1

A DFT-PBC study of infinite single-walled carbon nanotubes withvarious tubular diameters

Bo-Cheng Wanga, Wen-Hao Chena, Houng-Wei Wang*,b and Michitoshi Hayashib

aDepartment of Chemistry, Tamkang University, Tamsui 251, Taiwan

bCenter for Condensed Matter Sciences, Taipei 106, Taiwan

Abstract

A density functional theory (DFT) calculation with Gaussian orbital and periodic

boundary condition (PBC) simulation model has been used to determine the electronic

and optimized geometrical structure of zigzag and armchair type SWNTs up to

infinite tubular length. We shed light on the electronic structures of zigzag and

armchair SWNTs with various tubular diameters ([n, 0] zigzag type SWNT for n = 6

～ 20; and [n, n] armchair type SWNT for n = 4 ～ 10 ). Calculated Eg (band gap

between HOCO and LUCO) of the zigzag SWNTs have oscillated with Δn = 3 (the

repeat section is n = 3m - 1, 3m and 3m + 1, m = 1, 2 ..). We conclude that [n, 0]

zigzag SWNTs have narrow Eg being a metal when n = 3m, otherwise, the zigzag

SWNT is a moderate Eg as a semiconductor. For the [n, n] armchair SWNT, the

calculated Eg are between 0.013 eV to 0.030 eV with metallic property; these results

also have oscillation forΔn = 2. The optimized structures of SWNTs being generated

by DFT-PBC method mentioned that the C-C bond length decreased with increase the

tubular diameter. Up to [10, 10] SWNT, it has equal bond length and present the π

bond delocalization. The calculated Eg of SWNT in this study are consistent with the

experimental data.

2

Introduction

Due to the unique physical properties (elasticity, stiffness and deformation) and

the applications in various materials (semiconducting, H2 storage and the probes) of

carbon nanotubes, they have attracted considerable attention.1-5 Almost twenty years

ago, Smalley et al. discovered the truncated-icosahedral C60 carbon cluster by laser

vaporization of graphite in a high-pressure supersonic nozzle.6 In 1991, Ijima detected

the multi-wall carbon nanotubes in a plasma arc discharge apparatus.7 Two years late,

the single walled nanotubes (SWNTs) have been achieved by Iijima and Bethune.

Later, the large-scale purification process and the SEM, TEM and STM

characterization of SWNTs have been obtained.8,9 Although, much scientific interest

was focused on the physical and electronic properties, and commercial applications of

these new materials, there have been no experimental structural data sufficiently

accurate to generate the C-C bond length of SWNTs. In order to investigate the

physical properties of SWNT, they need the theoretical analysis to determine the real

nature of SWNTs and specify their properties.

The geometrical structure of SWNT is a rolling up 2-D graphite sheet as a

hollow cylindrical shape or a one-by-one layering of cyclic carbon array shape as 1-D

tube axis infinity extension.10 The defect free SWNTs have various types of

cylindrical shapes with respect to the array of benzenoids in carbon nano-tube.

According to the geometrical analysis, there exist armchair, zigzag and chiral tubules

among SWNTs. Recently, the theoretical and experimental work predicted that the

infinity length SWNTs are π -bonded aromatic molecules that can be either

semiconducting or metallic depending upon the tubular diameter and helical angle.11,

13

Since the limitation of the CPU time, the quantum chemistry calculations cannot

simulate the real infinite length SWNT model. In 1992, Saito and Hamada used the

3

tight binding model to generate the band structure of SWNTs.14 Almost the same year;

Nakamura et al. predicted the infinite length [5, 5] and [6, 6] armchair SWNT using

DFT calculation.15 Recently, Brus et al. used the DFT calculation to generate the

HOMO-LUMO gap from C20H20 to C210H20 and simulate the infinite length of [5, 5]

armchair SWNT; they concluded that [5, 5] armchair SWNT with infinite length

contains narrow Eg having the metallic property.11 Our previous work used the

semiempirical PM3 method to determine the electronic and optimized structures of

zigzag and armchair SWNT with finite length and various tubular diameters.16

Although lots of calculations have been presented for the finite model (small segment)

of SWNT; their results cannot provide the sufficient data to support the real SWNT

model (infinite length). In order to investigate the infinite system with periodic unit,

the fast multipole method have been proposed.17 Late, the periodic boundary

condition (PBC) model has been presented that could solve the discrete MO model

into the continuous bands.18-20 Thus, one can generate the finite SWNT segment with

DFT calculation with Gaussian type molecular orbital and extend to infinite length

SWNTs model. Scuseria et al. used the DFT-PBC method to optimize the geometrical

structure and to generate the energies of [5,0] zigzag SWNTs.17 But they did not apply

PBC model to the SWNTs with various tubular diameters and to investigate the

diameter effect in these SWNTs.

In the present study, the DFT-PBE and DFT-VSXC methods with 3-21G* and

6-31G* basis sets and PBC function were used to generate the geometry-optimized

structure, band structures and Eg of SWNTs of the zigzag (from [6, 0] to [20, 0]) and

armchair (from [4, 4] to [10, 10]) types with different tubular diameters up to infinite

tubular length. Calculated Eg of SWNTs allow us to predict some physical properties

of SWNT. The calculation results reveal that the [n, 0] zigzag SWNTs have the

oscillation band gap with △n = 3. If n is a multiple of 3 (n = 6, 9, 12, 15), the zigzag

4

SWNTs may have narrow Eg being a metal; otherwise, the SWNT is a moderate-gap

and has a semiconductor. For armchair SWNTs, they have the narrow Eg being

oscillation also.

Calculations

The geometrical structure of SWNT can be described in terms of a role up a

section of a 2-D graphite sheet, which denotes the unit vectors of hexagonal

honeycomb lattice. The roll up vector is denoted by Ch = na1 + ma2, where n and m

are integers; the translation vector T is perpendicular to Ch and direct along the

tubular length of SWNT (Fig. 1). Conveniently, SWNT could be presented by a [n, m]

pair of number; [n, 0] and [n, n] zigzag and armchair types designate SWNTs,

respectively. The carbon atoms of the zigzag SWNT arrange as cis-polyenes with a

single circular of carbon atoms. On the other hand, the armchair SWNT is obtained by

rolling up hexagons as the σv symmetry plane that the carbon atoms arrange as

trans-polyenes with a single circular plane of carbon atoms. For the zigzag type

SWNT, n denotes the number of benzenoids in the circumference of the tube and the

translation axis is the trans-polyene rings along the tubular length. The tubular

diameter (dt) of [n, 0] zigzag SWNT could be determined: dt = [rcos(π/6)]/[sin(π

/2n)]/2, r is the length of C-C bond in the SWNT. For the [n, n] armchair SWNT, dt =

r/(sin2nπ /3). In this paper, we consider the zigzag and armchair SWNTs for

calculations only.

For the DFT-PBC calculation for SWNTs, we start the single layer for the unit

cell and extend along the tubular axis to infinite length. For example the unit cell of [5,

0] SWNT contains 20 carbon atoms for single circumference, thus we used this 20

carbon atoms for the start unit extending to the infinite tube in this calculation. The

optimized structure and band structure of SWNT are generated by using the DFT-PBC

5

model with 6-31G* basis set. The DFT-PBC method is implemented in the Gaussian

03 program package.21

The PBC model in Gaussian 03 package is based on Gaussian type orbitals,[] to

transform GTOs into “Crystalline orbitals”for calculating with periodic boundary

conditions is modulated by a phase factor eikl. Those functions are known as Bloch

functions:[]

ll

lieN

kk

1(1)

where k=(kx,ky,kz) is the reciprocal-lattice vector, which classifies periodic orbitals by

their irreducible representations of the infinite translation group, ψl is an orbital ψ

located in cell l, and i is the imaginary unit. Orbitals belong to different k do not

interact directly with each other and this allows one to solve self-consistent-field (SCF)

equations separately for each k point. The equations are similar as non-periodic case:

Fk Ck = Sk Ck Ek (2)

We note that Eq. (2) is valid both for HF and DFT methods. The exponent in the

Bloch orbital definition (1) introduces complex factors and therefore all matrices in

Eq. (2) are, in general, complex. Matrix elements between periodic orbitals defined in

Eq. (1) can be easily computed from matrix elements for localized GTOs,

l

lil

l

lil eAeAA kk

kk0

0 (3)

In this equation, lA0 is a matrix element of operator A between the Gaussian atomic

orbitals located in the central cell 0 and ψ located in cell l. The Kohn-Sham

Hamiltonian matrix elements (or Fock matrix elements in the HF case), lF 0, include

several contributions:

xclllll EJUTF ,00000 (4)

where lT 0 is the electronic kinetic energy term, lU 0

is the electron-nuclear

6

attraction term, lJ 0 is the electron-electron repulsion term, and xclE ,0

is the

contribution from the DFT exchange-correlation potential. lT 0 and lU 0

terms do

not depend on the density matrix, while lJ 0 and xclE ,0

do. An important feature of

the Kohn-Sham Hamiltonian matrix elements, lF 0, is their exponential decay with

respect to the increasing separation between the and GTOs. Such behavior arises

from the individual decay of the kinetic energy term, the exchange-correlation

potential term, and the exponential decay of the combined electrostatic terms. Overall,

all terms in Eq. (4) are quite similar to analogous terms in molecular calculations. The

electrostatic terms ( lU 0 and lJ 0

) include interactions of a given pair of basis

functions with charge distributions in the system. The number of such interactions is

infinite, and this is indeed different from the molecular case. The infinite sums can be

handled using the Ewald summation techniques[] or by the periodic fast multipole

method.[] The real-space density-matrix elements lP0 required for the construction

of the Coulomb, exchange, and correlation contributions can be obtained by

integrating the complex density kP in reciprocal space,

kkk

k

dePV

P lil

10 (5)

where Vk is the volume of the unit cell in k space. The matrix Pk is obtained from the

orbital coefficients Ck, which are solutions to the eigenvalue Eq. (2). The

transformation described by equation (5) is the only coupling of different k points

during the SCF procedure. In practice, the integration is replaced by a weighted sum

and the reader is referred to Ref. [] for detailed discussions on this topic. The energy

per unit cell can be computed as

7

0

0000

21

l lNRxc

llll EEJUTPE (6)

where Exc is the exchange-correlation energy and ENR is the nuclear repulsion energy.

In the following, triple sums like the one in Eq. (6) will be abbreviated by l. In

order to avoid convergence problems and to maximize accuracy, it is important that

electrostatic terms be grouped together into electronic Ee and nuclear EN terms,

lNR

llN

l

llle

EPUE

PJUE

00

000

21

21

(7)

Once the converged density is available, it is possible to compute gradients of the

total energy with respect to nuclear displacements (forces).

Results and discussions

To investigate the influence of the tubular diameter of SWNTs, the geometric

optimized structures, band structures and Eg of infinite length zigzag and armchair

SWNTs with various tubular diameters were calculated by DFT VSXC method with

PBC function and 6-31G* basis set.

Zigzag type SWNT

According to the theoretical analysis, the zigzag SWNT contains the number of

benzenoid (n) in the circumference of the tube. We used DFT VSXC with PBC

function and 6-31G* basis sets to generate the optimized geometry and the band

structures of zigzag SWNT with various tubular diameters from [6, 0] to [20, 0]. The

calculated Eg of infinite length [n, 0] zigzag SWNT exhibits oscillation properties

with the repeat unit having n = 3m - 1, 3m and 3m + 1, m is integer (Fig. 2). Table 1

shows the calculated Eg of zigzag SWNTs up to infinite tubular length. According to

8

this table, the highest band gap was obtained for the [n (=3m + 1), 0] SWNTs while

the [n (=3m), 0] SWNTs have the lowest in each △n = 3 section. [6, 0], [9, 0], [12, 0],

[15,0] and [18, 0] have 0.0296 eV, 0.1466 eV, 0.0677 eV, 0.0387 eV and 0.0308 eV

calculated Eg, respectively, these are the narrow Eg and have the metallic property. For

the [n (= 3m + 1), 0] zigzag SWNT, [7, 0], [10, 0], [13, 0], [16, 0] and [19, 0] zigzag

SWNTs have 0.2043 eV, 0.7643 eV, 0.6363 eV, 0.5327 eV and 0.4579 eV calculated

Eg, respectively, these values are semiconductor or metal. Thus, we conclude that [n, 0]

SWNT with n is a multiple of 3 may have the metallic property and others have a

semiconductor. We also conclude that [8, 0], [11, 0], [14, 0] and [17, 0] have the

highest band gap in each △n = 3 section. Comparing with other calculation results,

we obtained Eg = 0.2043 eV in the present study for [7, 0] SWNT. While Ito et al.

reported Eg = 0.1304 eV for GGA and 0.1943 eV for LDA. Very early, the calculated

Eg by tight-binding model was reported to be 1.04 eV for this SWNT. Our calculation

is very closed to that of LDA results. Hamada et al. used the tight-binding model to

determine the Eg = 0.697 eV for [13, 0] SWNT Our calculation for this SWNT is

0.6363 eV that is very closed to previous calculation data. Although tight-bind and

LDA methods can determine the larger scale carbon tube than other quantum

chemistry methods, they restrict the C-C bond length in the same distance for the

whole system. In present study, the DFT-PBC model optimized the tube structure up

to two different C-C bond lengths that may more close to the real carbon tube

structure than other methods.

The DFT-PBC calculated geometrical parameters and tubular diameter for the

infinite length zigzag SWNTs are presented in Table 2. The calculated tubular

diameters increased from 4.860 A ([6, 0] SWNT) to 15.796 A ([20, 0] SWNT); there

are nearly 0.8 A tubular diameter difference between any two neighboring SWNT.

Experimentally, the tubular diameters of SWNTs can vary ranging from 10 Å to 16 Å

9

with a peak maximum at 12 Å. We may predict that the most possible product is [15,

0] or [16, 0] SWNT in experiment.

Although several papers have been presented by using tight-bind model to

investigate the large scale of SWNTs, they assume that SWNT has the same C-C bond

lengths in the whole nano-tube system. In the present study, the C-C bond length of

SWNT has been optimized to two different magnitudes. According to Table 2, there

are two different calculated C-C bond lengths for [6, 0] SWNT 1.450 A and 1.416 A,

respectively. Up to [13, 0] SWNT, the two C-C bond lengths of the optimized

structure are very close (1.431 A and 1.428 A), then, these lengths keep the same

magnitude from [14, 0] to [20, 0] SWNTs. Concludly, increasing the tubular diameter

of infinite zigzag SWNTs may increase the π electron delocalization; the tubular

diameter may not affect the C-C bond length after [13, 0] SWNT. Thus, these SWNTs

are the nearly rolled up graphite sheet.

Fig. 3 shows the sketch of HOMO and LUMO with double layer unit cell of the

infinite tubular length tube of [6, 0] to [8, 0] zigzag SWNTs. Since there are △n = 3

oscillation period in the Eg of zigzag SWNT, the frontier orbital may have this

oscillation property, thus, the HOMO and LUMO of [6, 0] and [9, 0], [7, 0] and [10,

0], [8, 0] and [11,0] have very similar sketches, respectively. Due to the above

consideration, we determined the HOMO and LUMO for [6, 0] to [8, 0] zigzag

SWNTs only (Fig. 4). In this study, we simulate carbon nano-tube with infinite tubular

length, and ignored the influence of tubular length. So that, the influence of tubular

diameter of infinite length zigzag SWNT should be viewed as the armchair SWNT. Li

et al. were used DFT and the semiempirical PM3 computational techniques to

generate the electronic wave functions of same shortened [5, 5] and [6, 6] armchair

SWNTs.22 Our sketch of HOMO and LUMO are the same as those of Li’s results.

For the electronic band structure, we have systematically studied. The band

10

structures of [9. 0], [10, 0] and [11, 0] zigzag SWNTS are shown in Fig. 5. Bands

higher than EF (the Fermi level) are antibonding π* bands; π and σ (sp2)

bonding bands are located lower than the EF. Fig. 5 (a) shows the band structure of [9,

0] SWNT, the calculated HOCO and LUCO are –3.9825 eV and –3.8360 eV at Γ

point, respectively, its energy gap is 0.1466 eV. The calculated HOCO and LUCO

bands for [10, 0] SWNT are–4.3603 eV and–3.5960 eV and exhibit an energy gap is

0.7643 eV. The X point in these structures has a wave number near π/a, with a the

graphite lattice constant. Two bands stick together at the X point due to the screw

symmetry.

Armchair type SWNT

Recently, the electronic and geometrical structures of armchair of SWNTs using

computational methods have been proposed. The semiempirical calculation

mentioned that the HOMO/LUMO and Eg of the finite tubular length of armchair type

SWNTs have the oscillation properties with the repeat unit having 3m-1, 3m and 3m +

1 models (m is integer) in the carbon section along their latitudes. In this study, we

simulate the infinite tubular length of armchair type SWNTs, thus their tubular length

influence shall be ignored. The DFT-PBC calculated Eg and related geometrical

parameters of selected armchair SWNTs are shown in Tables 2 and 3, respectively.

Particularly, the calculated tubular diameters are from 5.545 A ([4, 4]) to 13.701 ([10,

10]), the distance between any two neighboring armchair SWNTs are not consistent in

this series. All of the calculated Eg are between 0.01 to 0.03 eV for these SWNTs and

they should be a metal. Calculated Eg for [n, n] armchair SWNTs exhibit oscillations

with Δn = 2 (Fig. 3). This calculation result metioned that the calculated Eg of the

tube with even n is lower than that of tube with odd n. For example, the difference in

the calculated Eg for [5, 5] and [6, 6], [6,6] and [7,7] armchair SWNTs are 0.023 eV

11

and –0.017 eV, respectively. Recently, Nakamura et al. have presented the structures

and aromaticity of [5, 5] and [6, 6] armchair SWNTs by used DFT method. They

predict that [5, 5] zigzag SWNT possess higher HOMO and lower LUMO than that of

[6, 6] SWNT. Thus, [5, 5] SWNT contains higher Eg than that of [6, 6] SWNT. Our

calculations have the same trend as that of Nakamura’s data.

Table 4 shows the calculated optimized geometrical parameters of armchair

SWNT by DFT-PBC calculation. According to the calculation results, there are two

different C-C bond lengths in armchair SWNTs that is similar to that of zigzag

SWNTs. For [4, 4] armchair SWNT, it has 1.436 A and 1.432 A calculated C-C bond

length. Apparently, the C-C bond length decreases to 1.428 A for [10, 10] SWNT, that

is the only C-C bond length in this SWNT. This calculation results mentioned that the

C-C bond length decrease while the tubular diameter increase. Whereas, [n, n]

armchair SWNT. The structures and aromaticity of finite [5, 5] and [6, 6] armchair

SWNT have been generated by Nakamura et al.

The calculated sketch of HOMO and LUMO with double layer of the infinite

tubular length of [4, 4] and [5, 5] armchair SWNTs are shown in Fig. 4. The band

structures of [4, 4] and [5, 5] armchair SWNTs are presented in Fig. 6, HOCO and

LUCO are crossing in the K point, thus, both of [4, 4] and [5, 5] armchair SWNTs are

metal. This DFT-PBC calculation result is consistent with previous tight-binding

calculation.

Conclusion

In this study, DFT-PBC calculations were used to investigate the geometrical and

electronic structures of zigzag and armchair types of SWNT up to infinite tubular

length. The calculations reveal that the Eg of [n, 0] zigzag SWNTs have an oscillation

with △n = 3. The [n, 0] zigzag SWNTs with n is multiple by 3 that have narrow Eg

12

and have a metal, otherwise they have moderate Eg being the semiconducting bulk

materials. According to the calculated Eg for the [n, n] armchair SWNT, they have

narrow Eg and contain metal property. The calculated Eg in these SWNTs also have

oscillation with △n = 2, although there are very small Eg difference. For the

optimized structure of SWNTs, there are two different calculated C-C bond length,

those are decreasing with increasing the tubular diameter. This calculation provides

important information on the properties of SWNT needed for the design of new

nano-electronic devices.

Acknowledgment

We thank the National Science Council of ROC for supporting.

13

References

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8. S. Iijima, T. Ichihashi, Nature, 363 (1993), 603.

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14

Chienese Chemical Society, 50 (2003), 939.

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15

Fig. 1

[1,0,0]

[2,1,0]Zigzag SWNT

Armchair SWNT

Chiral SWNT

m

n

m

n

16

Table 1 Calculated HOCO, LUCO and Eg of [n, 0] zigzag SWNTs by DFT-PBCmethod

[n, 0] HOCO LUCO Eg

n = 6 -4.0890 -4.0594 0.02967 -4.4539 -4.2496 0.20438 -4.5062 -3.8665 0.63979 -3.9826 -3.8360 0.1466

10 -4.3603 -3.5960 0.764311 -4.3824 -3.4191 0.963312 -3.9756 -3.9078 0.067713 -4.3028 -3.6665 0.636314 -4.3200 -3.5770 0.742915 -3.9893 -3.9505 0.038716 -4.2664 -3.7336 0.532717 -4.2704 -3.6668 0.603518 -4.0081 -3.9773 0.030819 -4.2333 -3.7753 0.457920 -4.2347 -3.7251 0.5096

17

Table 2 Calculated geometrical parameters for [n, 0] zigzag SWNTs

[n, 0] Diameter (Å) C1-C2 (Å) C2-C3 (Å)

n = 6 4.860 1.450 1.4167 5.662 1.441 1.4248 6.395 1.440 1.4229 7.210 1.437 1.425

10 7.947 1.435 1.42711 8.755 1.435 1.42512 9.514 1.433 1.42613 10.344 1.432 1.42814 11.083 1.432 1.42715 11.907 1.432 1.42716 12.649 1.431 1.42817 13.453 1.431 1.42718 14.222 1.431 1.42819 15.022 1.430 1.42820 15.796 1.431 1.428

tube axis

18

Table 3 Calculated HOCO, LUCO and Eg of [n, n] armchair SWNTs by DFT-PBCmethod

[n, n] HOCO LUCO Eg

n = 4 -3.9262 -3.8985 0.02765 -3.9396 -3.9217 0.01786 -3.9456 -3.9150 0.03067 -3.9545 -3.9409 0.0136

8 -3.9849 -3.9527 0.0322

9 -3.9707 -3.9569 0.013810 -3.9933 -3.9716 0.0216

19

Table 4 Calculated geometrical parameters for [n, n] armchair SWNTs

[n, n] Diameter (Å) C1-C2 (Å) C2-C3 (Å)

n = 4 5.545 1.436 1.4325-5 6.805 1.434 1.4306-6 8.249 1.432 1.4297-7 9.544 1.431 1.4298-8 10.962 1.430 1.4289-9 12.279 1.429 1.428

10-10 13.671 1.428 1.428

tube axis

20

tube

Figure captionsFig. 1 Hexagonal network of a single graphite sheet for zigzag, armchair and chiralSWNTFig. 2 Calculated Eg with oscillation property for [n, 0] zigzag SWNTs by DFT-PBCmethodFig. 3 Calculated Eg with oscillation property for [n, n] armchair SWNTs byDFT-PBC methodFig. 4 Sketch of HOMO and LUMO for [6, 0], [7, 0] and [8, 0] zigzag SWNTsFig. 5 Sketch of HOMO and LUMO for [4, 4] and [5, 5] armchair SWNTsFig. 6 Band structure of [9, 0], [10, 0] and [11, 0] zigzag SWNTsFig. 7 Band structure of [4, 4] and [5, 5] armchair SWNTs

21

Fig . 1

[1,0,0]

[2,1,0]Zigzag SWNT

Armchair SWNT

Chiral SWNT

m

n

m

n

22

Fig. 2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

(n, 0)

Eg

(eV

)

23

Fig. 3

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 1 2 3 4 5 6 7 8 9 10 11

(n, n)

Eg

(eV

)

24

Fig. 4

HOMO LUMO

(6,0)

(7,0)

25

(8,0)

Fig. 5

HOMO LUMO

(4,4)

26

(5,5)

Fig. 6

-9-8

-7-6

-5-4

-3-2

-10

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76

k-point

E(e

V)

HOCO-4 HOCO-3 HOCO-2 HOCO-1 HOCO

LUCO LUCO+1 LUCO+2 LUCO+3 LUCO+4

27

(9,0) SWNT

-8

-7

-6

-5

-4

-3

-2

-1

0

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76

k-point

E(e

V)

HOCO-4 HOCO-3 HOCO-2 HOCO-1 HOCO

LUCO LUCO+1 LUCO+2 LUCO+3 LUCO+4

(10,0) SWNT

-8-7-6-5-4-3-2-10

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

k-point

E(e

V)

HOCO-4 HOCO-3 HOCO-2 HOCO-1 HOCOLUCO LUCO+1 LUCO+2 LUCO+3 LUCO+4

(11,0) SWNTFig.5

28

-8

-7

-6

-5

-4

-3

-2

-1

0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66

k-point

E(e

V)

HOCO-4 HOCO-3 HOCO-2 HOCO-1 HOCO LUCO

LUCO+1 LUCO+2 LUCO+3 LUCO+4

(4,4) SWNT

-8

-7

-6

-5

-4

-3

-2

-1

0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66

k-point

E(e

V)

HOCO-4 HOCO-3 HOCO-2 HOCO-1 HOCO LUCO

LUCO+1 LUCO+2 LUCO+3 LUCO+4

(5,5) SWNT