Ab-initio study of electronic band structure of zigzag single wall carbon nanotubes
Manoj Sharma*, A. Tiwari** and U.S.Sharma***
*Dept. of Applied Mathematics, RJIT BSF Academy, Tekanpur, Gwalior (MP), India, 475005
**Dept. of Civil Engineering, RJIT BSF Academy, Tekanpur, Gwalior (MP), India, 475005
***Dept. of Applied Physics, RJIT BSF Academy, Tekanpur, Gwalior (MP), India, 475005
Abstract
The electronic and structural properties of single wall carbon nanotubes (SWCNTs) are presented. The abinitio calculations were performed using SIESTA code employing norm conserving pseudopotential. In these
computations exchange correlation energies were obtained using local density approximation (LDA). The
calculated electronic band structures and the studies of the effect of curvature on electronic structure show that
(9,0), (12,0) and (15,0) zigzag SWCNTs are small gap semiconductors in nature. The diameter dependency on
the energy bandgap has been studied. The calculated energy band gap data for semiconducting zigzag SWCNTS
are in good agreement with previously reported theoretical and experimental data.
S.Iijima1 discovered multiwall Carbon nanotubes
(CNTs) in 1991with few tens of nanometers in outer
diameter. Two year later single walled carbon nanotubes
(SWCNTs) were reported2,3. SWCNTs are basically
graphite sheets rolled up into a cylinder with diameter of
the order of few nanometers, which are characterized by
two integers (n,m) defining the rolling vector of
graphite4. Basic information about electronic structure of
CNTs provides by zone-folding model5, if n - m is an
integer multiple of three the nanotubes are metallic,
otherwise semiconducting in nature. So (9,0), (12,0) and
(15,0) SWCNTs are predicted to be metallic by this
model. The tight binding calculations6 has been reported
that these tubes become gapful if σ orbitals is also
included along with л orbitals. This shows that σ-л band
hybridization takes place in these tubes which lead to
their semiconducting nature.
In this work we present a consistent ab initio study of
electronic band structure of zigzag SWCNTs. Stress has
been given on the studies of the effect of curvature on
electronic structure. The dependencies of energy band
gap on diameter also study.
COMPUTATIONAL DETAIL
Self-consistent density functional theory (DFT)
calculations were performed with the SIESTA7 ab initio
code.
The
exchange-correlation
energy
was
approximated by the LDA8,9 using the Ceperley-Alder
formula. The norm-conserving pseudopotential was used
and the valance electrons were described by localized
pseudoatomic orbitals (PAOs) with a double-ζ singly
polarized (DZP) basis set. As the confinement of the
atomic orbitals leads to rising of the energy levels, it has
been accounted for by choosing an energy shift of 200
meV for PAOs that minimize the total energy of the
system. The convergence of the total energy in the real
space mesh size was investigated and it has been found
that, in all cases, the total energy converges at planewave mesh cutoff of 250 Ry.
In present calculations the tetragonal supercell
geometry has been chosen and the relaxed atomic
positions were used. To relax atomic positions the
conjugate gradients (CG) method in molecular dynamics
have been used. Maximum force tolerance in CG
coordinate optimization move was set at 0.04 eV/Å.
The convergence of total energy with respect to
number of k-points was investigated and it has been
found that total energy converges at number of k-point
equal to 13 which corresponds to 1x1x25 MonkhorstPack. The band energies were calculated along Г
(gamma) and X high-symmetry directions.
RESULTS AND DISCUSSION
The convergence of the total energy in the real space
mesh size was investigated. A graph between total
energy and mesh cutoff plotted as shown in figure1. It
has been found that, in all cases, the total energy
converges at plane-wave mesh cutoff of 250 Ry. We take
this value for further calculations.
-6 8 3 4 .3 0
-6 8 3 4 .3 2
Total Energy (eV)
INTRODUCTION
-6 8 3 4 .3 4
-6 8 3 4 .3 6
-6 8 3 4 .3 8
-6 8 3 4 .4 0
-6 8 3 4 .4 2
-6 8 3 4 .4 4
0
50
10 0
150
200
2 50
300
3 50
400
Mesh cutof f (Ry)
Figure1: Plot between total energy (eV) and Mesh
cutoff (Ry).
Electronic and structural properties of an isolated
(9,0), (12,0) and (15,0) SWCNTs have been studied in
this work. The calculated electronic band structures of
these tubes have been given in figure. The effect of
curvature on electronic structure of these tubes has also
been studied. To account for this we calculate the band
structure of (n,0) zigzag SWCNTs ( 3  n  15 ). From
calculated electronic band structure we found that when
n is increased by one, an extra state appear both in
valance band (VB) and conduction band (CB) .
These newly appeared extra states in VB and CB tend
to continuously shift toward Fermi level as n was
increased till a minimum distance from Fermi level was
achieved. After achieving this minimum distance from
Fermi level these states start shifting away from Fermi
level as n was further increased and other newly
appeared states starts shifting towards Fermi level to
achieve a minimum distance. It was (9,0), (12,0) and
(15,0) SWCNTs where newly appeared states in VB and
CB attain minimum distance from Fermi level and make
them small gap semiconducting in nature. This shows
that curvature of CNTs has prominent effect on their
electronic band structure.
The calculated energy band gaps at Γ-point in
SWCNTs have been given in table and compare with
experimental and theoretical result. From table it is
evident that in comparison to the GGA results our
computed values of energy band gaps with LDA are in
excellent agreement with the STS experimental results.
Table: Energy band gap of isolated SWCNTs.
CNT
index
(n,0)
Energy band gap (eV)
Present
Ref.[11]
Ref. [10]
work
(GGA)
(Exp.)
(LDA)
(3,0)
(4,0)
(5,0)
(6,0)
(7,0)
(8,0)
(9,0)
(10,0)
(11,0)
(12,0)
(13,0)
(14,0)
(15,0)
Metallic
Metallic
Metallic
Metallic
0.219
0.580
0.075
0.764
0.924
0.035
0.669
0.711
0.022
Metallic
Metallic
Metallic
Metallic
0.243
0.643
0.093
0.764
0.939
0.078
0.625
0.736
0.028
0.080
0.042
0.029
Finally, we discuss the diameter dependence of the
fundamental gap. We calculated the LDA gap of (n,0)
tubes with n=3-15. Thin tubes (n=3,6) are metallic, as
discussed above. For n  7 , the tube is a moderate-gap
semiconductor when n is not divisible by three, as has
been suggested by the 1/3 rule. When n is a multiple of
three, it is a narrow-gap semiconductor. The gap is
decreased as the diameter is increased.
The calculated values of energy band gaps for these
tubes can be fitted in to following equation
E g  0.0321x d 2.47 04
where Eg is the band gap in eV and d is the diameter of
nanotube in nm. The diameter dependence discussed
earlier employing effective mass theory12 as well as tight
binding approximation13 show that the energy band gap
E g  d 2 .
(9,0)
Γ
(12,0)
X
Γ
(15,0)
X
Γ
X
Figure2: Electronic band structure of (9,0), (12,0) and
(15,0) SWCNTs.
CONCLUSION
In this work, we found that, zigzag SWCNTs with
are metallic in nature and (9,0), (12,0) and (15,0)
are small gap semiconducting in nature. The large
downshift of conduction band edge due to curvature
induced σ-л band hybridization at high curvature results
to their metallic nature. There is no regular trend in
increase or decrease in the value of energy band gap with
the decrease in curvature in semiconducting zigzag
SWCNTs. The curvature effect is responsible for their
small energy band gap. The (11,0) tube has highest value
of energy band gap (0.924 eV) and (15,0) SWCNT has
lowest value of energy band gap (0.022 eV) among these
tubes. The energy band gap Eg is found to be
proportional to d -2.4704 where d is diameter of the tube.
The calculated values of band gap for (9,0) (12,0) and
(15,0) tubes are in excellent agreement with previous
reported theoretical as well as experimental results.
3 n  6
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Acknowledgements: The authors would like to thank
Prof. U P Verma SOS in Physics, Jiwaji University
Gwalior and Prof. Renu Jain, SOMASS Jiwaji
University, Gwalior for the useful suggestions.