Dependence of exciton transition energy of single-walled carbon nanotubes on
surrounding dielectric materials
Y. Miyauchia, R. Saitob, K. Satob, Y. Ohnoc, S. Iwasakic, T. Mizutanic, J. Jiangd, S. Maruyamaa∗
Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan
Department of Physics, Tohoku University and CREST, Sendai 980-8578, Japan
Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan and
Center for High Performance Simulation and Department of Physics,
North Carolina State University, Raleigh, North Carolina 27695-7518, USA
(Dated: November 19, 2018)
We theoretically investigate the dependence of exciton transition energies on dielectric constant
of surrounding materials. We make a simple model for the relation between dielectric constant of
environment and a static dielectric constant describing the effects of electrons in core states, σ bonds
and surrounding materials. Although the model is very simple, calculated results well reproduce
experimental transition energy dependence on dielectric constant of various surrounding materials.
PACS numbers: 78.67.Ch; 78.67.-n; 71.35.-y
I. INTRODUCTION
Photoluminescence (PL) of single-walled carbon nan-
otubes (SWNTs) has been intensively studied for elu-
cidating their unusual optical and electronic properties
due to one dimensionality1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16.
Since both of electron-electron repulsion and electron-
hole binding energies for SWNTs are considerably large
compared with those for conventional three-dimensional
materials, the Coulomb interactions between electron-
electron and electron-hole play an important role in
optical transition of SWNTs17,18,19,20,21,22. Optical
transition energies of SWNTs are strongly affected
by the change of environment around SWNTs such
as bundling23, surfactant suspension7,14,24 and DNA
wrapping25. Lefebvre et al.7 reported that the tran-
sition energies for suspended SWNTs between two pil-
lars fabricated by the MEMS technique are blue-shifted
relative to the transition energies for micelle-suspended
SWNTs. Ohno et al.14 have compared the PL of sus-
pended SWNTs directly grown on a grated quartz sub-
strate using alcohol CVD technique6 with SDS-wrapped
SWNTs2. The energy differences between air-suspended
and SDS-wrapped SWNTs depend on (n,m) and type of
SWNTs [type I ((2n+m) mod 3 = 1) or type II ((2n+m)
mod 3 = 2)26,27,28].
Recently, Ohno et al. studied E11 transition energies of
SWNTs in various surrounding materials with different
dielectric constant, κenv
29. Observed dependence of E11
on κenv for a (n,m) nanotube showed a tendency that
can be roughly expressed as
E11 = E
11 +A
env (1)
where E∞11 denotes a transition energy when κenv is in-
finity, Aexpnm is the maximum value of an energy change
∗Corresoponding author. FAX: +81-3-5841-6421.
E-mail: maruyama@photon.t.u-tokyo.ac.jp (S. Maruyama)
of E11 by κenv, and α is a fitting coefficient in the or-
der of 1, respectively. At this stage, the reason why the
experimental curve follows Eq.(1) is not clear.
In the previous theoretical studies of excitonic transi-
tion energies for SWNTs17,19,21,22, a screening effect of a
surrounding material is mainly described using a static
dielectric constant κ. However, since κ consists of both
κenv and screening effect by nanotube itself, κtube, ex-
perimental dependence of transition energies on dielectric
constants of environment can not directly compared with
calculations17,19,21,22 using the static dielectric constant
κ. In this study, we make a simple model for the relation
between κenv and κ. The calculated results of excitons
for different κenv reproduced well the experimental tran-
sition energy dependence on dielectric constant of various
surrounding materials.
II. THEORETICAL METHOD
A. Exciton transition energy
Within the extended tight-binding model21,22,28, we
calculated transition energies from the ground state to
the first bright exciton state by solving the Bethe-
Salpeter equation,
[E(kc)− E(kv)]δ(k
c,kc)δ(k
v,kv)
v,kckv)
Ψn(kckv) = ΩnΨ
where kc and kv denote wave vectors of the conduc-
tion and valence energy bands and E(kc) and E(kv)
are the quasi-electron and quasi-hole energies, respec-
tively. Ωn is the energy of the n-th excitation of the
exciton (n = 0, 1, 2, · · · ), and Ψn(kckv) are the excitonic
wavefunctions. The kernel K(k
v,kckv) describes the
Coulomb interaction between an electron and a hole. De-
tails of the exciton calculation procedure is the same as
presented in Refs21,22,28.
http://arxiv.org/abs/0704.1380v1
The exciton wavefunction |Ψnq > with a center-of-mass
momentum q(= kc − kv) can be expressed as
|Ψnq >=
Znkc,(k−q)vc
c(k−q)v|0 >, (3)
where Zn
kc,(k−q)v is the eigenvector of the n-th (n =
0, 1, 2, · · · ) state of the Bethe-Salpeter equation, and |0〉
is the ground state. Due to momentum conservation, the
photon-excited exciton is an exciton with q ≈ 0 for par-
allel excitations to the nanotube axis. In this Letter, we
calculate the n = 0 state of q = 0 exciton for each (n,m)
SWNT.
B. Dielectric screening effect
In our calculation, the unscreened Coulomb potential
V between carbon π orbitals is modeled by the Ohno
potential19. We consider the dielectric screening effect
within the random phase approximation (RPA). In the
RPA, the static screened Coulomb interaction W is ex-
pressed as17
W = V/κǫ(q), (4)
where ǫ(q) is the dielectric function describing effects of
the polarization of the π bands. κ is a static dielec-
tric constant describing the effects of electrons in core
states, σ bonds, and surrounding materials. In the cal-
culation, we directly calculate only the polarization for
the π band, and the effects of electrons in core states,
σ bands, and surrounding materials are represented by
a single constant κ. In the most accurate expression,
the inhomogeneous and nonlocal dielectric response of
the nanotube itself and the surrounding materials should
be considered. However, it is not easy within extended
tight binding method. In this study, instead of treating
the complicated dielectric response including surround-
ing materials, we make a simple model for a relation be-
tween the static dielectric constant κ and κenv to obtain
the E11 dependence on κenv.
C. Relationship between κ and κenv
Figure 1 shows a schematic view for the model relation-
ship between κ and κenv. Here we consider the screening
effect related to κ as a linear combination of the screening
of nanotube itself and the surrounding material
Ctube
κtube
, (5)
where κtube is the dielectric constant within a nanotube
except for the π bands, and Ctube and Cenv are coef-
ficients for the inside and outside of a nanotube, re-
spectively. As shown in Eq.(1), the transition energies
observed in the experiment29 indicate that there is a
FIG. 1: Schematic of the connection of the net dielectric con-
stant κ and the dielectric constant of the surrounding material
κenv and the nanotube itself κtube
limit value19 when κenv → ∞. Hence, when κenv →
∞, Cenv/κenv can be removed from Eq.(5), and 1/κ is
expressed by the limit value as
Ctube
κtube
κ∞tube
, (κenv → ∞) (6)
where κ∞tube is the limit value of the net dielectric con-
stant κ when κenv is infinity. Since electric flux lines
through inside of the nanotube remain even when κenv
→ ∞, we assume there is a certain value of κ (κ∞tube)
that corresponds to the situation when dielectric screen-
ing by surrounding material is perfect and only dielectric
response of the nanotube itself contributes to the net
screening effect.
Replacing Ctube/κtube by κ
tube, Eq.(5) is modified as
κ∞tube
. (7)
Next, we imagine that the SWNT is placed in the vac-
uum, which corresponds to κ = κvac and κenv = 1, and
then Cenv can be expressed as
Cenv =
κ∞tube
, (8)
where κvac is the static dielectric constant not for the vac-
uum, but for the situation that the nanotube is placed
in the vacuum. We now express κ as a function of κenv
through two parameters κ∞tube and κ
vac, whose values can
be estimated from the following discussions. In the pre-
vious papers17,19,21,22, κ value is put around 2 to obtain a
0 20 40
0 50 100
1 2 3
FIG. 2: (a) The E11 energy for a (9, 8) SWNT as a function
of κ. (b) δE11 dependence on κenv. Inset in (a) shows the E11
dependence up to κ = 100. In (b), circles denote the experi-
mental data and solid curves denote the calculated results of
Eq.(10) for κvac = 1.0 (black), 1.5 (red) and 2.0 (blue).
good fit with experiments for SWNTs with surrounding
materials. Jiang et al.21 have compared the calculated
results with the results for the two photon absorption
experiments10, and obtained the best fit using κ = 2.22
for SWNTs in a polymer matrix. Here, since κvac is for
nanotubes without surrounding materials, κvac should be
less than about 2 and close to 1 due to vacancy of inside
of the tubes. With regard to κ∞tube, according to the ex-
perimental results7,14,29, transition energy change due to
change of surrounding materials is at most 30-100meV.
Fig.2(a) shows the calculated E11 energy dependence on
κ for a (9,8) SWNT in a small κ region, while the inset
shows the E11 dependence up to κenv = 100. As shown
in Fig.2(a), variation of κ that yields the transition en-
ergy change of 30 to 100 meV is about 1 to 3 when κ is
around 2. Therefore, the value of κ∞tube should be around
2 to 3 and that of κvac should be around 1 to 2.
D. Dependence of excitation energy on κenv
As shown in Fig.2(a), the calculated E11 energies de-
crease with increasing κ. This is mainly due to the fact
that the self energy (e-e repulsion) always exceeds to
the e-h binding energy and that the both interactions
(e-e and e-h) decrease with increasing κ. The E11 al-
most linearly depends on κ around the small κ region.
We checked that the linear dependence is universal for
all (n,m)’s for diameters more than 0.7 nm. Assuming
the linear dependence, variation of the excitation energy
δE11 ≡ E11 − E11(κenv = 1) for the small κ region is
approximated by
δE11 = −Anm(κ− κ
vac), (9)
where Anm is the gradient of δE11 near the small κ region
for each (n,m) type. After we transform κ using the
relationship of Eq.(5), Eq.(9) is modified as
δE11 = −Anm(κ
tube − κ
vac)(
κenv − 1
κenv + (κ
tube − κ
vac)/κvac
Anm(κ
tube − κ
vac) corresponds to the maximum value
of δE11 when κenv → ∞, which corresponds to the value
of coefficient Aexpnm in the fitting curve of Eq.(1). For
(9, 8) SWNT, the fitted value to the calculated results
for Anm is 33 meV and A
nm obtained by the fit to the
experiment29 using Eq.(1) is 36 meV, and κ∞tube − κ
should be around 1. The values for κ∞tube and κ
are consistent with the values conventionally used for
SWNTs in dielectric materials17,19,21,22.
III. RESULTS AND DISCUSSION
Figure 2(b) compares δE11 for a (9,8) SWNT depend-
ing on κenv by the experiment (solid circles) and the cal-
culated results (lines) for κvac = 1, 1.5, 2.0 using Eq.(10).
As shown in Fig.2(b), the qualitative shape of theoretical
curves are in good agreement with the experiment and
not affected so much by the change of κvac. Since the ex-
act value of κvac is unknown, we hereafter set κ
vac = 1.5
for each (n,m) SWNT. For the (9, 8) SWNT in Fig.2(b),
κ∞tube = 2.7 and κ
vac = 1.5 are fitting values. These
values are consistent with the discussion in the previous
section. After setting κvac = 1.5, Eq.(10) turns to be
δE11 =
−Anm(κenv − 1)
κenv/(κ
tube − κ
vac) + 1/1.5
. (11)
Thus, we express δE11 as a function of κenv with one
parameter (κ∞tube − κ
vac).
Figure 3(a) shows the calculated values of Anm for each
(n,m)’s. Family pattern of (2n + m = const.) family
is drawn with the 2n + m values by dotted lines. We
found a slight diameter dependence and relatively large
chiral angle dependence of Anm for type II SWNTs (blue)
compared with type I SWNTs (red). The type II SWNTs
with larger chiral angles tend to have larger value of Anm.
For a convenient use of Eq.(10), we give a fitting function
of Anm meV as
Anm = A+Bdt + (C +D/dt) cos 3θ, (12)
which gives the average (maximum) error of
±2meV(8meV) for type I, and ±2meV(5meV) for
type II SWNTs. The fit curve is shown in Fig.3(a) by
solid lines. Here dt (nm) is the diameter of nanotube
and θ is the chiral angle26. The values of (A, B, C, D)
are (36, -4, 0, 0) and (33, -3, 6, 7) for type I and for type
II SWNTs, respectively.
In order to expand our result to many (n,m) SWNTs,
we need a function to describe (κ∞tube−κ
vac). It is impor-
tant to note that κ∞tube should depend on the diameter.
0.8
 1.2
 1.6
d
t 
(nm)
19
 22
34
 37
2n+m=20
35
 38
0 0.5 1
FIG. 3: (a) Calculated values of Anm for each (n,m) SWNT.
Open (red) and solid (blue) circles correspond to type I and
type II SWNTs, respectively. Solid lines denote the fit curve
by Eq.(12). (b) κ∞tube−κ
vac vs 1/d2t . The values of κ
tube−κ
are obtained by the fit of Eq.(10) to the experimental data
for each (n,m).
An exact function should be calculated by taking into ac-
count the Coulomb interaction considering induced sur-
face charge at the boundary of the nanotube and sur-
rounding materal for an e-e or e-h pair for each (n,m)
SWNT. Instead of calculating this complicated function,
here we roughly estimate the (κ∞tube − κ
vac) as a simple
function of diameter dt, since (κ
tube−κ
vac) should depend
on the cross section of a SWNT. As shown in Fig.3(b),
(κ∞tube − κ
vac) is roughly proportional to 1/d2t ,
(κ∞tube − κ
vac) =
, (13)
with the coefficient E = 1.5±0.3 nm2. Here (κ∞tube−κ
is obtained by the fit using Eq.(10) and Anm calculated
for each chirality. Fig.3(b) clearly shows that our cal-
culated Anm well describes the chiral angle dependence
of δE11 and that the remaining diameter dependence is
understood by (κ∞tube − κ
vac) through 1/d2t . This 1/d
dependence implies that κ∞tube depends on the volume of
inner space of the nanotube. Although the number of
experimental data available for the fit is small and se-
lection of this function is arbitrary to some extent, it is
reasonable that 1/κ∞tube increase with the increase of the
diameter, since 1/κ∞tube corresponds to the Coulomb in-
teraction through the inner space of the nanotube. In
order to find an accurate form of the function, future ex-
periments and theoretical studies are definitely needed.
Figure 4 shows δE11 as a function of κenv for (a) the ex-
periment and (b) the calculation using Eq.(11) and (13).
Fig.4(c) compares δE11 for the experiment and that for
the calculation with the same κenv values. The same sym-
bols for an (n,m) are used in three figures of Fig.4. De-
tails of experimental data will be published elsewhere29.
Although our treatment is very simple, the calculated
(a) (b)
(8,6)
(9,4)
(8,7)
(9,7)
(10,5)
(11,3)
(12,1)
(9,8)
(10,6)
60 40 20 0
δE11(theory) (meV)
- - -0 20 40
0 20 40
FIG. 4: The transition energy dependence plotted as a func-
tion of κenv. (a) experiment and (b) calculated results are
indicated by (a) symbols and (b) solid curves. In (b), (n,m)
for each curve is indicated by a symbol on the curve. (c)
Comparison of δE11 for the experiment (δE11(experiment))
and calculation (δE11(theory)). A dotted line indicates the
line of δE11(experiment) = δE11(theory). Open (red) and
solid (blue) symbols correspond to type I and type II SWNTs,
respectively. The data in the dotted circle are the data for
κenv = 1.9
29 (see text).
curves for various (n,m) SWNTs well reproduce the ex-
perimentally observed tendency for each (n,m) SWNT,
and the degree of difference between each (n,m) type is
also in good agreement with the experiment. As shown
in Fig.4(c), δE11(theory) is in a good agreement with
δE11(experiment) except for several points indicated by
a dotted circle in the figure, which correspond to a case
for the smallest κenv = 1.9 (hexane) except for κenv = 1
(air) in the experimental data29. The value of κenv = 1.9
for hexane is adopted as the dielectric constant for the
material, in which the dipole moments of liquid hexane
are not aligned perfectly even in the presence of the elec-
tric field. Since κenv = 1.9 is a macroscopic value, a local
dielectric response might be different from the averaged
macroscopic response. If the local dielectric constant
near SWNTs becomes large (for example, κenv ≈ 3), the
fitting of Fig.4(c) becomes better. We expect that the
dipole moments of a dielectric material might be aligned
locally for a strong electric field near an exciton, which
makes the local dielectric constant relatively large. This
will be an interesting subject for exciton PL physics.
Since the difference of Anm between each (n,m) type de-
creases with increasing the diameter, it is predicted that
the amount of variation due to the change of κenv mostly
depend on diameter in the larger diameter range. Thus a
PL experiment for nanotubes with large diameters would
be desirable for a further comparison.
IV. SUMMARY
In summary, the dependence of exciton transition en-
ergies on dielectric constant of surrounding materials are
investigated. We proposed a model for the relation be-
tween dielectric constant of the environment and a static
dielectric constant κ in the calculation. Although the
model is quite simple, calculated results well reproduce
the feature of experimentally observed transition energy
dependence on dielectric constant of various surrounding
materials, and various dt and θ.
Acknowledgments
Y.M. is supported by JSPS Research Fellowships for
Young Scientists (No. 16-11409). R.S. acknowledges a
Grant-in-Aid (No. 16076201) from the Ministry of Edu-
cation, Japan.
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http://arxiv.org/abs/0704.1018

  We theoretically investigate the dependence of exciton transition energies on
dielectric constant of surrounding materials. We make a simple model for the
relation between dielectric constant of environment and a static dielectric
constant describing the effects of electrons in core states, $\sigma$ bonds and
surrounding materials. Although the model is very simple, calculated results
well reproduce experimental transition energy dependence on dielectric constant
of various surrounding materials.

Dependence of exciton transition energy of single-walled carbon nanotubes on
surrounding dielectric materials
Y. Miyauchia, R. Saitob, K. Satob, Y. Ohnoc, S. Iwasakic, T. Mizutanic, J. Jiangd, S. Maruyamaa∗
Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan
Department of Physics, Tohoku University and CREST, Sendai 980-8578, Japan
Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan and
Center for High Performance Simulation and Department of Physics,
North Carolina State University, Raleigh, North Carolina 27695-7518, USA
(Dated: November 19, 2018)
We theoretically investigate the dependence of exciton transition energies on dielectric constant
of surrounding materials. We make a simple model for the relation between dielectric constant of
environment and a static dielectric constant describing the effects of electrons in core states, σ bonds
and surrounding materials. Although the model is very simple, calculated results well reproduce
experimental transition energy dependence on dielectric constant of various surrounding materials.
PACS numbers: 78.67.Ch; 78.67.-n; 71.35.-y
I. INTRODUCTION
Photoluminescence (PL) of single-walled carbon nan-
otubes (SWNTs) has been intensively studied for elu-
cidating their unusual optical and electronic properties
due to one dimensionality1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16.
Since both of electron-electron repulsion and electron-
hole binding energies for SWNTs are considerably large
compared with those for conventional three-dimensional
materials, the Coulomb interactions between electron-
electron and electron-hole play an important role in
optical transition of SWNTs17,18,19,20,21,22. Optical
transition energies of SWNTs are strongly affected
by the change of environment around SWNTs such
as bundling23, surfactant suspension7,14,24 and DNA
wrapping25. Lefebvre et al.7 reported that the tran-
sition energies for suspended SWNTs between two pil-
lars fabricated by the MEMS technique are blue-shifted
relative to the transition energies for micelle-suspended
SWNTs. Ohno et al.14 have compared the PL of sus-
pended SWNTs directly grown on a grated quartz sub-
strate using alcohol CVD technique6 with SDS-wrapped
SWNTs2. The energy differences between air-suspended
and SDS-wrapped SWNTs depend on (n,m) and type of
SWNTs [type I ((2n+m) mod 3 = 1) or type II ((2n+m)
mod 3 = 2)26,27,28].
Recently, Ohno et al. studied E11 transition energies of
SWNTs in various surrounding materials with different
dielectric constant, κenv
29. Observed dependence of E11
on κenv for a (n,m) nanotube showed a tendency that
can be roughly expressed as
E11 = E
11 +A
env (1)
where E∞11 denotes a transition energy when κenv is in-
finity, Aexpnm is the maximum value of an energy change
∗Corresoponding author. FAX: +81-3-5841-6421.
E-mail: maruyama@photon.t.u-tokyo.ac.jp (S. Maruyama)
of E11 by κenv, and α is a fitting coefficient in the or-
der of 1, respectively. At this stage, the reason why the
experimental curve follows Eq.(1) is not clear.
In the previous theoretical studies of excitonic transi-
tion energies for SWNTs17,19,21,22, a screening effect of a
surrounding material is mainly described using a static
dielectric constant κ. However, since κ consists of both
κenv and screening effect by nanotube itself, κtube, ex-
perimental dependence of transition energies on dielectric
constants of environment can not directly compared with
calculations17,19,21,22 using the static dielectric constant
κ. In this study, we make a simple model for the relation
between κenv and κ. The calculated results of excitons
for different κenv reproduced well the experimental tran-
sition energy dependence on dielectric constant of various
surrounding materials.
II. THEORETICAL METHOD
A. Exciton transition energy
Within the extended tight-binding model21,22,28, we
calculated transition energies from the ground state to
the first bright exciton state by solving the Bethe-
Salpeter equation,
[E(kc)− E(kv)]δ(k
c,kc)δ(k
v,kv)
v,kckv)
Ψn(kckv) = ΩnΨ
where kc and kv denote wave vectors of the conduc-
tion and valence energy bands and E(kc) and E(kv)
are the quasi-electron and quasi-hole energies, respec-
tively. Ωn is the energy of the n-th excitation of the
exciton (n = 0, 1, 2, · · · ), and Ψn(kckv) are the excitonic
wavefunctions. The kernel K(k
v,kckv) describes the
Coulomb interaction between an electron and a hole. De-
tails of the exciton calculation procedure is the same as
presented in Refs21,22,28.
http://arxiv.org/abs/0704.1380v1
The exciton wavefunction |Ψnq > with a center-of-mass
momentum q(= kc − kv) can be expressed as
|Ψnq >=
Znkc,(k−q)vc
c(k−q)v|0 >, (3)
where Zn
kc,(k−q)v is the eigenvector of the n-th (n =
0, 1, 2, · · · ) state of the Bethe-Salpeter equation, and |0〉
is the ground state. Due to momentum conservation, the
photon-excited exciton is an exciton with q ≈ 0 for par-
allel excitations to the nanotube axis. In this Letter, we
calculate the n = 0 state of q = 0 exciton for each (n,m)
SWNT.
B. Dielectric screening effect
In our calculation, the unscreened Coulomb potential
V between carbon π orbitals is modeled by the Ohno
potential19. We consider the dielectric screening effect
within the random phase approximation (RPA). In the
RPA, the static screened Coulomb interaction W is ex-
pressed as17
W = V/κǫ(q), (4)
where ǫ(q) is the dielectric function describing effects of
the polarization of the π bands. κ is a static dielec-
tric constant describing the effects of electrons in core
states, σ bonds, and surrounding materials. In the cal-
culation, we directly calculate only the polarization for
the π band, and the effects of electrons in core states,
σ bands, and surrounding materials are represented by
a single constant κ. In the most accurate expression,
the inhomogeneous and nonlocal dielectric response of
the nanotube itself and the surrounding materials should
be considered. However, it is not easy within extended
tight binding method. In this study, instead of treating
the complicated dielectric response including surround-
ing materials, we make a simple model for a relation be-
tween the static dielectric constant κ and κenv to obtain
the E11 dependence on κenv.
C. Relationship between κ and κenv
Figure 1 shows a schematic view for the model relation-
ship between κ and κenv. Here we consider the screening
effect related to κ as a linear combination of the screening
of nanotube itself and the surrounding material
Ctube
κtube
, (5)
where κtube is the dielectric constant within a nanotube
except for the π bands, and Ctube and Cenv are coef-
ficients for the inside and outside of a nanotube, re-
spectively. As shown in Eq.(1), the transition energies
observed in the experiment29 indicate that there is a
FIG. 1: Schematic of the connection of the net dielectric con-
stant κ and the dielectric constant of the surrounding material
κenv and the nanotube itself κtube
limit value19 when κenv → ∞. Hence, when κenv →
∞, Cenv/κenv can be removed from Eq.(5), and 1/κ is
expressed by the limit value as
Ctube
κtube
κ∞tube
, (κenv → ∞) (6)
where κ∞tube is the limit value of the net dielectric con-
stant κ when κenv is infinity. Since electric flux lines
through inside of the nanotube remain even when κenv
→ ∞, we assume there is a certain value of κ (κ∞tube)
that corresponds to the situation when dielectric screen-
ing by surrounding material is perfect and only dielectric
response of the nanotube itself contributes to the net
screening effect.
Replacing Ctube/κtube by κ
tube, Eq.(5) is modified as
κ∞tube
. (7)
Next, we imagine that the SWNT is placed in the vac-
uum, which corresponds to κ = κvac and κenv = 1, and
then Cenv can be expressed as
Cenv =
κ∞tube
, (8)
where κvac is the static dielectric constant not for the vac-
uum, but for the situation that the nanotube is placed
in the vacuum. We now express κ as a function of κenv
through two parameters κ∞tube and κ
vac, whose values can
be estimated from the following discussions. In the pre-
vious papers17,19,21,22, κ value is put around 2 to obtain a
0 20 40
0 50 100
1 2 3
FIG. 2: (a) The E11 energy for a (9, 8) SWNT as a function
of κ. (b) δE11 dependence on κenv. Inset in (a) shows the E11
dependence up to κ = 100. In (b), circles denote the experi-
mental data and solid curves denote the calculated results of
Eq.(10) for κvac = 1.0 (black), 1.5 (red) and 2.0 (blue).
good fit with experiments for SWNTs with surrounding
materials. Jiang et al.21 have compared the calculated
results with the results for the two photon absorption
experiments10, and obtained the best fit using κ = 2.22
for SWNTs in a polymer matrix. Here, since κvac is for
nanotubes without surrounding materials, κvac should be
less than about 2 and close to 1 due to vacancy of inside
of the tubes. With regard to κ∞tube, according to the ex-
perimental results7,14,29, transition energy change due to
change of surrounding materials is at most 30-100meV.
Fig.2(a) shows the calculated E11 energy dependence on
κ for a (9,8) SWNT in a small κ region, while the inset
shows the E11 dependence up to κenv = 100. As shown
in Fig.2(a), variation of κ that yields the transition en-
ergy change of 30 to 100 meV is about 1 to 3 when κ is
around 2. Therefore, the value of κ∞tube should be around
2 to 3 and that of κvac should be around 1 to 2.
D. Dependence of excitation energy on κenv
As shown in Fig.2(a), the calculated E11 energies de-
crease with increasing κ. This is mainly due to the fact
that the self energy (e-e repulsion) always exceeds to
the e-h binding energy and that the both interactions
(e-e and e-h) decrease with increasing κ. The E11 al-
most linearly depends on κ around the small κ region.
We checked that the linear dependence is universal for
all (n,m)’s for diameters more than 0.7 nm. Assuming
the linear dependence, variation of the excitation energy
δE11 ≡ E11 − E11(κenv = 1) for the small κ region is
approximated by
δE11 = −Anm(κ− κ
vac), (9)
where Anm is the gradient of δE11 near the small κ region
for each (n,m) type. After we transform κ using the
relationship of Eq.(5), Eq.(9) is modified as
δE11 = −Anm(κ
tube − κ
vac)(
κenv − 1
κenv + (κ
tube − κ
vac)/κvac
Anm(κ
tube − κ
vac) corresponds to the maximum value
of δE11 when κenv → ∞, which corresponds to the value
of coefficient Aexpnm in the fitting curve of Eq.(1). For
(9, 8) SWNT, the fitted value to the calculated results
for Anm is 33 meV and A
nm obtained by the fit to the
experiment29 using Eq.(1) is 36 meV, and κ∞tube − κ
should be around 1. The values for κ∞tube and κ
are consistent with the values conventionally used for
SWNTs in dielectric materials17,19,21,22.
III. RESULTS AND DISCUSSION
Figure 2(b) compares δE11 for a (9,8) SWNT depend-
ing on κenv by the experiment (solid circles) and the cal-
culated results (lines) for κvac = 1, 1.5, 2.0 using Eq.(10).
As shown in Fig.2(b), the qualitative shape of theoretical
curves are in good agreement with the experiment and
not affected so much by the change of κvac. Since the ex-
act value of κvac is unknown, we hereafter set κ
vac = 1.5
for each (n,m) SWNT. For the (9, 8) SWNT in Fig.2(b),
κ∞tube = 2.7 and κ
vac = 1.5 are fitting values. These
values are consistent with the discussion in the previous
section. After setting κvac = 1.5, Eq.(10) turns to be
δE11 =
−Anm(κenv − 1)
κenv/(κ
tube − κ
vac) + 1/1.5
. (11)
Thus, we express δE11 as a function of κenv with one
parameter (κ∞tube − κ
vac).
Figure 3(a) shows the calculated values of Anm for each
(n,m)’s. Family pattern of (2n + m = const.) family
is drawn with the 2n + m values by dotted lines. We
found a slight diameter dependence and relatively large
chiral angle dependence of Anm for type II SWNTs (blue)
compared with type I SWNTs (red). The type II SWNTs
with larger chiral angles tend to have larger value of Anm.
For a convenient use of Eq.(10), we give a fitting function
of Anm meV as
Anm = A+Bdt + (C +D/dt) cos 3θ, (12)
which gives the average (maximum) error of
±2meV(8meV) for type I, and ±2meV(5meV) for
type II SWNTs. The fit curve is shown in Fig.3(a) by
solid lines. Here dt (nm) is the diameter of nanotube
and θ is the chiral angle26. The values of (A, B, C, D)
are (36, -4, 0, 0) and (33, -3, 6, 7) for type I and for type
II SWNTs, respectively.
In order to expand our result to many (n,m) SWNTs,
we need a function to describe (κ∞tube−κ
vac). It is impor-
tant to note that κ∞tube should depend on the diameter.
0.8
 1.2
 1.6
d
t 
(nm)
19
 22
34
 37
2n+m=20
35
 38
0 0.5 1
FIG. 3: (a) Calculated values of Anm for each (n,m) SWNT.
Open (red) and solid (blue) circles correspond to type I and
type II SWNTs, respectively. Solid lines denote the fit curve
by Eq.(12). (b) κ∞tube−κ
vac vs 1/d2t . The values of κ
tube−κ
are obtained by the fit of Eq.(10) to the experimental data
for each (n,m).
An exact function should be calculated by taking into ac-
count the Coulomb interaction considering induced sur-
face charge at the boundary of the nanotube and sur-
rounding materal for an e-e or e-h pair for each (n,m)
SWNT. Instead of calculating this complicated function,
here we roughly estimate the (κ∞tube − κ
vac) as a simple
function of diameter dt, since (κ
tube−κ
vac) should depend
on the cross section of a SWNT. As shown in Fig.3(b),
(κ∞tube − κ
vac) is roughly proportional to 1/d2t ,
(κ∞tube − κ
vac) =
, (13)
with the coefficient E = 1.5±0.3 nm2. Here (κ∞tube−κ
is obtained by the fit using Eq.(10) and Anm calculated
for each chirality. Fig.3(b) clearly shows that our cal-
culated Anm well describes the chiral angle dependence
of δE11 and that the remaining diameter dependence is
understood by (κ∞tube − κ
vac) through 1/d2t . This 1/d
dependence implies that κ∞tube depends on the volume of
inner space of the nanotube. Although the number of
experimental data available for the fit is small and se-
lection of this function is arbitrary to some extent, it is
reasonable that 1/κ∞tube increase with the increase of the
diameter, since 1/κ∞tube corresponds to the Coulomb in-
teraction through the inner space of the nanotube. In
order to find an accurate form of the function, future ex-
periments and theoretical studies are definitely needed.
Figure 4 shows δE11 as a function of κenv for (a) the ex-
periment and (b) the calculation using Eq.(11) and (13).
Fig.4(c) compares δE11 for the experiment and that for
the calculation with the same κenv values. The same sym-
bols for an (n,m) are used in three figures of Fig.4. De-
tails of experimental data will be published elsewhere29.
Although our treatment is very simple, the calculated
(a) (b)
(8,6)
(9,4)
(8,7)
(9,7)
(10,5)
(11,3)
(12,1)
(9,8)
(10,6)
60 40 20 0
δE11(theory) (meV)
- - -0 20 40
0 20 40
FIG. 4: The transition energy dependence plotted as a func-
tion of κenv. (a) experiment and (b) calculated results are
indicated by (a) symbols and (b) solid curves. In (b), (n,m)
for each curve is indicated by a symbol on the curve. (c)
Comparison of δE11 for the experiment (δE11(experiment))
and calculation (δE11(theory)). A dotted line indicates the
line of δE11(experiment) = δE11(theory). Open (red) and
solid (blue) symbols correspond to type I and type II SWNTs,
respectively. The data in the dotted circle are the data for
κenv = 1.9
29 (see text).
curves for various (n,m) SWNTs well reproduce the ex-
perimentally observed tendency for each (n,m) SWNT,
and the degree of difference between each (n,m) type is
also in good agreement with the experiment. As shown
in Fig.4(c), δE11(theory) is in a good agreement with
δE11(experiment) except for several points indicated by
a dotted circle in the figure, which correspond to a case
for the smallest κenv = 1.9 (hexane) except for κenv = 1
(air) in the experimental data29. The value of κenv = 1.9
for hexane is adopted as the dielectric constant for the
material, in which the dipole moments of liquid hexane
are not aligned perfectly even in the presence of the elec-
tric field. Since κenv = 1.9 is a macroscopic value, a local
dielectric response might be different from the averaged
macroscopic response. If the local dielectric constant
near SWNTs becomes large (for example, κenv ≈ 3), the
fitting of Fig.4(c) becomes better. We expect that the
dipole moments of a dielectric material might be aligned
locally for a strong electric field near an exciton, which
makes the local dielectric constant relatively large. This
will be an interesting subject for exciton PL physics.
Since the difference of Anm between each (n,m) type de-
creases with increasing the diameter, it is predicted that
the amount of variation due to the change of κenv mostly
depend on diameter in the larger diameter range. Thus a
PL experiment for nanotubes with large diameters would
be desirable for a further comparison.
IV. SUMMARY
In summary, the dependence of exciton transition en-
ergies on dielectric constant of surrounding materials are
investigated. We proposed a model for the relation be-
tween dielectric constant of the environment and a static
dielectric constant κ in the calculation. Although the
model is quite simple, calculated results well reproduce
the feature of experimentally observed transition energy
dependence on dielectric constant of various surrounding
materials, and various dt and θ.
Acknowledgments
Y.M. is supported by JSPS Research Fellowships for
Young Scientists (No. 16-11409). R.S. acknowledges a
Grant-in-Aid (No. 16076201) from the Ministry of Edu-
cation, Japan.
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