Effects of environmental dielectric screening on optical absorption in carbon nanotubes

Transcription

1 Effects of environmental dielectric screening on optical absorption in carbon nanotubes Tsuneya Ando Department of Physics, Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo , Japan Abstract Effects of environmental dielectric screening on optical absorption spectrum are studied in carbon nanotubes within a k p scheme. The screening is sensitive to the effective distance between the nanotube and the dielectric medium. For material surrounding the nanotube, the band gap decreases with the increase of the dielectric constant, but the energy of the ground exciton exhibits only a slight decrease for effective distance comparable to interlayer spacing of bulk graphite. For dielectric material inside the nanotube, effects are much weaker because of anti-screening. Keywords: carbon nanotube, exciton, image potential, effective-mass theory, dielectric constant, anti-screening 1. Introduction In carbon nanotubes, exciton effects are known to play dominant roles in their electronic properties because of the quasione-dimensionality and of strong electron-electron interaction. The interaction effect can be modified by environmental dielectric material covering or being inserted into a nanotube. The purpose of this paper is to theoretically clarify effects of environmental dielectric screening on electronic and optical properties of nanotubes based on a k p scheme. Important optical properties of nanotubes were first demonstrated within the lowest order k p scheme for describing electronic states in graphene [1, 2, 3, 4, 5, 6, 7, 8]. In fact, it was used for the demonstration of the suppression of the absorption of light polarized perpendicular to the axis [9, ] and the role of interaction effects on the band gap and on excitons [11, 12]. The same scheme was also used for various exciton-related properties such as excitons for cross polarization [13, 14], excitons in metallic nanotubes [15], two-photon absorption spectra [16], exciton fine structure [17], family behavior [18], and most recently effects of environmental screening [19]. In this paper, we study environmental effects on the effective Coulomb potential, clarify the screening and anti-screening behavior depending on outside or inside dielectric material, and calculate optical absorption spectrum within a continuum model. A brief review is given on the k p description of electronic states in Sec. 2 and the Coulomb potential in the presence of environmental dielectric medium in Sec. 3. Explicit numerical results on optical absorption spectra are presented in Secs. 4 and 5. A short summary and conclusion are given in Sec Electronic States Carbon nanotubes are rolled up graphene sheets. In graphene sheet, the conduction and valence bands consisting of π states cross at K and K points of the Brillouin zone, where the Fermi level is located [, 21]. In the lowest-order effective-mass approximation, electronic states of the π-bands near a K point are described by the k p equation: γ(σ xˆk x + σ yˆk y )F(r) = εf(r), (1) where γ is a band parameter, σ x and σ y are the Pauli spin matrices, and ˆk = (ˆk x, ˆk y ) is a wave-vector operator. The structure of a carbon nanotube is specified by chiral vector L in the circumference direction, corresponding to a lattice translation vector in the graphene sheet. For tubes with a sufficiently large diameter, the energy bands are obtained by imposing periodic boundary condition F(r + L) = F(r) exp( 2πiν/3), where ν is an integer determined uniquely for L as ν = or ±1. A nanotube becomes a semiconductor for ν = ±1 and a metal with vanishing gap for ν = [5, 7]. The energy bands of a semiconducting tube are specified by α = (s, n, k), where n is an integer specifying the wave vector κ νn (2π/L)[n (ν/3)] along the circumference, s = +1 for the conduction and 1 for the valence band, respectively, and k is the wave vector in the axis direction. The corresponding wave function is written as F α (r) = (LA) 1/2 exp(iκ νn x + iky) F α, with κ κ κ κ Figure 1: A schematic illustration of a nanotube (CN) with radius R, surrounded by a dielectric medium with a hollow cylinder with radius R and containing a cylindrical dielectric medium with radius R. The dielectric constant for r >R is denoted by κ out and that for r <R by κ in. Preprint submitted to Physica E June 21,

2 . -.1 Energy (units of e 2 /κinr) κ out /κ in = κ out =κ in -.5 κ out /κ in = κ out =κ in Distance (units of L) κ out /κ in = κ out =κ in Figure 2: Some examples of the effective potential in the real space for n = dominant in the exciton binding, when the nanotube is surrounded by a dielectric medium with a hollow cylinder with radius R. κ out /κ in = Energy (units of e 2 /κoutr) κ in /κ out = κ in =κ out -.5 κ in /κ out = κ in =κ out Distance (units of L) κ in /κ out = κ in =κ out Figure 3: Some examples of the effective potential in the real space for n = dominant in the exciton binding, when a dielectric medium with a hollow cylinder with radius R is inserted in the nanotube. κ in /κ out =... A being the tube length, where x axis is chosen in the circumference direction and the y axis in the axis direction. The energy is given by ε snk = sε n (k) with ε n (k) = γ κ 2 νn + k 2. In the absence of interaction, the first and second gaps are given by 4πγ/3L and 8πγ/3L, respectively. The energy bands for the K point are obtained by making time reversal operation F T K = e ψ σ z F K and F T K = e iψ σ z F K with ψ being an arbitrary phase and σ z the Pauli spin matrix, where F K and F K are the wave functions at the K and K points, respectively [23, 24]. In actual nanotubes with small diameters, higher-order terms giving rise to trigonal warping of the bands and effective flux due to curvature and associated lattice distortion can play significant roles and cause the so-called family behavior [18]. Such terms are not important in clarifying essential features of effects of environmental dielectric material and not considered in this paper Environmental Dielectric Material We consider effects of dielectric material surrounding or filling inside of a tube as illustrated in Fig. 1. When the nanotube with radius R is surrounded by material with dielectric constant κ out having a hollow and coaxial cylinder with radius R, the Fourier coefficient of the Coulomb potential becomes V n (q) = 2e2 κ in F n (qr) [ 1 δ κg n (qr ) 1 + δ κ G n (qr ) τ n(qr, qr ) ], (2) F n (qr) = I n (qr)k n (qr), (3) G n (qr ) = qr I n (qr )K n(qr ), (4) τ n (qr, qr ) = I n(qr) K n (qr ) K n (qr) I n (qr ), (5) where I n (t) and K n (t) are the modified Bessel function of the first and the second kind, respectively, δ κ = (κ out /κ in ) 1, n is

3 Excitation = γ =2.7eV ε c (2πγ/L) -1 = κ=2.5 κ in = 1 Exciton Band Gap No Interaction... Exp. by Ohno et al Dielectric Constant Outside of Nanotube Excitation = γ =2.7eV ε c (2πγ/L) -1 = κ=2.5 κ out = Exciton Band Gap No Interaction Dielectric Constant Inside of Nanotube Figure 4: Calculated excitation energies of semiconducting carbon nanotubes as a function of dielectric constant κ out of outside material for various values of together with normalized experimental results of Ohno et al. [22]. Figure 5: Calculated excitation energies as a function of dielectric constant κ in of inside material for various values of. an integer specifying the wave number along the circumference, and q is the wave number in the axis direction [19, 25]. The term proportional to δ κ represents effects of the image charge due to the polarization of the surrounding dielectric medium. When a dielectric material with cylindrical shape having dielectric constant κ in and radius R is inserted into the nanotube, the Coulomb potential becomes V n (q) = 2e2 F n (qr) [ 1 δ κ G n(qr ) κ out 1 + δ κ G n(qr ) τ n(qr, qr ) ], (6) G n(qr ) = qr I n(qr )K n (qr ) = 1 G n (qr ), (7) τ n(qr, qr ) = I n(qr ) K n (qr ) K n (qr) I n (qr) = τ n(qr, qr), (8) with δ κ = (κ in /κ out ) 1. Figure 2 shows some examples of the effective real-space potential for n = dominant in the exciton binding over the distance of the order of the circumference L roughly corresponding to the size of the ground-state exciton. The potential is very sensitive to the radius of the hollow region, i.e.,, in the small-distance region < L. In the large distance > L, the potential is well screened for κ out > for < 2. Figure 3 shows similar examples when a dielectric material is inserted into the nanotube. In this case the potential is not well screened and exhibits anti-screening behavior in the longdistance region, i.e., the potential is enhanced rather than reduced for large distance. This is a direct consequence of the fact that all electric-force lines pass through the outer region for n = and q, i.e., V n (q) V n (q), where V n (q) is the potential in the absence of dielectric material. In fact, the potential averaged over the distance ( V ()) remains unaffected by the presence of dielectric material in the nanotube and therefore the reduction of the potential due to dielectric screening in the short-distance region is canceled by the enhancement in the long-distance region. 3 The exciton wave function for an electron and a hole at the K point satisfies ε u ψ u n(k) = [ 2ε n (k) + ε nk ] ψ u n (k) 1 A V n ( q ) ε n m ( q ) (F +,n,k F +,m,k+q)(f,m,k+q F,n,k)ψ u m(k+q), (9) with ε nk = Σ +,n,k Σ,n,k, where Σ α is the self-energy calculated in the screened Hartree-Fock approximation as previously discussed [11, 12]. The dielectric function is written as ε n (q) = 1 + V n (q) [ Π n (q) + Π n(q) ], () where Π n (q) represents contributions of electrons in the vicinity of the K and K points described by the present k p scheme and Π n(q) represents those of electrons in the π bands away from the K and K points, electrons in the σ bands, and electrons in the core states. In previous studies without environmental dielectric screening [11, 12], we have replaced effects of Π n(q) with a phenomenological dielectric constant κ in such a way that 2e 2 F n (q)π n(q) κ 1 corresponding to the replacement of V n (q) with V n (q)/κ. We use this approximation for all n and q values. 4. Excitation Energies Figure 4 shows calculated excitation energies as a function of κ out for κ in = 1 and various values of the radius R of the hollow cylinder of surrounding dielectric medium. The energy of the first-excited exciton level is shown only for the first gap. In actual calculation of the self-energy, cut-off energy ε c has to be introduced because of logarithmic divergence [26, 27]. Here, we show results for ε c corresponding to typical tube diameters. m q

4 = κ in =1 κ out =1. ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in =1 κ out = ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in =1 κ out = ε c (2πγ/L) -1 = ΓL/2πγ=.1 Figure 6: Calculated dynamical conductivity in the presence of dielectric material outside a nanotube with = 1. κ out /κ in = 1... = κ in =1 κ out = ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in =1 κ out = ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in =1 κ out = ε c (2πγ/L) -1 = ΓL/2πγ=.1 Figure 7: Calculated dynamical conductivity in the presence of dielectric material outside a nanotube with = 1.5. κ out /κ in =... For different values of ε c the absolute energy values are slightly different, but the dependence on κ out is essentially the same. When = 1, both exciton and band-gap energies of the first gap approach the horizontal dotted line corresponding to the band gaps in the absence of the Coulomb interaction for sufficiently large κ out. The same is true for the second gap. The binding energy of excited exciton states diminishes with κ out faster than that of the ground exciton state. For excited exciton states, the long-range part of the attractive electron-hole interaction is dominant, i.e., the region qr 1. In this case the Coulomb potential rapidly decreases with the increase of κ out. Therefore, the special feature leading to the strong binding of excited exciton states [11, 16, 28] disappears in the presence of environmental material with large κ out. 4 The dependence on κ out is quite sensitive to. In fact, with the increase of, the asymptotic values of the band gap and the exciton ground-state energy for sufficiently large κ out become larger than those for = 1 and the deviation is appreciable even for = 1.5. Further, for > 1, the band gap and exciton ground state approach different asymptotic values. The difference becomes considerable with the increase of. This feature is qualitatively the same for the second gap. On the other hand, the binding energy of excited exciton states of the first gap rapidly disappears with κ out independent of. Figure 5 shows some examples of results in the case that a cylindrical dielectric medium with κ in is inserted coaxially into a nanotube. In this case, effects of κ in are much weaker due

5 =1. κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 =1. κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 =1. κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 Figure 8: Calculated dynamical conductivity in the presence of dielectric material in a nanotube with = 1. κ out /κ in =... = κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 = κ in = κ out =1 ε c (2πγ/L) -1 = ΓL/2πγ=.1 Figure 9: Calculated dynamical conductivity in the presence of dielectric material in a nanotube with = 1.5. κ out /κ in =... to the anti-screening discussed above. In fact, effects become visible only when κ in > and do not saturate even for κ in. One notable feature appears in the first-excited exciton states associated with the first gap, i.e., the binding energy is not affected for κ in < and its energy is almost parallel to the band gap as a function of κ in. The binding energy of the ground exciton state decreases much more rapidly with κ in. spectrum approaches that in the absence of interaction. Figure 7 shows results for = 1.5, i.e., the spacing between the tube and the dielectric material is the half of the tube radius. In this case, the position of the ground-state exciton with dominant absorption intensity is only slightly shifted to the low energy side and therefore the gross feature of the absorption spectrum remains almost independent of the dielectric constant. 5. Optical Absorption Figure 6 shows some examples of calculated optical absorption spectra when the tube is surrounded by dielectric material, i.e., = 1. With increase of κ out both band-gap enhancement and exciton binding energies diminish and the absorption 5 Figure 8 shows some examples when cylindrical dielectric medium with radius R = R is inserted into the tube. Effects of the dielectric screening are generally weak because of the antiscreening effect and several exciton bound states are present even for κ in =. When there is spacing between the tube and the dielectric material, i.e., = 1.5, as shown in Fig. 9, screening effects are further reduced.

6 Optical transition energies were experimentally investigated in the presence of some environmental dielectric material [29,, 31, 32, 33, 34, 35, 36, 37, 38, 39,, 41, 22, 42, 43, 44, 45, 46, 47, 48]. In ref. [22], in particular, the dependence on the dielectric constant was systematically studied in the range of κ out from 1 to 37, by immersing nanotubes in various organic solvents by means of the photoluminescence and excitation spectroscopy. The results show that with increasing κ out, both E 11 and E 22, corresponding to the ground exciton energies of the first and second gap, respectively, exhibit redshift by several tens mev without indication of significant dependence on their chiral structure. Figure 4 contains these experimental results by symbols such as filled circles, filled squares, filled triangles, etc. They are normalized in such a way that the transition energies for κ out = 1 are just at the corresponding theoretical results for both E 11 and E 22. The effect of environmental dielectric constant is sensitive to. For the (,) nanotube the diameter is R 6.8 Å. The effective radius of surrounding dielectric material is likely to depend on its microscopic or molecular structure and therefore it is very difficult to estimate exact value. If we choose the interlayer distance (3.34 Å) of bulk graphite as a typical distance, we have 1.5, for which the theory gives redshift only about a few percent of the transition energy itself, in reasonable agreement with the experiments. There remain various fundamental problems concerning the use of static dielectric constant, the continuum model with a constant effective distance, etc. Those problems are out of the scope of this paper and left for a future study. 6. Conclusion Effects of environmental dielectric screening have been theoretically studied within the k p scheme for surrounding material or cylindrical material inserted in a nanotube. The dielectric screening depends sensitively on the effective distance between the nanotube and the dielectric medium. For material surrounding the nanotube, the band gap decreases with the increase of the dielectric constant, but the energy of the ground exciton exhibits only a slight decrease except when the distance is negligibly small. The binding energy of excited exciton states disappears rapidly with the increase of the dielectric constant. For dielectric material inside the nanotube, effects of dielectric screening are much weaker due to the anti-screening and excited exciton states remain as bound states even for very large κ in such as. 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