surface of the cathode is not smooth (like the metal
emitters) and consists of hollow tubes in curved shapes
and with certain spacings. An added complexity is the
realignment of individual CNTs due to electrodynamic
interaction between the neighbouring CNTs during field
emission. Analysis of these processes requires the de-
termination of the current density by considering the
individual geometry of the CNTs, their dynamic ori-
entations and the variation in the electric field during
electronic transport.
Based on our previously developed model [7], which
describes the degradation of CNTs and the CNT geom-
etry and orientation, the rate of degradation of CNTs is
defined as
vburn = Vcell
dn1(t)
s(s− a1)(s− a2)(s− a3)
n2a21 +m
2a22 + nm(a
1 + a
2 − a
where Vcell is the representative volume element, n1 is
the concentration of carbon atoms in the cluster form
in the cell, a1, a2, a3 are lattice constants, s =
(a1 +
a2 + a3),n and m are integers (n ≥ |m| ≥ 0). The pair
(n,m) defines the chirality of the CNT. Therefore, at
a given time, the length of a CNT can be expressed as
h(t) = h0− vburnt, where h0 is the initial average height
of the CNTs and d is the distance between the cathode
substrate and the anode (see Fig. 2).
The effective electric field component for field emis-
sion calculation in Eq. (1) is expressed as
Ez = −e
−1 dV(z)
, (3)
where e is the positive electronic charge and V is the
electrostatic potential energy. The total electrostatic
potential energy can be expressed as
V(x, z) = −eVs−e(Vd−Vs)
G(i, j)(n̂j−n) , (4)
where Vs is the constant source potential (on the sub-
strate side), Vd is the drain potential (on the anode side),
G(i, j) is the Green’s function [8] with i being the ring
position, n̂j denotes the electron density at node po-
sition j on the ring, and (n,m) denotes the chirality
parameter of the CNT. The field emission current (Icell)
from the anode surface associated with the elemental
volume Vcell of the film is obtained as
Icell = Acell
Jj , (5)
where Acell is the anode surface area and N is the num-
ber of CNTs in the volume element. The toal current is
obtained by summing the cell-wise current (Icell). This
formulation takes into account the effect of CNT tip ori-
entations, and one can perform statistical analysis of the
device current for randomly distributed and randomly
oriented CNTs. However, due to the deformation of the
CNTs due to electrodynamic forces, the evolution pro-
cess requires a much more detailed treatment from the
mechanics point of view.
Based on the studies reported in published literature,
it is reasonable to expect that a major contribution is by
the Lorentz force due to the flow of electron gas along
the CNT and the ponderomotive force due to electrons
in the oscillatory electric field . The oscillatory electric
field could be due to hopping of the electrons along the
CNT surfaces and the changing relative distances be-
tween two CNT surfaces. In addition, the electrostatic
force and the van der Waals force are also important.
The net force components acting on the CNTs parallel
to the Z and the X directions are calculated as [9]
(flz + fvsz )ds+ fcz + fpz , (6)
(flx + fvsx)ds+ fcx + fpx . (7)
where fl, fvs, fc and fp are Lorentz, van der Waals,
coulomb and ponderomotive forces, respectively and ds
is the length of a small segment of CNTs. Next, we em-
ploy these force components in the expression of work
done on the ensemble of CNTs and formulate an en-
ergy conservation law. Due to their large aspect ratio,
the CNTs have been idealized as one-dimensional elastic
members (as in Euler-Bernoulli beam). By introducing
the strain energy density, the kinetic energy density and
the work density, and applying the Hamilton principle,
we obtain the governing equations in (ux′ , uz′) for each
CNT, which can be expressed as
+ρA0ü
x′ −ρA2
πCvs[(r
(m+1))2
−(r(m))2]
θ(z′)
− flx′ − fcx′ = 0 , (8)
−E′A0
E′A0α
∂∆T (z′)
+ ρA0ü
z′0 − πCvs
[(r(m+1))2−(r(m))2]
θ(z′)
−flz′−fcz′ = 0 ,
where ux′ and uz′ are lateral and longitudinal dispace-
ments of the oriented CNTs, E′ is the effective modulus
of elasticity of CNTs, A0 is the effective cross-sectional
area, A2 is the second moment of cross-sectional area
about Z-axis, ∆T (z′) = T (z′)− T0 is the difference be-
tween the absolute temperature (T ) during field emis-
sion and a reference temperature (T0), α is the effective
coefficient of thermal expansion (longitudinal), Cvs is
the van der Waals coefficient, superscript (m) indicates
the mth wall of the MWNT with r(m) as its radius, ∆x′
is the lateral displacement due to pressure and ρ is the
mass per unit length of CNT. We assume fixed bound-
ary conditions (u = 0) at the substrate-CNT interface
(z = 0) and forced boundary conditions at the CNT tip
(z = h(t)).
The governing equation in temperature is obtained
by the thermodynamics of electron-phonon interaction.
By considering the Fourier heat conduction and ther-
mal radiation from the surface of CNT, the energy rate
balance equation in T can be expressed as
dqF − πdtσSB(T
4 − T 40 )dz
′ = 0 , (10)
where dQ is the heat flux due to Joule heating over a
segment of a CNT, qF is the Fourier heat conduction,
dt is the diameter of the CNT and σSB is the Stefan-
Boltzmann constant. Here, we assume the emissivity
to be unity. At the substrate-CNT interface (z′ = 0),
the boundary condition T = T0 is applied and at the tip
we assign a reported estimate of the power dissipated by
phonons exiting the CNT tip [10] to the conductive flux.
We first compute the electric field at the nodes and then
solve all the governing equations simultaneously at each
time step and the curved shape s(x′ + ux′ , z
′ + uz′) of
each of the CNTs is updated. The angle of orientation θ
between the nodes j+1 and j at the two ends of segment
∆sj is expressed as
θ(t) = tan−1
(xj+1 + uj+1x )− (x
j + ujx)
(zj+1 + u
z )− (zj + u
, (11)
= [Γ(θ(t−∆t)j)]
, (12)
where Γ is the usual coordinate transformation matrix
which maps the displacements (ux′ , uz′) defined in the
local (X ′, Z ′) coordinate system into the displacements
(ux, uz) defined in the cell coordinate system (X,Z).
For this transformation, we employ the angle θ(t −∆t)
obtained at the previous time step and for each node
j = 1, 2, 3, . . ..
3 RESULTS AND DISCUSSIONS
The CNT film considered in this study consists of
randomly oriented multiwalled CNTs. The film was
grown on a stainless steel substrate. The film surface
area (projected on anode) is 49.93 mm2 and the aver-
age thickness of the film (based on randomly distributed
CNTs) is 10-14 µm. In the simulation and analysis,
the constants B and C in Eq. (1) were taken as B =
(1.4× 10−6)× exp((9.8929)×Φ−1/2) and C = 6.5× 107,
respectively [11]. It has been reported in the literature
(e.g., [11]) that the work function Φ for CNTs is smaller
than the work functions for metal, silicon, and graphite.
However, there are significant variations in the exper-
imental values of Φ depending on the types of CNTs
(i.e., SWNT/MWNT), geometric parameters. The type
of substrate materials have also significant influence on
the electronic band-edge potential. The results reported
in this paper are based on computation with Φ = 2.2eV .
Following sample configuration has been used in this
study: average height of CNTs h0 = 12µm, uniform
diameter dt = 3.92nm and uniform spacing between
neighboring CNTs at the substrate contact region in the
film d1 = 2µm. The initial height distribution h and the
orientation angle θ are randomly distributed. The elec-
trode gap (d) is maintained at 34.7µm. The orientation
of CNTs is parametrized in terms of the upper bound of
the CNT tip deflection (denoted by h0/m
′, m′ >> 1).
Several computational runs are performed and the out-
put data are averaged out at each sampling time step.
For a constant bias voltage (650V in this case), as the
initial state of deflection of the CNTs increases (from
h0/50 to h0/25), the average current reduces until the
initial state of deflection becomes large enough that the
electrodynamic interaction among CNTs produces sud-
den pull in the deflected tips towards the anode result-
ing in current spikes (see Fig. 3). As mentioned earlier,
0 20 40 60 80 100
Time (s)
Figure 3: Field emission current histories for various
initial average tip deflections and under bias voltage of
650V. The current I is in Ampere unit.
0 20 40 60 80 100
CNT number
t=100
Figure 4: Comparison of tip orientation angles at t=0
and t=100s.
0 20 40 60 80 100
CNT number
Figure 5: Maximum temperature of CNT tips during
100s of field emission.
spikes in the current have also been observed experimen-
tally. Fig. 4 reveals that after experiencing the elec-
trodynamic pull and Coulombic repulsion, some CNTs
reorient themselves. In Fig. 5, maximum tip tempera-
ture distribution over an array of 100 CNTs during field
emission over 100 s duration is plotted. The maximum
temperature rises up to approximately 350 K.
4 CONCLUSION
In this paper, a model has been developed from the
device design point of view, which sheds light on the
coupling issues related to the mechanics, the thermo-
dynamics, and the process of collective field emission
from CNTs in a thin film, rather than a single isolated
CNT. The proposed modeling approach handles several
complexities at the device scale. While the previous
works by the authors mainly dealt with decay, kinemat-
ics and dynamics of CNTs during field emission, this
work includes some more aspects that were assumed
constant earlier. These include: (i) non-local nature of
the electric field, (ii) non-linear relationship between the
electronic transport and the electric field, and (iii) non-
linear relationship between the electronic transport and
the heat conduction. The trend in the simulated results
matches qualitatively well with the results of published
experimental studies.
REFERENCES
[1] A.G. Rinzler, J.H. Hafner, P. Nikolaev, L. Lou, S.G.
Kim, D. Tomanek, D. Colbert and R.E. Smalley,
Science 269, 1550, 1995.
[2] W.A. de Heer, A. Chatelain, and D. Ugrate, Science
270, 1179, 1995.
[3] J.M. Bonard, J.P. Salvetat, T. Stockli, L. Forro and
A. Chatelain, Appl. Phys. A 69, 245, 1999.
[4] Y. Saito and S. Uemura, Carbon 38, 169, 2000.
[5] J.M. Bonard, J.P. Salvetat, T. Stockli, L. Forro and
A. Chatelain, Phys. Rev. B 67, 115406, 2003.
[6] R.H. Fowler and L. Nordheim, Proc. Royal Soc.
London A 119, 173, 1928.
[7] N. Sinha, D. Roy Mahapatra, J.T.W. Yeow, R. Mel-
nik and D. A. Jaffray, Proc. IEEE Nano 2006.
[8] A. Svizhenko, M.P. Anantram and T.R. Govindan,
IEEE Trans. Nanotech. 4, 557, 2005.
[9] N. Sinha, D. Roy Mahapatra, J.T.W. Yeow, R. Mel-
nik and D. A. Jaffray, J. comp. Theor. Nanosci.
(Accepted).
[10] H.-Y. Chiu, V.V. Deshpande, H.W.Ch. Postma,
C.N. Lau, C. Mikó, L. Forró and M. Bockrath,
Phys. Rev. Lett. 95, 226101, 2005.
[11] Z.P. Huang, Y. Tu, D.L. Carnahan and Z.F. Ren,
“Field emission of carbon nanotubes,” Encyclope-
dia of Nanoscience and Nanotechnology (Ed. H.S.
Nalwa) 3, 401-416, 2004.

  Owing to their distinct properties, carbon nanotubes (CNTs) have emerged as
promising candidate for field emission devices. It has been found
experimentally that the results related to the field emission performance show
variability. The design of an efficient field emitting device requires the
analysis of the variabilities with a systematic and multiphysics based modeling
approach. In this paper, we develop a model of randomly oriented CNTs in a thin
film by coupling the field emission phenomena, the electron-phonon transport
and the mechanics of single isolated CNT. A computational scheme is developed
by which the states of CNTs are updated in time incremental manner. The device
current is calculated by using Fowler-Nordheim equation for field emission to
study the performance at the device scale.

arXiv:0704.1682v1  [cond-mat.mtrl-sci]  13 Apr 2007
Modeling the Field Emission Current Fluctuation in Carbon Nanotube
Thin Films
N. Sinha*, D. Roy Mahapatra**, J.T.W. Yeow* and R. Melnik***
* Department of Systems Design Engineering, University of Waterloo,Waterloo, ON, Canada
** Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India
*** Mathematical Modeling and Computational Sciences,Wilfrid Laurier University, Waterloo, ON, Canada
jyeow@engmail.uwaterloo.ca
ABSTRACT
Owing to their distinct properties, carbon nanotubes
(CNTs) have emerged as promising candidate for field
emission devices. It has been found experimentally that
the results related to the field emission performance
show variability. The design of an efficient field emit-
ting device requires the analysis of the variabilities with
a systematic and multiphysics based modeling approach.
In this paper, we develop a model of randomly oriented
CNTs in a thin film by coupling the field emission phe-
nomena, the electron-phonon transport and the mechan-
ics of single isolated CNT. A computational scheme is
developed by which the states of CNTs are updated in
time incremental manner. The device current is calcu-
lated by using Fowler-Nordheim equation for field emis-
sion to study the performance at the device scale.
Keywords: carbon nanotube, field emission, electro-
dynamics, current density.
1 INTRODUCTION
Field emission from carbon nanotubes (CNTs) was
first reported in 1995 [1],[2]. With advancement in syn-
thesis techniques, application of CNTs in field emis-
sion devices, such as field emission displays, gas dis-
charge tubes and X-ray tube sources has been success-
fully demonstrated [3], [4]. Field emission performance
of a single isolated CNT is found to be remarkable due to
its structural integrity, geometry, chemical stability and
high thermal conductivity. One can use a single CNT
to produce an electron beam in a single electron beam
device. However, in many applications (such as X-ray
imaging systems), a continuous or patterned film is re-
quired to produce several independent electron beams.
However, the situation in these cases becomes complex
due to coupling among (i) the ballistic electron- phonon
transport at moderate to high temperature range, (ii)
field emission from each of the CNT tip and (iii) electro-
dynamic forces causing mechanical srain and deforming
CNTs (and thus changing the electron density and dy-
namic conductivity). In such cases, the individual CNTs
are not always inclined normal to the substrate surface
(as shown in Fig. 1, where CNT tips are oriented in a
random manner). This is the most common situation,
which can evolve from an initially ordered state of uni-
formly distributed and vertically oriented CNTs. Such
evolution process must be analyzed accurately from the
view point of long-term performance of the device. The
interest of the authors’ towards such an analysis and de-
sign studies stem from the problem of precision biomed-
ical X-ray generation.
In this paper, we focus on a diode configuration,
where the cathode contains a CNT thin film grown on a
metallic substrate and the anode is a copper plate acting
as emission current collector. Here, the most important
requirement is to have a stable field emission current
without compromising the lifetime of the device. As
the CNTs in the film deplete with time, which is due
to burning and fragmentation that result in a decreas-
ing number of emitting sites, one observes fluctuation
in the output current. Small spikes in the current have
also been observed experimentally [5], which can gener-
ally be attributed to the change in the gap between the
elongated CNT tip and the anode, and also possibly a
dynamic contact of pulled up tip with the anode under
high voltage. As evident from the reported studies [5],
it is important to include various coupled phenomena in
a numerical model, which can then be employed to un-
derstand the effects of various material parameters and
also geometric parameters (e.g. CNT geometry as well
as thin film patterns) on the collective field emission
performance of the thin film device. Another aspect of
interest in this paper is the effect of the angle of orien-
tation of the CNT tips on the collective performance.
A physics based modeling approach has been developed
here to analyze the device-level performance of a CNT
thin film.
2 MODEL FORMULATION
The physics of field emission from metallic surfaces
is fairly well understood. The current density (J) due to
field emission from a metallic surface is usually obtained
by using the Fowler-Nordheim (FN) equation [6]
CΦ3/2
, (1)
where E is the electric field, Φ is the work function of
the cathode material, and B and C are constants. How-
http://arxiv.org/abs/0704.1682v1
Figure 1: SEM image showing randomly oriented tips
of CNTs in a thin film.
E(x,y)
Cathode
Anode
Figure 2: CNT array configuration.
ever, in the case of a CNT thin film acting as cathode,
the surface of the cathode is not smooth (like the metal
emitters) and consists of hollow tubes in curved shapes
and with certain spacings. An added complexity is the
realignment of individual CNTs due to electrodynamic
interaction between the neighbouring CNTs during field
emission. Analysis of these processes requires the de-
termination of the current density by considering the
individual geometry of the CNTs, their dynamic ori-
entations and the variation in the electric field during
electronic transport.
Based on our previously developed model [7], which
describes the degradation of CNTs and the CNT geom-
etry and orientation, the rate of degradation of CNTs is
defined as
vburn = Vcell
dn1(t)
s(s− a1)(s− a2)(s− a3)
n2a21 +m
2a22 + nm(a
1 + a
2 − a
where Vcell is the representative volume element, n1 is
the concentration of carbon atoms in the cluster form
in the cell, a1, a2, a3 are lattice constants, s =
(a1 +
a2 + a3),n and m are integers (n ≥ |m| ≥ 0). The pair
(n,m) defines the chirality of the CNT. Therefore, at
a given time, the length of a CNT can be expressed as
h(t) = h0− vburnt, where h0 is the initial average height
of the CNTs and d is the distance between the cathode
substrate and the anode (see Fig. 2).
The effective electric field component for field emis-
sion calculation in Eq. (1) is expressed as
Ez = −e
−1 dV(z)
, (3)
where e is the positive electronic charge and V is the
electrostatic potential energy. The total electrostatic
potential energy can be expressed as
V(x, z) = −eVs−e(Vd−Vs)
G(i, j)(n̂j−n) , (4)
where Vs is the constant source potential (on the sub-
strate side), Vd is the drain potential (on the anode side),
G(i, j) is the Green’s function [8] with i being the ring
position, n̂j denotes the electron density at node po-
sition j on the ring, and (n,m) denotes the chirality
parameter of the CNT. The field emission current (Icell)
from the anode surface associated with the elemental
volume Vcell of the film is obtained as
Icell = Acell
Jj , (5)
where Acell is the anode surface area and N is the num-
ber of CNTs in the volume element. The toal current is
obtained by summing the cell-wise current (Icell). This
formulation takes into account the effect of CNT tip ori-
entations, and one can perform statistical analysis of the
device current for randomly distributed and randomly
oriented CNTs. However, due to the deformation of the
CNTs due to electrodynamic forces, the evolution pro-
cess requires a much more detailed treatment from the
mechanics point of view.
Based on the studies reported in published literature,
it is reasonable to expect that a major contribution is by
the Lorentz force due to the flow of electron gas along
the CNT and the ponderomotive force due to electrons
in the oscillatory electric field . The oscillatory electric
field could be due to hopping of the electrons along the
CNT surfaces and the changing relative distances be-
tween two CNT surfaces. In addition, the electrostatic
force and the van der Waals force are also important.
The net force components acting on the CNTs parallel
to the Z and the X directions are calculated as [9]
(flz + fvsz )ds+ fcz + fpz , (6)
(flx + fvsx)ds+ fcx + fpx . (7)
where fl, fvs, fc and fp are Lorentz, van der Waals,
coulomb and ponderomotive forces, respectively and ds
is the length of a small segment of CNTs. Next, we em-
ploy these force components in the expression of work
done on the ensemble of CNTs and formulate an en-
ergy conservation law. Due to their large aspect ratio,
the CNTs have been idealized as one-dimensional elastic
members (as in Euler-Bernoulli beam). By introducing
the strain energy density, the kinetic energy density and
the work density, and applying the Hamilton principle,
we obtain the governing equations in (ux′ , uz′) for each
CNT, which can be expressed as
+ρA0ü
x′ −ρA2
πCvs[(r
(m+1))2
−(r(m))2]
θ(z′)
− flx′ − fcx′ = 0 , (8)
−E′A0
E′A0α
∂∆T (z′)
+ ρA0ü
z′0 − πCvs
[(r(m+1))2−(r(m))2]
θ(z′)
−flz′−fcz′ = 0 ,
where ux′ and uz′ are lateral and longitudinal dispace-
ments of the oriented CNTs, E′ is the effective modulus
of elasticity of CNTs, A0 is the effective cross-sectional
area, A2 is the second moment of cross-sectional area
about Z-axis, ∆T (z′) = T (z′)− T0 is the difference be-
tween the absolute temperature (T ) during field emis-
sion and a reference temperature (T0), α is the effective
coefficient of thermal expansion (longitudinal), Cvs is
the van der Waals coefficient, superscript (m) indicates
the mth wall of the MWNT with r(m) as its radius, ∆x′
is the lateral displacement due to pressure and ρ is the
mass per unit length of CNT. We assume fixed bound-
ary conditions (u = 0) at the substrate-CNT interface
(z = 0) and forced boundary conditions at the CNT tip
(z = h(t)).
The governing equation in temperature is obtained
by the thermodynamics of electron-phonon interaction.
By considering the Fourier heat conduction and ther-
mal radiation from the surface of CNT, the energy rate
balance equation in T can be expressed as
dqF − πdtσSB(T
4 − T 40 )dz
′ = 0 , (10)
where dQ is the heat flux due to Joule heating over a
segment of a CNT, qF is the Fourier heat conduction,
dt is the diameter of the CNT and σSB is the Stefan-
Boltzmann constant. Here, we assume the emissivity
to be unity. At the substrate-CNT interface (z′ = 0),
the boundary condition T = T0 is applied and at the tip
we assign a reported estimate of the power dissipated by
phonons exiting the CNT tip [10] to the conductive flux.
We first compute the electric field at the nodes and then
solve all the governing equations simultaneously at each
time step and the curved shape s(x′ + ux′ , z
′ + uz′) of
each of the CNTs is updated. The angle of orientation θ
between the nodes j+1 and j at the two ends of segment
∆sj is expressed as
θ(t) = tan−1
(xj+1 + uj+1x )− (x
j + ujx)
(zj+1 + u
z )− (zj + u
, (11)
= [Γ(θ(t−∆t)j)]
, (12)
where Γ is the usual coordinate transformation matrix
which maps the displacements (ux′ , uz′) defined in the
local (X ′, Z ′) coordinate system into the displacements
(ux, uz) defined in the cell coordinate system (X,Z).
For this transformation, we employ the angle θ(t −∆t)
obtained at the previous time step and for each node
j = 1, 2, 3, . . ..
3 RESULTS AND DISCUSSIONS
The CNT film considered in this study consists of
randomly oriented multiwalled CNTs. The film was
grown on a stainless steel substrate. The film surface
area (projected on anode) is 49.93 mm2 and the aver-
age thickness of the film (based on randomly distributed
CNTs) is 10-14 µm. In the simulation and analysis,
the constants B and C in Eq. (1) were taken as B =
(1.4× 10−6)× exp((9.8929)×Φ−1/2) and C = 6.5× 107,
respectively [11]. It has been reported in the literature
(e.g., [11]) that the work function Φ for CNTs is smaller
than the work functions for metal, silicon, and graphite.
However, there are significant variations in the exper-
imental values of Φ depending on the types of CNTs
(i.e., SWNT/MWNT), geometric parameters. The type
of substrate materials have also significant influence on
the electronic band-edge potential. The results reported
in this paper are based on computation with Φ = 2.2eV .
Following sample configuration has been used in this
study: average height of CNTs h0 = 12µm, uniform
diameter dt = 3.92nm and uniform spacing between
neighboring CNTs at the substrate contact region in the
film d1 = 2µm. The initial height distribution h and the
orientation angle θ are randomly distributed. The elec-
trode gap (d) is maintained at 34.7µm. The orientation
of CNTs is parametrized in terms of the upper bound of
the CNT tip deflection (denoted by h0/m
′, m′ >> 1).
Several computational runs are performed and the out-
put data are averaged out at each sampling time step.
For a constant bias voltage (650V in this case), as the
initial state of deflection of the CNTs increases (from
h0/50 to h0/25), the average current reduces until the
initial state of deflection becomes large enough that the
electrodynamic interaction among CNTs produces sud-
den pull in the deflected tips towards the anode result-
ing in current spikes (see Fig. 3). As mentioned earlier,
0 20 40 60 80 100
Time (s)
Figure 3: Field emission current histories for various
initial average tip deflections and under bias voltage of
650V. The current I is in Ampere unit.
0 20 40 60 80 100
CNT number
t=100
Figure 4: Comparison of tip orientation angles at t=0
and t=100s.
0 20 40 60 80 100
CNT number
Figure 5: Maximum temperature of CNT tips during
100s of field emission.
spikes in the current have also been observed experimen-
tally. Fig. 4 reveals that after experiencing the elec-
trodynamic pull and Coulombic repulsion, some CNTs
reorient themselves. In Fig. 5, maximum tip tempera-
ture distribution over an array of 100 CNTs during field
emission over 100 s duration is plotted. The maximum
temperature rises up to approximately 350 K.
4 CONCLUSION
In this paper, a model has been developed from the
device design point of view, which sheds light on the
coupling issues related to the mechanics, the thermo-
dynamics, and the process of collective field emission
from CNTs in a thin film, rather than a single isolated
CNT. The proposed modeling approach handles several
complexities at the device scale. While the previous
works by the authors mainly dealt with decay, kinemat-
ics and dynamics of CNTs during field emission, this
work includes some more aspects that were assumed
constant earlier. These include: (i) non-local nature of
the electric field, (ii) non-linear relationship between the
electronic transport and the electric field, and (iii) non-
linear relationship between the electronic transport and
the heat conduction. The trend in the simulated results
matches qualitatively well with the results of published
experimental studies.
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