Dielectric constant model for environmental effects on the exciton energies of single wall carbon nanotubes
(supporting online material, A. R. T. Nugraha et al., Appl. Phys. Lett., 2010)
Supplement 1: Calculated Eii using the κ function for (n,m) SWNTs of the super-growth (SG) sample.
1.6
0.4


0.8  1   1 





Here the explicit formula of κ for the SG sample is given by:   0.84 p      1.39   1.00 ; ...(S.1)


 d t   lk 
where we have used the same origin for all the κ functions for different sample, that is 1.39 (see Fig. 2 of the paper).
In the table below, the data are arranged in columns and categorized into metallic SWNTs, type-I semiconducting SWNTs, and type-II semiconducting SWNTs
where mod( 2n  m,3)  0, 1, 2 , respectively. The Eii data for different subbands p are listed, namely, E11M lower branch and E11M higher branch for metallic
SWNTs, and E11S up to E44S for both type-I and type-II semiconducting SWNTs. Note that for metallic SWNTs, the density of states are splitted into lower and
higher E11M. The diameter (dt) range in this table is from 0.7 to 3.0 nm.
Metallic SWNTs
Type-I Semiconducting SWNTs
Type-II Semiconducting SWNTs
Energy
subband
n
m
p
dt [nm]
Eii [eV]
Energy
subband
n
m
p
dt [nm]
Eii [eV]
Energy
subband
n
m
p
dt [nm]
Eii [eV]
E11M
6
6
3
0.819
3.114
E11S
7
5
1
0.823
1.254
E11S
6
5
1
0.753
1.310
(lower)
7
4
3
0.761
3.080
8
3
1
0.777
1.391
7
3
1
0.703
1.311
7
7
3
0.953
2.771
8
6
1
0.956
1.093
7
6
1
0.887
1.138
8
2
3
0.725
2.864
9
1
1
0.754
1.564
8
4
1
0.834
1.158
8
5
3
0.894
2.779
9
4
1
0.908
1.183
8
7
1
1.021
1.006
8
8
3
1.088
2.473
9
7
1
1.091
0.976
9
2
1
0.800
1.151
9
0
3
0.712
2.741
10
2
1
0.877
1.311
9
5
1
0.966
1.026
9
3
3
0.852
2.702
10
5
1
1.039
1.059
9
8
1
1.156
0.905
9
6
3
1.027
2.521
10
8
1
1.225
0.878
10
0
1
0.789
1.171
9
9
3
1.223
2.250
11
0
1
0.867
1.500
10
3
1
0.927
1.023
Note: Double click and scroll down the above box to view the complete list as an excel spreadsheet. You can also copy and save it to a separate excel file.
You can calculate your own environmental transition energy ( E iienv ) for other surrounding materials. Use the following equation, which is related to Eq. (2) of the
paper:
E iienv

E iiSG
 p
~ env 
 C   A  B

 dt


 p
  C

d

 t




2

;


...(S.2)
where A, B, and C are the fitting parameters given in the paper and EiiSG is given in the table above.
~ env
As given in the paper, the C values for the super-growth (as-grown), alcohol-assisted CVD (as-grown), and HiPco (SDS) are 1.00, 1.42, and 1.52, respectively,
where we fitted the optimized κ from Eii data based on resonance Raman spectroscopy (RRS) experiments.
~
We have also calculated the C
env
values for other surrounding materials and measurements by photoluminescence spectroscopy (not only from resonance RRS as
~
discussed in the present paper). The C
env
values are listed in the following table:
Measurement
Synthesis method
(Environment)
SG
(as-grown)
RRS
AA-CVD
(as-grown)
HiPCO
(SDS) a)
HiPCO
(SDS) a)
PL
AA-CVD
(Hexane)b)
AA CVD
(Chloroform)b)
C env
0.84  0.03 (R2=0.71)
1.19  0.02 (R2=0.90)
1.28  0.05 (R2=0.73)
1.29  0.05
1.49  0.04
1.73  0.06
~ env
C
1.00  0.03 (R2=0.71)
1.42  0.02 (R2=0.90)
1.52  0.05 (R2=0.73)
1.54  0.05
1.77  0.04
2.06  0.06
a)
~
The C env values from two different measurements (PL and RRS) are within their error bars.
b)
The experimental Eii values used in the optimized κ calculations are taken from Y. Ohno et al., phys. stat. sol. (b) 244, 4002 (2007).
Supplement 2: Eii calculation from κ functions and comparison with experiment.
In the following table, we give a simple illustration for a calculation of κ for the SG sample using Eq. (1) of the paper, or explicitly Eq. (S.1) of this supplemental
material. The κ value thus obtained then specifies a unique E ii   value from our exciton program (see J. Jiang et al., Phys. Rev. B 75, 035407, 2007 and K. Sato
et al., Phys. Rev. B 76, 195446, 2007). Experimental data, Eii exp  , are taken by RRS measurements (see P. T. Araujo et al., Phys. Rev. Lett. 103, 146802, 2009)
SWNT type
Metal
Energy
type
n
m
p
2n + m
dt [nm]
lk [1/nm]
pdtlk
kappa
Eq. (1)
Eii(kappa)
[eV]
Eii (exp)
[eV]
diff [meV]
E11M
10
7
3
27
1.161
0.1375
4.1930
3.355
2.119
2.112
-7
(lower)
11
5
3
27
1.113
0.1289
4.6047
3.700
2.115
2.123
8
11
8
3
30
1.295
0.1306
3.5931
2.851
1.957
1.951
-6
12
6
3
30
1.245
0.1251
3.8946
3.104
1.962
1.978
16
13
4
3
30
1.208
0.1185
4.1779
3.342
1.952
1.981
29
13
13
3
39
1.763
0.1123
2.3316
1.791
1.566
1.549
-17
14
5
3
33
1.337
0.1149
3.5938
2.851
1.826
1.867
41
14
8
3
36
1.511
0.1127
2.9795
2.335
1.712
1.713
1
15
6
3
36
1.468
0.1090
3.1620
2.488
1.714
1.732
18
15
9
3
39
1.645
0.1093
2.6339
2.045
1.606
1.624
18
15
12
3
42
1.834
0.1045
2.2520
1.724
1.501
1.495
-6
15
15
3
45
2.033
0.1026
1.9239
1.448
1.396
1.412
16
16
4
3
36
1.437
0.1039
3.3367
2.635
1.707
1.743
36
Note: Double click and scroll down the above box to view the complete list as an excel spreadsheet. You can also copy and save it to a separate excel file.
In the above table, "lk" denotes the exciton size, "pdtlk" stands for p 0.8 1 / d t 
1.6
paper; and "diff" gives EiiSG exp  EiiSG   .
1 / l k 0.4
as shown in Fig. 2 of the paper; "kappa" is calculated using Eq. (1) of the