Systematic ab initio study of curvature effects in carbon nanotubes
O. Gu
¨
lseren,
1,2
T. Yildirim,
1
and S. Ciraci
3
1
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
2
Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
3
Department of Physics, Bilkent University, Ankara 06533, Turkey
共Received 11 December 2001; published 28 March 2002兲
We investigate curvature effects on geometric parameters, energetics, and electronic structure of zigzag
nanotubes with fully optimized geometries from first-principle calculations. The calculated curvature energies,
which are inversely proportional to the square of radius, are in good agreement with the classical elasticity
theory. The variation of the band gap with radius is found to differ from simple rules based on the zone folded
graphene bands. Large discrepancies between tight binding and first-principles calculations of the band gap
values of small nanotubes are discussed in detail.
DOI: 10.1103/PhysRevB.65.153405 PACS number共s兲: 73.22.⫺f, 62.25.⫹g, 61.48.⫹c, 71.20.Tx
I. INTRODUCTION
Single wall carbon nanotubes 共SWNT’s兲 are basically
rolled graphite sheets, which are characterized by two inte-
gers (n,m) defining the rolling vector of graphite.
1
There-
fore, electronic properties of SWNT’s, at first order, can be
deduced from that of graphene by mapping the band struc-
ture of two dimensional 共2D兲 hexagonal lattice on a
cylinder.
1–5
Such analysis indicates that the (n,n) armchair
nanotubes are always metal and exhibit one dimensional
quantum conduction.
6
The (n,0) zigzag nanotubes are gen-
erally semiconductor and only are metal if n is an integer
multiple of three. However, recent experiments
7
indicate
much more complicated structural dependence of the band
gap and electronic properties of SWNT’s. The semiconduct-
ing behavior of SWNT’s has been of particular interest, since
the electronic properties can be controlled by doping or
implementing defects in a nanotube-based optoelectronic
devices.
8–14
It is therefore desirable to have a good under-
standing of electronic and structural properties of SWNT’s
and the interrelations between them.
Band calculations of SWNT’s were initially performed by
using a one-band
-orbital tight binding model.
2
Subse-
quently, experimental data
15–18
on the band gaps were ex-
trapolated to confirm the inverse proportionality with the ra-
dius of the nanotube.
5
Later, first-principles calculation
19
within local density approximation 共LDA兲 showed that the
*
-
*
hybridization becomes significant at small R 共or at
high curvature兲. Such an effect were not revealed by the
-orbital tight-binding bands. Recent analytical studies
20–22
showed the importance of curvature effects in carbon nano-
tubes. Nonetheless, band calculations performed by using
different methods have been at variance on the values of the
band gap. While recent studies predict interesting effects,
such as strongly local curvature dependent chemical
reactivity,
14
an extensive theoretical analysis of the curvature
effects on geometric and electronic structure has not been
carried out so far.
In this paper, we present a systematic ab initio analysis of
the band structure of zigzag SWNT’s showing interesting
curvature effects. Our analysis includes a large number of
zigzag SWNT’s with n ranging from 4 to 15. The fully op-
timized structural and electronic properties of SWNT’s are
obtained from extensive first-principle calculations within
the generalized gradient approximation
23
共GGA兲 by using
pseudopotential planewave method.
24
We used plane waves
up to an energy of 500 eV and ultrasoft pseudopotentials.
25
The calculated total energies converged within 0.5 meV/
atom. More details about the calculations can be found in
Refs. 26,27.
II. GEOMETRIC STRUCTURE
First, we discuss effects of curvature on structural param-
eters such as bond lengths and angles. Figure 1 shows a
schematic side view of a zigzag SWNT which indicates two
types of C-C bonds and C-C-C bond angles, respectively.
The curvature dependence of the fully optimized structural
parameters of zigzag SWNT’s are summarized in Fig. 2. The
variation of the normalized bond lengths 共i.e., d
C-C
/d
0
where
d
0
is the optimized C-C bond length in graphene兲 and the
bond angles with tube radius R 共or n) are shown in Figs. 2共a兲
FIG. 1. A schematic side view of a zigzag SWNT, indicating
two types of C-C bonds and C-C-C bond angles. These are labeled
as d
1
, d
2
,
1
, and
2
. Radius dependence of these variables are
important in tight-binding description of SWNT’s as discussed in
the text.
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and 2共b兲, respectively. Both the bond lengths and the bond
angles display a monotonic variation and approach the
graphene values as the radius increases. As pointed out ear-
lier for the armchair SWNT’s,
28
the curvature effects, how-
ever, become significant at small radii. The zigzag bond
angle (
1
) decreases with decreasing radius. It is about 12°
less than 120°, namely, the bond angle between sp
2
bonds of
the graphene, for the (4,0) SWNT, the smallest tube we stud-
ied. The length of the corresponding zigzag bonds (d
2
), on
the other hand, increases with decreasing R. On the other
hand, the length of the parallel bond (d
1
) decreases to a
lesser extent with decreasing R, and the angle involving this
bond (
2
) is almost constant.
An internal strain is implemented upon the formation of
tubular structure from the graphene sheet. The associated
strain energy, which is specified as the curvature energy E
c
is
calculated as the difference of total energy per carbon atom
between the bare SWNT and the graphene 共i.e., E
c
⫽ E
T,SWNT
-E
T,graphene
) for 4⭐n⭐15. The calculated curva-
ture energies are shown in Fig. 2共c兲. As expected E
c
is posi-
tive and increases with increasing curvature. Consequently,
the binding 共or cohesive兲 energy of carbon atom in a SWNT
decrease with increasing curvature. We note that in the clas-
sical theory of elasticity the curvature energy is given by the
following expression:
29–31
E
c
⫽
Yh
3
24
⍀
R
2
⫽
␣
R
2
. 共1兲
Here Y is the Young’s modulus, h is the thickness of the tube,
and ⍀ is the atomic volume. Interestingly, the ab initio cur-
vature energies yield a perfect fit to the relation
␣
/R
2
as seen
in Fig. 2共c兲. This situation suggests that the classical theory
of elasticity can be used to deduce the elastic properties of
SWNT’s. In this fit
␣
is found to be 2.14 eV Å
2
/atom,
wherefrom Y can be calculated with an appropriate choice of
h.
III. ELECTRONIC STRUCTURE
An overall behavior of the electronic band structures of
SWNT’s has been revealed from zone folding of the
graphene bands.
2–4
Accordingly, all (n,0) zigzag SWNT
were predicted to be metallic when n is multiple of 3, since
the double degenerate
and
*
states, which overlap at the
K point of the hexagonal Brillouin zone 共BZ兲 of graphene
folds to the ⌫ point of the tube.
2,4
This simple picture pro-
vides a qualitative understanding, but fails to describe some
important features, in particular for small radius or metallic
nanotubes. This is clearly shown in Table I, where the band
gaps calculated in the present study are summarized and
compared with results obtained from other methods in the
literature. For example, our calculations result in small but
non-zero energy band gaps of 93, 78, and 28 meV for (9,0),
(12,0), and (15,0) SWNT’s, respectively 共see Table I兲. Re-
cently, these gaps were measured by scanning tunneling
spectroscopy 共STS兲 experiments
7
as 80, 42, and 29 meV, in
the same order. The biggest discrepancy noted in Table I is
between the tight-binding and the first-principles values of
FIG. 2. 共a兲 Normalized bond lengths (d
1
/d
0
and d
2
/d
0
) versus
the tube radius R (d
0
⫽ 1.41 Å). 共b兲 The bond angles (
1
and
2
)
versus R. 共c兲 The curvature energy, E
c
per carbon atom with respect
to graphene as a function of tube radius. The solid lines are the fit to
the data as 1/R
2
.
TABLE I. Band gap E
g
as a function of radius R of (n,0) zigzag nanotubes. M denotes the metallic state. Present results for E
g
were
obtained within GGA. First row of Ref. 19 is LDA results while all the rest are tight-binding 共TB兲 results. Two rows of Ref. 33 are for two
different TB parametrization.
n 4 5 6 7 8 9 10 11 12 13 14 15
R 共Å兲 1.66 2.02 2.39 2.76 3.14 3.52 3.91 4.30 4.69 5.07 5.45 5.84
E
g
共eV兲 M M M 0.243 0.643 0.093 0.764 0.939 0.078 0.625 0.736 0.028
Ref. 19 M 0.09 0.62 0.17
Ref. 19 0.05 1.04 1.19 0.07
Ref. 2 0.21 1.0 1.22 0.045 0.86 0.89 0.008 0.697 0.7 0.0
Ref. 33 0.79 1.12 0.65 0.80
Ref. 33 1.11 1.33 0.87 0.96
BRIEF REPORTS PHYSICAL REVIEW B 65 153405
153405-2
the gaps for small radius tubes such as (7,0). These results
indicate that curvature effects are important and the simple
zone folding picture has to be improved. Moreover, the
analysis of the LDA bands of the (6,0) SWNT calculated by
Blase et al.
19
brought another important effect of the curva-
ture. The antibonding singlet
*
and
*
states mix and repel
each other in curved graphene. As a result, the purely
*
state of planar graphene is lowered with increasing curva-
ture. For zigzag SWNT’s, the energy of this singlet
*
state
is shifted downwards with decreasing R 共or increasing cur-
vature兲. Here, we extended the analysis of Blase et al.
19
to
the (n,0) SWNT’s with 4⭐n⭐15 by performing GGA cal-
culations.
In Fig. 3共a兲, we show the double degenerate
states
共which are the valence band edge at the ⌫ point兲, the double
degenerate
*
states 共which become the conduction band
edge at ⌫ for large R), and the singlet
*
state 共which is in
the conduction band for large R). As seen, the shift of the
singlet
*
state is curvature dependent, and below a certain
radius determines the band gap. For tubes with radius greater
than 3.3 Å 共i.e., n⬎ 8), the energy of the singlet
*
state at
the ⌫ point of the BZ is above the doubly degenerate
*
states 共i.e., bottom of the conduction band兲, while it falls
between the valence and conduction band edges for n⫽ 7,8,
and eventually dips even below the double degenerate va-
lence band
states for the zigzag SWNT with radius less
than 2.7 Å共i.e., n⬍ 7). Therefore, all the zigzag tubes with
radius less than 2.7 Å are metallic. For n⫽ 7,8, the edge of
the conduction band is made by the singlet
*
state, but not
by the double degenerate
*
state. The band gap derived
from the zone folding scheme is reduced by the shift of this
singlet
*
state as a result of curvature induced
*
-
*
mix-
ing. This explains why the tight binding calculations predict
band gaps around 1 eV for n⫽ 7,8 tubes while the self-
consistent calculations predict much smaller value.
Another issue we next address is the variation of the band
gap E
g
as a function of tube radius. Based on the
-orbital
tight binding model, it was proposed
5
that E
g
behaves as
E
g
⫽
␥
0
d
0
R
, 共2兲
which is independent from helicity. Within the simple
-orbital tight binding model,
␥
0
is taken to be equal to the
hopping matrix element V
pp
.(d
0
is the bond length in
graphene.兲 However, as seen in Fig. 3共b兲, the band gap dis-
plays a rather oscillatory behavior up to radius 6.0 Å. The
relation given in Eq. 共2兲 was obtained by a second order
Taylor expansion of one-electron eigenvalues of the
-orbital tight binding model
5
around the K point of the BZ,
and hence it fails to represent the effect of the helicity. By
extending the Taylor expansion to the next higher order,
Yorikawa and Muramatsu
32,33
included another term in the
empirical expression of the band gap variation
E
g
⫽ V
pp
d
0
R
冋
1⫹
共
⫺ 1
兲
p
␥
cos
共
3
兲
d
0
R
册
, 共3兲
which depends on the chiral angle
as well as an index p.
Here
␥
is a constant and the index p is defined as the integer
from k⫽ n⫺ 2m⫽ 3q⫹ p. The factor (⫺ 1)
p
comes from the
fact that the allowed k is nearest to either the K or K
⬘
point
of the hexagonal Brillouin zone. For zigzag nanotubes stud-
ied here, the chiral angle is zero, so the second term just
gives R
⫺ 2
dependence as ⫾
␥
V
pp
(d
0
/R)
2
. Hence, the solid
lines in Fig. 3共b兲 are fits to the empirical expression E
g
⫽ V
pp
d
0
/R⫾ V
pp
␥
d
0
2
/R
2
, obtained from Eq. 共3兲 for
⫽ 0
by using the parameters V
pp
⫽ 2.53 eV and
␥
⫽ 0.43. The
experimental data obtained by STS 共Refs. 17,18兲 are shown
by open diamonds in the same figure. The agreement be-
tween our calculations and the experimental data is very
good considering the fact that there might be some uncertain-
ties in identifying the nanotube 关i.e., assignment of (n,m)
indices兴 in the experiment. The fit of this data to the empiri-
cal expression given by Eq. 共2兲 are also presented by a
dashed line for comparison.
The situation displayed in Fig. 3 indicates that the varia-
tion of the band gap with the radius is not simply 1/R, but
additional terms incorporating the chirality dependence are
required. Most importantly, the mixing of the singlet
*
state with the the singlet
*
state due to the curvature, and
its shift towards the valence band with increasing curvature
is not included in neither the
orbital tight binding model,
nor the empirical relations expressed by Eqs. 共2兲 and 共3兲.
This behavior of the singlet
*
states is of particular impor-
tance for the applied radial deformation that modifies the
curvature and in turn induces metallization.
12,27,34
FIG. 3. 共a兲 Energies of the double degenerate
states 共VB兲, the
double degenerate
*
states 共CB兲, and the singlet
*
state as a
function of nanotube radius. Each data point corresponds to n rang-
ing from 4 to 15 consecutively. 共b兲 The calculated band gaps as a
function of the tube radius shown by filled symbols. Solid 共dashed兲
lines are the plots of Eq. 共3兲关Eq. 共2兲兴. The experimental data are
shown by open diamonds 共Refs. 7, 17,18兲.
BRIEF REPORTS PHYSICAL REVIEW B 65 153405
153405-3
In conclusion, we investigated structural and electronic
properties that result from the tubular nature of the SWNT’s.
The first-principles total energy calculations indicated that
significant amount of strain energy is implemented in a
SWNT when the radius is small. However, the elastic prop-
erties can be still described by the classical theory of elastic-
ity. We showed how the singlet
*
state in the conduction
band of a zigzag tube moves and eventually enters in the
band gap between the doubly degenerate
*
-conduction and
-valence bands. As a result, the energy band structure and
the variation of the gap with radius 共or n) differs from what
one derived from the zone folded band structure of graphene
based on the simple tight binding calculations.
ACKNOWLEDGMENTS
This work was partially supported by the NSF under
Grant No. INT01-15021 and TU
¨
BI
´
TAK under Grant No.
TBAG-U/13共101T010兲.
1
M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, Science of
Fullerenes and Carbon Nanotubes 共Academic Press, San Diego,
1996兲; R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical
Properties of Carbon Nanotubes 共Imperial College Press, Lon-
don, 1998兲.
2
N. Hamada, S. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68,
1579 共1992兲.
3
M.S. Dresselhaus, G. Dresselhaus, and R. Saito, Phys. Rev. B 45,
6234 共1992兲.
4
J.W. Mintmire, B.I. Dunlap, and C.T. White, Phys. Rev. Lett. 68,
631 共1992兲.
5
C.T. White, D.H. Robertson, and J.W. Mintmire, Phys. Rev. B 47,
5485 共1993兲.
6
S. Frank, P. Poncharal, Z.L. Wang, and W.A. Heer, Science 280,
1744 共1998兲; for a recent review, S. Ciraci, A. Buldum, and I.
Batra, J. Phys.: Condens. Matter 13, 537 共2001兲.
7
M. Ouyang, J. Huang, C.L. Cheung, and C.M. Lieber, Science
292, 702 共2001兲.
8
L. Chico, M.P. Lopez Sancho, and M.C. Munoz, Phys. Rev. Lett.
81, 1278 共1998兲.
9
P.G. Collins, A. Zettl, H. Bando, A. Thess, and R.E. Smalley,
Science 278, 100 共1997兲.
10
M. Bockrath, D.H. Cobden, P.L. McEuen, N.G. Chopra, A. Zettl,
A. Thess, and R.E. Smalley, Science 275, 1922 共1997兲.
11
A. Bezryadin, A.R.M. Verschueren, S.J. Tans, and C. Dekker,
Phys. Rev. Lett. 80, 4036 共1998兲.
12
C¸ . Kılıc¸, S. Ciraci, O. Gu
¨
lseren, and T. Yildirim, Phys. Rev. B 62,
16 345 共2000兲.
13
H.S. Sim, C.J. Park, and K.J. Chang, Phys. Rev. B 63, 073402
共2001兲.
14
O. Gu
¨
lseren, T. Yildirim, and S. Ciraci, Phys. Rev. Lett. 87,
116802 共2001兲.
15
J.W.G. Wildo
¨
er, L.C. Venema, A.G. Rinzler, R.E. Smalley, and C.
Dekker, Nature 共London兲 391,59共1998兲.
16
L.C. Venema, J.W. Janssen, M.R. Buitelaar, J.W.G. Wildo
¨
er, S.G.
Lemay, L.P. Kouwenhoven, and C. Dekker, Phys. Rev. B 62,
5238 共2000兲.
17
T.W. Odom, J. Huang, P. Kim, and C.M. Lieber, Nature 共London兲
391,62共1998兲.
18
T.W. Odom, J. Huang, P. Kim, and C.M. Lieber, J. Phys. Chem. B
104, 2794 共2000兲.
19
X. Blase, L.X. Benedict, E.L. Shirley, and S.G. Louie, Phys. Rev.
Lett. 72, 1878 共1994兲.
20
C.L. Kane and E.J. Mele, Phys. Rev. Lett. 78, 1932 共1997兲.
21
A. Kleiner and S. Eggert, Phys. Rev. B 63, 073408 共2001兲.
22
A. Kleiner and S. Eggert, Phys. Rev. B 64, 113402 共2001兲.
23
J.P. Perdew and Y. Wang, Phys. Rev. B 46, 6671 共1992兲.
24
M.C. Payne, M.P. Teter, D.C. Allen, T.A. Arias, and J.D. Joan-
nopoulos, Rev. Mod. Phys. 64, 1045 共1992兲.
25
D. Vanderbilt, Phys. Rev. B 41, 7892 共1990兲.
26
T. Yildirim, O. Gu
¨
lseren, C¸ . Kılıc¸, and S. Ciraci, Phys. Rev. B 62,
12648 共2000兲; T. Yildirim, O. Gu
¨
lseren, and S. Ciraci, ibid. 64,
075404 共2001兲.
27
O. Gu
¨
lseren, T. Yildirim, S. Ciraci, and C¸ . Kılıc¸, Phys. Rev. B 65,
155410 共2002兲.
28
D. Sanchez-Portal, E. Artacho, J.M. Soler, A. Rubio, and P. Or-
dejon, Phys. Rev. B 59,12678共1999兲.
29
D.H. Robertson, D.W. Brenner, and J.W. Mintmire, Phys. Rev. B
45,12592共1992兲.
30
G.G. Tibbets, J. Cryst. Growth 66, 632 共1984兲.
31
K.N. Kudin, G.E. Scuseria, and B.I. Yakobson, Phys. Rev. B 64,
235406 共2001兲.
32
H. Yorikawa and S. Muramatsu, Solid State Commun. 94, 435
共1995兲.
33
H. Yorikawa and S. Muramatsu, Phys. Rev. B 52, 2723 共1995兲.
34
C.J. Park, Y.H. Kim, and K.J. Chang, Phys. Rev. B 60,10656
共1999兲.
BRIEF REPORTS PHYSICAL REVIEW B 65 153405
153405-4