Chiral and quantum size effects of single-wall carbon nanotubes on field emission

Abstract

The emission current of a single-wall carbon nanotube (SWNT) in field emission is studied by the tunneling theory with the tight-binding approach. The emission current is almost independent of the chiral angle of SWNT in low fields, but increases with increase of chiral angles in very high fields. We found a room-temperature quantum size effect of SWNT on field emission. As the diameters of SWNTs increase, the current densities decrease for metallic tubes, but increase for semiconducting tubes. When the diameters of SWNTs are larger than 2 nm the current densities of metallic and semiconducting tubes are very close. These chiral and quantum size effects are originated from the energy band structure of nanotubes.

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Chiral and quantum size effects of single-wall carbon nanotubes
on field emission
Shi-Dong Liang, N. Y. Huang, S. Z. Deng, and N. S. Xua)
Guangdong Province Key Laboratory of Display Materials and Technologies, and State Key Laboratory of
Optoelectronic Materials and Technologies, Zhongshan University, Guangzhou, 510275,
People’s Republic of China
(Received 1 December 2003; accepted 28 May 2004)
The emission current of a single-wall carbon nanotube (SWNT) in field emission is studied by the
tunneling theory with the tight-binding approach. The emission current is almost independent of the
chiral angle of SWNT in low fields, but increases with increase of chiral angles in very high fields.
We found a room-temperature quantum size effect of SWNT on field emission. As the diameters of
SWNTs increase, the current densities decrease for metallic tubes, but increase for semiconducting
tubes. When the diameters of SWNTs are larger than 2 nm the current densities of metallic and
semiconducting tubes are very close. These chiral and quantum size effects are originated from the
energy band structure of nanotubes. © 2004 American Institute of Physics
. [DOI: 10.1063/1.1776337]
As a kind of quasi-one-dimensional material, carbon
nanotubes (CNTs) promise a potential application to nano-
technologies, especially to the field-eimssion electron
sources.1,2Some of the experimental evidence indicates that
the behavior of field emission of CNTs is quite different from
common metals. Typically, the Fowler–Nordheim (FN) plot
of the I–V characteristics has a slight nonlinearity at the high
current range and both single- and multi peaks of the energy
distribution of emitted electrons were observed.2The multi-
peaks energy distributions strongly suggests that the elec-
trons are not emitted from a metallic continuum, but from
energy bands. Physically, for such quasi-one-dimensional
systems the energy band structure of CNT should play an
important role in the field emission. Using the tunneling
theory with the tight-binding approximation we developed a
formula to study the field emission behavior of CNTs.3We
found that the I–V characteristics still roughly follow FN
theory, especially in the low fields, and the current densities
of metallic CNTs are obviously larger than that of semicon-
ducting CNTs in low fields. For the group of CNTs ?12,m?
where m is from 0 to 12, the current density of the tube with
the large chiral angle is larger than that of the small chiral
angle at a high field.3Exactly speaking, the CNT is identified
by its chiral vector ?n,m?, namely the chiral angle and diam-
eter. The energy band structure of CNTs depends on the chi-
ral angle and diameter of CNT. For example, the single-wall
carbon nanotube (SWNT) is metallic if its chiral vector sat-
isfies mod ??n−m?/3?=0 and others are semiconducting.4
The energy gap of the density of states for semiconducting
CNTs is approximately inverse proportional to the diameter
of CNTs.4Theoretically, it is necessary to clarify the effects
of these intrinsic factors on field emission. In this letter we
intend to examine further the chiral and size effects in field
emission.
We start from a general formula in field emission to
calculate the density of the emission current3,5,6
j?F,T? =1
C?
q?
BZ
N?Eq?k??D?Eq?k?,F?dk,
?1?
where N?Eq?k??=?e/?? vgf?Eq?k?? is the supply function of
CNTs. vg=?1/???Eq?k?/?k?0 is the group velocity along
the tube axis and f?E?=1/?1+exp?E/kBT?? is the Fermi dis-
tribution function. The Eq?k? is the dispersion relation of
CNTs where k and q are quantum numbers and run over the
Brillouin zone of CNTs. The F is the applied electric field
and the C is the circumference of the CNT. Here we assure
that the current flows only in the tube layer. The tunneling
probability D?E,F? through the vacuum potential barrier
U?x,F? may be calculated by the Wentzel–Kramers–
Brillovin approach. The exact shape of the vacuum potential
barrier for a CNT U?x,F? is unknown. We use a typical
potential barrier: the triangular-shape barrier with the image
potential U?x,F?=?−e?Fx−e2/4x as an approximation,
where ? is the work function and ? is the field enhancement
factor. The triangular-shape barrier is verified to be valid for
some qualitatively physical features.3,5Thus, the tunneling
probability is obtained by an intergration6
D?Eq?k?,F? = exp?−4?2m
3?
?? − Eq?k??3/2
e?F
v?y??
?2?
for ?−Eq?k??0 and D?Eq?k?,F?=1 for ?−Eq?k??0, where
the function v?y? is related to certain elliptic function and
y=?e3?F/??-Eq?k??.6
In the ?-electronic tight-binding approximation, the dis-
persion relation of SWNT Eq?k? can be obtained by folding
the two-dimensional graphite due to the curvature effect to
be usually neglected approximately,4
Eq?k?= ± t?1 + 4 cos??3
2kxa? cos?kya
2? + 4cos2?kya
2?,
?3?
where
a)Author to whom correspondence should be addressed; electronic mail:
stsxns@zsu.edu.cn
APPLIED PHYSICS LETTERS VOLUME 85, NUMBER 5 2 AUGUST 2004
0003-6951/2004/85(5)/813/3/$20.00
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© 2004 American Institute of Physics813

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?kx
ky? =?
a
2L?m − n?k +?3?a
?3a
2L?m + n?k +?a
L2?n + m?q
L2?n − m?q?
,
?4?
where the a is lattice constant and q=0, 1,¯,N−1 and
−?/T??/T;N is the number of hexagons per unit cell; the
T is the length of the translational vector to be parallel to the
CNT axis. With help of Eqs. (2) and (3) we can obtain the
density of emission current of CNTs from Eq. (1).
The emission current is usually related to the emitter size
for common metal emitters at a given applied field because
the local field of the emitter depends on the geometric shape
of emitters. This is the well-known field enhancement factor
in field emission. The CNTs are also expected to have this
field enhancement effect induced by the tube size. However,
it is very difficult to consider this effect exactly for CNTs
because CNTs are a nanosystem. How to calculate the field
distribution of CNTs is another issue. Since our attention
here focuses on the intrinsic properties of CNTs, we neglect
the variation of the field enhancement factor for different
CNTs, namely using a fixed field enhancement factor ?
=1000 empirically for different CNTs.2The carbon–carbon
bond energy t=−2.7eV.4The work function is set to ?
=4.7eV2,7and the room temperature T=300K.
In principle, the emission current depends on two intrin-
sic factors, the chiral angle and diameter of CNT. In order to
examine the chiral effect of CNTs on field emission we in-
vestigate the current density versus the chiral angles at a
given applied field for a group of SWNTs with a very similar
diameter, D?2.0±0.03 nm. It can be seen from Fig. 1 that
the current densities for both metallic and semiconducting
CNTs are almost independent of the chiral angles in the low
fields F?8V/?m shown in Fig. 1 (a), but increase with the
chiral angles in the high fields F?14V/?m shown in Fig.
1(b). Physically, electrons are emitted from near Fermi en-
ergy in the low field region. For metallic tubes the density of
Downloaded 27 Sep 2004 to 141.52.232.84. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
states (DOS) near the Fermi surface is almost independent of
the chiral angle of CNT such that the current density is not
sensitive to the chiral angle of CNT. For semiconducting
tubes the energy gap of DOS is approximately inverse pro-
portional the the diameter of CNTs D.4Consequently, the
current density is also independent of the chiral angle of
CNT in low fields for a given diameter of CNT. With in-
crease of the applied field the potential barrier becomes thin-
ner allowing a higher tunneling probability for low energy
electrons. More electrons come out of different energy bands
below Fermi energy. The current density is expected to de-
pend on the chiral angle of CNT. Our results in Fig. 1(b)
reveal this chiral effect on field emission. The current density
of the tube with larger chiral angle is obviously greater than
that of the tube with the small chiral angle at F=14V/?m.
The difference between the current densities of armchair and
zigzag tubes is about 1?A/nm. However, in practice, CNTs
could be melted in such high currents due to the thermal
effect. The chiral effect on the emission current will not be
easy to observe experimentally under the normal field-
emission condition. Nevertheless, if one can find a cooling
method to control temperature in CNTs he/she should be able
to obverse this chiral effect by field emission. Actually, there
has been some evidence indicating that electrons in CNTs
exhibit the ballistic transport behavior so that CNTs can
carry quite high currents.8Another possibility to observe the
chiral effect is that the tip of CNTs is chemically modified by
some atoms, which could suppress the potential barrier to
increase the tunneling probability of the low energy elec-
trons. Recently, the theoretical studies predict that the local-
ized states in the tip and cap of CNTs play an important role
in the field emission.9–12
On the other hand, we examine the relation between the
current density and the diameter of CNT at a given applied
field for several groups of SWNTs with different chiral
angles in Fig. 2. As the diameters increase the current density
decreases for metallic tubes, but increases for semiconduct-
ing tubes shown in Fig. 2. When the diameters of CNTs are
larger than 20 Å the changes of the current densities dimin-
ish and current densities of the metallic and semiconducting
tubes are very close. This is a new size effect of CNTs in
field emission, which originates from the intrinsic properties
of the energy band structure. Its influences to metallic and
semiconducting tubes are different. From DOS point of
views, DOS depends on the diameter of CNT. For metallic
tubes, the DOS of large diameter tubes near Fermi energy is
lower than that of small one shown in Fig. 3(a). This is why
the current density decreases with increase of the diameter.
For the semiconducting CNTs, since the energy gap of DOS
is approximately inverse proportional to the diameter of
CNTs shown in Fig. 3(b), there are more electrons coming
out for large diameter tubes at a given applied field. This can
explain the increase of the current density with increase of
the diameter for semiconducting tubes. This is a quantum
size effect in field emission, which is fundamentally different
from the field enhancement effect of common metals. The
field enhancement effect is induced by the dependence be-
tween the local field of emitter and the shape and size of
emitter. From the phenomenological point of views, these
two size effects are consistent for metallic tubes, but com-
peted for semiconducting tubes. General speaking, the cur-
rent density depends on two intrinsic factors, chiral angle
FIG. 1. (a) The current densities vs the chiral angles ? for a given diameters
of SWNTs, D?2.0±0.03nm at the applied field F=6V/?m in (a), and F
=14V/?m in (b).
814 Appl. Phys. Lett., Vol. 85, No. 5, 2 August 2004Liang et al.

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and diameter at a given applied field, especially for small
tubes.
Moreover, the theoretical comparative study on the field-
emission properties of various types of CNTs found that the
behavior of field emission from the bundle of CNTs deviates
from the FN theory at high fields and the multiwall CNTs is
better than SWNTs in the field emission property.13
Our study of field emission from SWNT reveals two
intrinsic properties of SWNT in field emission, the chiral
effect and the quantum size effect on the emission current.
These intrinsic properties can be understood from the prop-
erties of DOS and the energy band structure. Our findings
suggest that the energy band structure plays an important
role in field emission, especially for the small tubes. These
results may provide a physical insight to understand the
field-emission properties of CNTs and a guideline to choose
the CNT as a field emitter.
The authors are thankful for financial support of the
project from the National Natural Science Foundation of
China, the Ministry of Education of China, the Ministry of
Science and Technology of China, and the Department of
Science and Technology of Guangdong Province.
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FIG. 2. The current densities vs the diameter of CNTs at the applied field
F=6V/?m in (a) and F=8V/?m in (b), where the metallic zigzag tubes are
?3u,0?, where u=2,3,...,9; the semiconducting zigzag tubes are ?7
+3u,0?, where u=0,1...,7; the metallic chiral tubes with ?=16.1° are
?5,2?,?10,4?, and ?20,8?; the semiconducting chiral tubes with ?=21.8° are
?5,3?,?10,6?, and ?20,12?; the armchair tubes are ?4+2u,4+2u?, where u
=0,1,...,7.
FIG. 3. The density of states for four SWNTs, two metallic tubes ?5,2? and
?10,4? in (a), and two semiconducting tubes ?5,0? and ?11,0? in (b).
Appl. Phys. Lett., Vol. 85, No. 5, 2 August 2004 Liang et al.
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