Dependence of exciton transition energy of single-walled carbon nanotubes on surrounding dielectric materials

Chemical Physics Letters 442 (2007) 394–399 www.elsevier.com/locate/cplett

Dependence of exciton transition energy of single-walled carbon nanotubes on surrounding dielectric materials Y. Miyauchi a, R. Saito b, K. Sato b, Y. Ohno c, S. Iwasaki c, T. Mizutani c, J. Jiang d, S. Maruyama a,* a

d

Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan b Department of Physics, Tohoku University and CREST, Sendai 980-8578, Japan c Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan Center for High Performance Simulation and Department of Physics, North Carolina State University, Raleigh, NC 27695-7518, USA Received 10 April 2007; in final form 28 May 2007 Available online 8 June 2007

Abstract We theoretically investigate the environmental effect for optical transition energies of single-walled carbon nanotubes (SWNTs), by calculating the exciton transition energies of SWNTs. The static dielectric constants used in the exciton calculation can be expressed as a function of the dielectric constants of the surrounding material and that of the SWNT, in which the static and dynamic dielectric constants of the SWNT represent the screening effect of core electrons and the valence p electrons, respectively. The calculated results reproduce the environmental effect of the experimental transition energies for various surrounding materials and for various diameters of SWNTs.  2007 Elsevier B.V. All rights reserved.

1. Introduction Photoluminescence (PL) of single-walled carbon nanotubes (SWNTs) has been intensively studied for elucidating their unusual optical and electronic properties due to one dimensionality [1–16]. Since the energies of electron–electron repulsion and electron–hole binding for SWNTs are both considerably larger than those for conventional three-dimensional materials, the electron–electron and electron–hole Coulomb interactions play an important role in the optical transitions of SWNTs [17–22]. Thus, optical transition energies of SWNTs are strongly affected by the change of environment around SWNTs such as bundling [23], surfactant suspension [7,14,24] and DNA wrapping [25]. Lefebvre et al. [7] reported that the transition energies for suspended SWNTs between two pil*

Corresponding author. Fax: +81 3 5800 6983. E-mail address: [email protected] (S. Maruyama). URL: http://www.photon.t.u-tokyo.ac.jp (S. Maruyama).

0009-2614/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.06.018

lars fabricated on a Si substrate are blue-shifted relative to those for micelle-suspended SWNTs. Ohno et al. [14] compared the PL of suspended SWNTs directly grown on a grated quartz substrate using the alcohol CVD technique [6] with SDS-wrapped SWNTs [2]. The energy differences between air-suspended and SDS-wrapped SWNTs depend on (n, m) and type of SWNTs [26–28] [type I ((2n + m) mod 3 = 1) and type II ((2n + m) mod 3 = 2)]. For E11 transition energies, E11 for air-suspended SWNTs is always larger (blue-shifted) than E11 for SDS-wrapped SWNTs. This blue shift is mainly attributed to the increase in the net Coulomb interaction, which consists of the exciton binding energy and the self energy [21], by decreasing the dielectric constant. The blue shift of E11 is larger for type II SWNT than that for type I with similar diameters, while the blue shift of E22 is larger for type I than type II [14]. The surrounding material may also change the exciton energy by inducing the pressure to the SWNT, which depends on n  m [29]. However, we will set aside the pressure effect in the present Letter for simplicity.

Y. Miyauchi et al. / Chemical Physics Letters 442 (2007) 394–399

Recently, Ohno et al. studied the E11 transition energies of SWNTs in various surrounding materials with different dielectric constants, jenv [30]. The observed dependence of E11 on jenv for a (n, m) nanotube showed a tendency that can be roughly expressed as exp a E11 ¼ E1 11 þ Anm jenv

ð1Þ

where E1 11 denotes the transition energy when jenv is infinity, Aexp nm is the maximum value of the energy change of E11 by jenv, and a is a fitting coefficient in the order of 1. The reason why the experimental curve follows Eq. (1) has not yet been elucidated. In previous theoretical studies of excitonic transition energies for SWNTs [17,19,21,22], the screening effect of a surrounding material is mainly described using a static dielectric constant j. However, since j consists of both jenv and the screening effect by nanotube itself, jtube, the experimental dependence of transition energies on the dielectric constants of environment cannot be directly compared with the calculations [17,19,21,22] based on the static dielectric constant j. In this study, we set up a simple model for the relation between jenv and j. The calculated results of excitons for different jenv values reproduced well the experimental dependence of the transition energy on the dielectric constant of various surrounding materials.

395

the nanotube axis. In this Letter, we calculate the n = 0 state of q = 0 exciton for each (n, m) SWNT. 2.2. Dielectric screening effect The unscreened Coulomb potential V between carbon p orbitals is modeled by the Ohno potential [19]. We consider the dielectric screening effect within the random phase approximation (RPA). In the RPA, the static screened Coulomb interaction W is expressed as [17] W ¼ V =jðqÞ;

ð4Þ

where (q) is the dielectric function describing effects of the polarization of the p bands; j is the static dielectric constant describing the effects of electrons in core states, the r bonds, and surrounding materials. We directly calculate only the polarization for the p band, while the effects of electrons in the core states, the r bands, and the surrounding materials are represented by a single constant j. 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 the extended tight-binding method. Therefore, we make a simple model in this study for a relation between the static dielectric constant j and jenv to obtain the E11 dependence on jenv instead of treating the complicated dielectric response including surrounding materials.

2. Theoretical method

2.3. Relationship between j and jenv

2.1. Exciton transition energy

Fig. 1 shows a schematic view of the model relationship between j and jenv. Here we consider the screening effect related to j as a linear combination of the screening of nanotube itself and the surrounding material

Within the extended tight-binding model [21,22,28], we calculated the transition energies from the ground state to the first bright exciton state by solving the Bethe–Salpeter equation:  ½Eðkc Þ  Eðkv Þdðk0c ; kc Þdðk0v ; kv Þ  þ Kðk0c k0v ; kc kv Þ Wn ðkc kv Þ ¼ Xn Wn ðk0c k0v Þ; ð2Þ where kc and kv denote wave vectors of the conduction and valence energy bands and E(kc) and E(kv) are the quasielectron and quasi-hole energies, respectively. Xn is the energy of the nth excitation of the exciton (n = 0, 1, 2,. . .), and Wn(kckv) are the excitonic wavefunctions. The kernel Kðk0c k0v ; kc kv Þ describes the Coulomb interaction between an electron and a hole. Details of the exciton calculation procedure is the same as those presented in Refs. [21,22,28]. The exciton wavefunction jWnq i with a center-of-mass momentum q(=kc  kv) can be expressed as X Z nkc;ðkqÞv cþ ð3Þ jWnq i ¼ kc cðkqÞv j0i; k

where Z nkc;ðkqÞv is the eigenvector of the nth (n = 0, 1, 2,. . .) state of the Bethe–Salpeter equation, and j0æ is the ground state. Due to momentum conservation, the photon-excited exciton is an exciton with q  0 for parallel excitations to

1 C tube C env ¼ þ ; j jtube jenv

ð5Þ

where jtube is the dielectric constant within a nanotube except for the p bands, and Ctube and Cenv are coefficients for the inside and outside of a nanotube, respectively. Since the size of an exciton in the axial direction is much larger than the diameter for a semiconductor SWNT [21], most of the electric flux lines go through the tube and the surrounding material. Then Eq. (5) is understood by analogy of the serial connection of two capacitors with different dielectric constants, in which the electric flux lines pass through the two dielectric materials. The inverse capacitance for the series-connected capacitors is calculated as a sum of their inverse capacitances, which gives Eq. (5). Although Eq. (5) is a simplified model, this formula seems to be reasonable as the first approximation. Here we treat Ctube and Cenv as functions of the diameter of SWNT, since these values should depend on its cross section. As shown in Eq. (1), the observed transition energies [30] indicate that there is a limiting case [19] at jenv ! 1. There Cenv/jenv can be removed from Eq. (5), and 1/j is expressed as

396

Y. Miyauchi et al. / Chemical Physics Letters 442 (2007) 394–399

Since jvac defined here refers to the SWNTs without surrounding materials, this jvac value should be less than two and close to one. 2.4. Dependence of excitation energy on jenv The calculated decrease in the E11 energy for a (9, 8) SWNT with increasing j in a small j region, shown in Fig. 2a, can be mainly ascribed to the difference between the self energy (e–e repulsion) and the e–h binding energy; the former always exceeds the latter and both e–e and e–h interactions decrease with increasing j. The E11 depends almost linearly on j in the small j region. We have confirmed that this linear dependence is universal for all (n, m) SWNTs with diameters more than 0.7 nm. Such a linear dependence being assumed, the observed variation in the excitation energy dE11 ” E11  E11 (jenv = 1) in the small j region can be approximated by dE11 ¼ Anm ðj  jvac Þ;

Fig. 1. A schematic of the connection of the net dielectric constant j and the dielectric constant of the surrounding material jenv and the nanotube itself jtube.

ð9Þ

where Anm is the gradient of dE11 near the small j region for each (n, m) species. After transformation of j using Eq. (5), Eq. (9) can be modified as   jenv  1 1 vac dE11 ¼ Anm ðjtube  j Þ : vac Þ=jvac jenv þ ðj1 tube  j ð10Þ

ðjenv ! 1Þ

ð6Þ 0.9

j1 tube

1 1 C env ¼ 1 þ : j jtube jenv

0.88

E11 (eV)

where is the limiting value of the net dielectric constant j when jenv is infinity. Since the electric flux lines through the nanotube remain even when jenv ! 1, we assume that there is a certain value of jðj1 tube Þ that corresponds to the situation where dielectric screening by the surrounding material is perfect and only the dielectric response of the nanotube itself contributes to the net screening effect. Replacing Ctube/jtube by j1 tube , Eq. (5) is modified as ð7Þ

vac

0.7 0

50

100

k

0.86

1

3

2

k 0

k vac =1.0 k vac =1.5

-0.01

ð8Þ

where j is the static dielectric constant under the condition that the nanotube is placed in the vacuum, instead of the free vacuum. We now express j as a function of jenv vac using two parameters, j1 . It is important to note tube and j vac 1 that both jtube and j depend on the diameter, as discussed in Section 3. The value of jvac can be estimated as follows. The j value is assumed to be 2 in previous papers [17,19,21,22], to obtain a good fit with experiments for the SWNTs placed in contact with surrounding materials.

δE11 (eV)

1 1  1 ; vac j jtube

0.8

0.84

Next, we assume that the SWNT is placed in a vacuum, which corresponds to j = jvac and jenv = 1, and then Cenv can be expressed as C env ¼

E11 (eV)

1 C tube 1 ¼  ; j jtube j1 tube

k vac =2.0 -0.02 -0.03

0

20

40

kenv Fig. 2. (a) The E11 energy for a (9, 8) SWNT as a function of j. (b) dE11 dependence on jenv. Inset in (a) shows the E11 dependence up to j = 100. In (b), circles denote the experimental data and solid curves denote the calculated results of Eq. (10).

Y. Miyauchi et al. / Chemical Physics Letters 442 (2007) 394–399 60 2n+m=20

23

Anm (meV)

vac The factor Anm ðj1 Þ corresponds to the maximum tube  j value of dE11 when jenv ! 1, which corresponds to the value of coefficient Aexp nm in the fitting curve of Eq. (1). For (9, 8) SWNT, the value fitted to the calculated results for Anm is 33 meV and Aexp nm obtained by the fit to the experiment [30] using Eq. (1) is 36 meV, and hence vac j1 should be 1. Since jvac is estimated to be tube  j 1–2 as discussed in Section 2.3, j1 tube is likely to be 2–3. vac These values for j1 are consistent with those tube and j used conventionally for the SWNTs in dielectric materials [17,19,21,22].

ð11Þ

Thus, we express dE11 as a function of jenv with a single vac parameter, ðj1 Þ, that depends on the diameter of tube  j SWNT, dt. The calculated values of Anm for several (n, m)’s, shown in Fig. 3a, display systematic patterns (family pattern) for (2n + m = const). We note a slight dependence on the diameter and a larger dependence of Anm on the chiral angle h [26] for type II SWNTs (blue) than for type I (red). The type II SWNTs with larger chiral angles tend to have larger values of Anm. For a convenient use of Eq. (10), we give a fitting function of Anm as Anm ¼ A þ Bd t þ ðC þ D=d t Þ cos 3h;

ð12Þ

which gives the average error of ±2 meV and the maximum errors of 8 and 5 meV for SWNTs of types I and II, respectively. The fit curve is shown in Fig. 3a by solid lines. The values of A (meV), B (meV nm1), C (meV), D (meV nm) are (36, 4, 0, 0) and (33,3, 6, 7) for type I and II, respectively. In order to expand our results to other (n, m) SWNTs, we need to describe the diameter dependence of vac ðj1 Þ. An exact function should be calculated by tube  j taking into account the Coulomb interaction considering the induced surface charge at the boundary of the nanotube and the surrounding material for an e–e or e–h pair for each (n, m) SWNT. Instead of calculating this complivac cated function, we roughly estimate ðj1 Þ as a simtube  j ple function of diameter dt. As shown in Fig. 3b, vac ðj1 Þ is roughly proportional to 1=d 2t , tube  j vac Þ¼ ðj1 tube  j

E ; d 2t

29

32 35

38

40

19

22

25 28

31

34

37

0.8

1.2 dt (nm)

1.6

2

ð13Þ

k vac k tube

As shown in Fig. 2b, qualitative shapes of the theoretical curves of dE11 based on Eq. (10) agree well with the experiment, being hardly affected by the change in jvac. Since the exact value of jvac is yet unknown, we hereafter assume that jvac = 1.5 for all (n, m) SWNTs. The fitting values vac for (9, 8) SWNT in Fig. 2b, j1 = 1.5, are tube ¼ 2:7 and j consistent with those discussed in Section 2. After setting jvac = 1.5, Eq. (10) turns to Anm ðjenv  1Þ : vac Þ þ 1=1:5 jenv ðj1 tube  j

26

20

3. Results and discussion

dE11 ¼

397

1

0 0

0.5 2 2 1/dt (nm )

1

Fig. 3. (a) Calculated values of Anm for each (n, m) SWNT. Open (red) and solid (blue) circles correspond to types I and II SWNTs, respectively. Solid vac lines denote the curve by Eq. (12). (b) Dependence of j1 on 1=d 2t . tube  j A dotted line is a fitting function of E=d 2t (Eq. (13)). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

vac Þ is with the coefficient E = 1.5 ± 0.3 nm2. Here ðj1 tube  j obtained by a fit using Eq. (10) and Anm calculated for each chirality. Fig. 3b clearly shows that our calculated Anm well describes the chiral angle dependence of dE11 and that the remaining diameter dependence is understood by vac ðj1 Þ in terms of 1=d 2t . This 1=d 2t dependence imtube  j plies that j1 tube depends on the volume of the inner space of SWNT. Although the number of experimental data available for this fit is small and the selection of this function is somewhat arbitrary, it seems reasonable to conclude that 1=j1 tube increases with the increase in the diameter, since 1=j1 tube corresponds to the Coulomb interaction through the inner space of the nanotube. Future experimental and theoretical studies are definitely needed to find a more accurate form of this function. Fig. 4 shows dE11 as a function of jenv for (a) the experiment and (b) the calculation using Eqs. (11) and (13). Fig. 4c compares dE11 for the experiment and that for the calculation with the same jenv values. Details of experimental data will be published elsewhere [30]. The calculated curves for various (n, m) SWNTs well reproduce the observed tendency for each (n, m) SWNT, and the degree of difference between each (n, m) type also agrees well with the experiment. As shown in Fig. 4c, dE11 (theory) agrees with dE11 (experiment) except for several points indicated by a dotted ellipse in the figure, which correspond to a case for the smallest jenv = 1.9 (hexane) except for jenv = 1 (air) in the experimental data [30].

Y. Miyauchi et al. / Chemical Physics Letters 442 (2007) 394–399

(9,4) (8,6) (12,1) (11,3) (10,5) (9,7) (8,7) (9,8) (10,6)

-20

0

0

-20

δE11 (meV)

δE11 (meV)

0

δE11(experiment) (meV)

398

-40

-40

-60

-60 0

20 k env

40

0

20 k env

40

-20

-40

- 60 -60

-20 δE11(theory) (meV)

- 40

0

Fig. 4. Transition energy dependence plotted as a function of jenv: (a) experimental values, for (n, m) with symbols; (b) calculated results in solid curves with the same symbols; (c) two-dimensional plots of dE11(experiment) and dE11(theory), with a dotted line indicating equality. Open (red) and solid (blue) symbols correspond to SWNTs of types I and II, respectively. Data enclosed in the dotted ellipse represent those for jenv = 1.9 [30]; see text. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The value of jenv = 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 electric field. Since jenv = 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, jenv  3), the fitting of Fig. 4c 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 decreases with increasing diameter, it is predicted that the amount of variation due to the change of jenv mostly depends on the diameter in the larger diameter range. Thus a PL experiment for nanotubes with large diameters is awaited for a further comparison. 4. Summary The dependence of exciton transition energies of SWNTs on dielectric constant of surrounding materials is investigated. We have proposed a simple model to describe a surrounding dielectric material around SWNTs through a static dielectric constant j in the calculation. Based on the model and our calculation of excitons, a formula to reproduce the feature of experimentally observed transition energy dependence on dielectric constant of various surrounding materials, and various dt and h is presented. Acknowledgements Y.M. is supported by JSPS Research Fellowships for Young Scientists (No. 16-11409). R.S. acknowledges a Grant-in-Aid (No. 16076201) from MEXT, Japan. S.M.

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