Edge-state signature in optical absorption of nanographenes: Tight ...

5 downloads 8661 Views 232KB Size Report
Sep 26, 2006 - capture the edge-state signature by optical experiments with visible light. ... exhibit unique electronic structures that do not exist for bulk.
RAPID COMMUNICATIONS

PHYSICAL REVIEW B 74, 121409共R兲 共2006兲

Edge-state signature in optical absorption of nanographenes: Tight-binding method and time-dependent density functional theory calculations Takahiro Yamamoto, Tomoyuki Noguchi, and Kazuyuki Watanabe Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan and CREST, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan 共Received 12 June 2006; published 26 September 2006兲 The optical absorption of triangular nanographenes with well-defined edge structure, that is, zigzag- and armchair-type edges, is studied using the tight-binding method within the Hückel approximation. The absorption spectra of zigzag triangular nanographenes exhibit rich peak structures originating from the excitations associated with the edge states, while those of armchair ones do not. The main feature in the absorption spectra is reproduced by time-dependent density functional theory calculations within a real-time scheme. By varying the size of the nanographenes, we elucidate the role of the edge states in the optical absorption, which becomes conspicuous when the graphene size is on a nanometer scale. Our finding in this paper provides a way to capture the edge-state signature by optical experiments with visible light. DOI: 10.1103/PhysRevB.74.121409

PACS number共s兲: 78.67.⫺n, 81.07.Nb

Nanographene is a hexagonal network of carbon atoms in two dimensions 共2D兲 with open edges. Because of the large fraction of carbon atoms sited on the edge, these compounds exhibit unique electronic structures that do not exist for bulk graphite. Fujita et al. proposed theoretically that graphitic ribbons with zigzag-type 共trans-polyacetylene-type兲 edges show ␲ states localized at the ribbon edges with flat bands close to the Fermi level, but the edge states do not appear in those with armchair-type 共cis-polyacetylene-type兲 edges.1–3 Since their prediction of the edge states, large efforts have been devoted to experimental confirmation of the edge states. Very recently the edge states were observed directly in scanning tunneling microscopy of graphite surfaces near monatomic step edges.4–7 Although a considerable amount of theoretical work has been done on various physical properties of graphitic ribbons with respect to the edge states, they have not been confirmed by experiments as it is difficult to synthesize the graphitic ribbons. In contrast to graphitic ribbons, nanometer-sized graphitic fragments exist widely in nature. For an example, activated carbon fibers consist of an assembly of nanographites of 2–3 nm size.8 Also, recent experiments succeeded in preparing a single graphitic fragment by heat treatment of nanodiamond particles at high temperatures.9 Despite the experimental activity, the physical properties of nanographenes have not yet been elucidated. Optical spectroscopy is a powerful way to characterize electronic structures of material. In this paper, we study the optical absorption of triangular nanographenes 共TNGs兲 with well-defined boundaries, that is, zigzag- and armchair-type edges, as shown in Fig. 1. The TNG is a simple and suitable model to investigate edge effects of nanographenes,10 although real nanographenes may have more complex edge structures. The dangling ␴-bond states on the edge carbon atoms are assumed to be terminated by hydrogen atoms in the present model. As illustrated in Fig. 1, the edge atoms 共dimers兲 on a side of the zigzag 共armchair兲 TNGs are numbered from 1 to Nz 共Na兲. The total numbers of carbon atoms in the zigzag and armchair TNGs are counted as Nz2 + 4Nz 1098-0121/2006/74共12兲/121409共4兲

+ 1 and 3Na共Na + 1兲, respectively. The smallest zigzag TNG with Nz = 2 corresponds to the phenalenyl C13H9 and the smallest armchair TNG with Na = 2 is known to be the triphenylene C18H12. We previously studied the optical absorption of zigzag TNG with Nz = 5 by time-dependent density functional theory 共TDDFT兲 calculations within a real-time scheme,11 which is a reliable method for addressing the problems associated with electronic excitations.12–14 We found a clear peak originating from the edge states in the absorption spectra of zigzag TNGs. The TDDFT method, however, is not adequate to clarify all the physical origins of every peak and structure in the optical absorption spectra. To study the size dependence of the edge effect on the optical absorption of TNGs is also a hard task for a TDDFT calculation because of the high computational cost for large systems. The objectives of this paper are to investigate the optical absorption spectra of the zigzag and armchair TNGs with various sizes and elucidate the physical origins of major peaks including edge states by adopting the orthogonal tightbinding method within the Hückel approximation. The nearest-neighbor transfer integral and the on-site energy of the 2pz orbital are chosen to be t = −3.0 eV and ⑀2pz = 0, re-

FIG. 1. Structures of 共a兲 zigzag nanographene with Nz = 6 and 共b兲 armchair nanographene with Na = 4. The edge carbon atoms are indicated by solid circles on each side.

121409-1

©2006 The American Physical Society

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 74, 121409共R兲 共2006兲

YAMAMOTO, NOGUCHI, AND WATANABE TABLE I. Parameters for the Gaussian expansion of ␲ orbitals 共Ref. 15兲. The units for Ik and ␴k are a . u.−5/2 and a.u., respectively. k

Ik ␴k

1

2

3

4

0.05 2.16

0.41 0.91

1.06 0.13

1.05 0.39

spectively. The result for the optical absorption spectra for a small system obtained by the tight-binding method will be compared to that obtained by the TDDFT method. The absorption for photoirradiation with energy ប␻ and electric polarization along the ␣ direction is characterized by the optical absorption cross section

␴ ␣共 ␻ 兲 =

␲e2 ប 兺 f fi␦共E f − Ei − ប ␻兲, 2c⑀0m f,i ␣

共1兲

where e and m are the charge and mass of electrons, c the speed of light, and ⑀0 the dielectric constant in vacuum. The summation indices i and f run over the occupied and unoccupied molecular orbitals, respectively. f ␣fi is the oscillator strength defined as f ␣fi =

2 円具⌿ f 兩p␣兩⌿i典円2 m E f − Ei

共2兲

with the momentum operator p␣ along ␣ = x , y , z. Here, 兩⌿i共f兲典 is the initial 共final兲 state with energy Ei共f兲. Within the tight-binding scheme based on the Hückel approximation, the molecular orbital 兩⌿m典 is given as a linear combination of 2pz orbitals: Ntot

兩⌿m典 = 兺 csm␾共r − Rs兲

共m = 1,2, . . . ,Ntot兲,

共3兲

s=1

where Ntot is the total number of carbon atoms, which is given previously in the text, and the coefficient set 兵csm其 is determined by tight-binding calculations. ␾共r − Rs兲 is the wave function of the 2pz orbital of the carbon atom at the position Rs. To calculate the oscillator strength in Eq. 共2兲, ␾共r − Rs兲 is expanded with Gaussian basis functions,

␾共r − Rs兲 =

共z − Zs兲

冑C

n



Ikexp 兺 k=1



共r − Rs兲2 2␴2k



,

共4兲

up to n = 4.15 The parameters Ik and ␴k are listed in Table I, and C is a normalization constant. We first discuss the electronic structures of zigzag TNGs. The density of states 共DOS兲 of zigzag TNGs with three different sizes 共Nz = 4, 10, and 50兲 is shown in Fig. 2. The lengths of each side of the TNG with Nz = 4, 10, and 50 are about L = 1.0, 2.5, and 12.3 nm, respectively. The continuous DOS curves 关共red兲 solid curves in Fig. 2兴 are given by smoothing discontinuous energy spectra of TNGs with a Lor-

FIG. 2. 共Color online兲 Density of states of zigzag TNGs with Nz⫽共a兲 4, 共b兲 10, and 共c兲 50. Dashed curves represent the DOS of 2D graphene. These DOSs are normalized by the number of carbon atoms in the system.

entz function with a broadening factor of 0.05 eV. The dashed curves represent the DOS of 2D bulk graphene. The DOS curves for TNGs have a prominent peak at E = 0 arising from 共Nz − 1兲-degenerate states with zero energy, which are not expected from the electronic states of 2D graphene. The 共Nz − 1兲-degenerate states are occupied by 共Nz − 1兲 electrons and show the electron density only on the sublattice including edge atoms in the zigzag TNGs. A particular state among the 共Nz − 1兲 states has electron density that is completely localized on the edge atoms. Therefore, these 共Nz − 1兲-degenerate states are called edge states as a generic name. Because the ratio of the number of the edge states to all states of the zigzag TNG is given by 共Nz − 1兲 / 共Nz2 + 4Nz + 1兲, the ratio exhibits a maximum value at Nz = 3 , 4 and decays as Nz−1 with increasing Nz. We thus expect that the edge effect on various physical quantities will become remarkable for zigzag TNGs with Nz = 3 and 4 共L = 0.75 and 1.0 nm on a side兲. On the other hand, no edge-localized states appear in the case of armchair TNGs. However, the armchair TNGs show another interesting nanometer-sized effect. Their DOSs exhibit two sharp peaks at E = ± 3.0 eV arising from 关共3Na − 1兲 / 2兴-degenerate states when Na is an odd number, but not when Na is an even number 共the DOSs of armchair TNGs are not shown兲. As the system size increases, the even-odd effect disappears and the DOSs reproduce the DOS of 2D graphene. Similar behaviors are also known in the zigzag carbon nanotubes16 and armchair graphitic ribbons.17 We are not going into the details of the even-odd effect because the discussion would deviate from the main subject of this paper. We now focus on the role of edge states in the optical absorption spectra of the zigzag TNGs. Figure 3 shows the averaged cross sections ␴共␻兲 = 关␴x共␻兲 + ␴y共␻兲兴 / 2Ntot for the zigzag TNGs with three different sizes calculated by the tight-binding method. The inset of Fig. 3共a兲 shows ␴共␻兲 for a zigzag TNG with Nz = 4 calculated by the TDDFT method for comparison with the tight-binding result. The cross sections generated by the electromagnetic field with the electric polarization along the x and y directions are almost the same

121409-2

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 74, 121409共R兲 共2006兲

EDGE-STATE SIGNATURE IN OPTICAL ABSORPTION…

FIG. 4. The ratio of the oscillator strength associated with the edge states to the total oscillator strength of zigzag TNGs as a function of Nz. The curve is a guide to the eye.

FIG. 3. 共Color online兲 Optical absorption cross sections of zigzag TNGs with Nz⫽共a兲 4, 共b兲 10, and 共c兲 50 calculated by the tight-binding method. Dashed curves represent the DOS of 2D graphene. The inset of 共a兲 is the cross section of zigzag TNGs with Nz = 4 calculated by the TDDFT method.

关␴x共␻兲 ⬇ ␴y共␻兲兴 and that by the electromagnetic field with the electric polarization along the z direction is zero 共␴z = 0兲. In Fig. 3, the solid and dashed curves represent the cross sections for the TNGs and 2D graphene, respectively. Here, we make the cross sections for the TNGs smooth using a Lorentz function with a broadening factor of 0.05 eV. The 共red兲 shaded area represents the cross section associated with optical excitations from the edge states to others, and vice versa. When the system size is small, the optical absorption in the visible light region 共1–3 eV兲 is dominated by the edge-state-related absorption indicated by the 共red兲 shaded area. With increasing size of the TNGs, the weight of the edge-state-related absorption becomes small and the overall features of the cross section curve reproduce that of 2D graphene. These results reflect that the ratio of the number of edge states and that of all states has a maximum at Nz = 3 , 4 as described previously. To quantify the size dependence of the role of the edge states in the optical absorption of zigzag TNGs, we introduce the ratio f edge / f all corresponding to the ratio of the 共red兲 shaded area to the total cross section in Fig. 3. Here, f edge 共all兲 is a sum of the oscillator strengths in Eq. 共2兲 with respect to the edge states 共all states兲. Figure 4 shows the ratio f edge / f all

as a function of Nz. The ratio exhibits a maximum value at Nz = 4 共1.0 nm on a side兲, and decreases with increasing Nz. Finally, the cross section for a small TNG with Nz = 4 calculated by the tight-binding method is compared with the TDDFT result in the inset in Fig. 3共a兲. The computational details of the present TDDFT calculations are available in our previous paper.11 The overall features of the two spectra are similar in the low-energy region below ⬃9 eV. Now we confirm that the peak at 3 eV in the optical absorption cross section calculated by the TD-DFT originates from optical transition from/to the edge states. The absorption band above ⬃9 eV in the TDDFT result is attributed to the excitation between ␴ orbitals, which is neglected in our tight-binding calculations. Here it is noted that the energy resolution for the TDDFT spectra is lower than that for the tight-binding spectra, i.e., the broadening factor of 0.2 eV for the TDDFT calculations is larger than that of 0.05 eV for the tightbinding calculations. For this reason, as seen in Fig. 3共a兲, the tight-binding spectrum exhibits fine structures reflecting the discrete energy spectra of TNGs in Fig. 2, which do not exist in the TDDFT spectrum. In principle, the energy resolution of the TDDFT spectrum could be increased by performing a long-time evolution of wave functions that cost computational time, because the energy resolution ⌬E of the TDDFT spectrum is related to the computation time T by the uncertainty principle ⌬E ⬃ h / T.18 Although we cannot directly discuss the quantitative aspects of the absorption spectra using the tight-binding and TDDFT methods, the qualitative features caused by the edge states can be captured by the tightbinding calculation, as also obtained by the TDDFT calculation. In conclusion, we have studied the electronic structures of nanographenes and their optical absorption using the tightbinding method. It is quite important to experimentally capture electronic states that are unique to nanostructures such as the edge state in nanographenes. We chose the TNGs with zigzag- and armchair-type edges as good models to explore the edge states in nanographenes and show that the edgestate signature can be observed by optical experiments with visible light. The optical absorption spectra of zigzag TNGs exhibit rich peak structures associated with the edge states in the visible light region 共1–3 eV兲, but not in the case of arm-

121409-3

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 74, 121409共R兲 共2006兲

YAMAMOTO, NOGUCHI, AND WATANABE

chair TNGs. Moreover, the effect of the edge states on the optical absorption is most remarkable for zigzag TNGs with Nz = 4 共1.0 nm on a side兲. Two complementary calculation methods, i.e., that tight-binding method and TDDFT method, give clear indication of edge states in the optical spectra.

1 M.

Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 共1996兲. 2 K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 共1996兲. 3 Y. Miyamoto, K. Nakada, and M. Fujita, Phys. Rev. B 59, 9858 共1999兲. 4 Y. Kobayashi, K. I. Fukui, T. Enoki, K. Kusakabe, and Y. Kaburagi, Phys. Rev. B 71, 193406 共2005兲. 5 Y. Kobayashi, K. I. Fukui, T. Enoki, and K. Kusakabe, Phys. Rev. B 73, 125415 共2006兲. 6 Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H. Fukuyama, Appl. Surf. Sci. 241, 43 共2005兲. 7 Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H. Fukuyama, Phys. Rev. B 73, 085421 共2006兲. 8 Y. Shibayama, H. Sato, T. Enoki, and M. Endo, Phys. Rev. Lett. 84, 1744 共2000兲. 9 A. M. Affoune, B. L. V Prasad, H. Sato, T. Enoki, Y. Kaburagi,

This work was supported in part by the “Academic Frontier” Project of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Some of the numerical calculations were performed on the Hitachi SR11000s at ISSP, the University of Tokyo.

and Y. Hishiyama, Chem. Phys. Lett. 348, 17 共2001兲. K. Tagami and M. Tsukada, e-J. Surf. Sci. Nanotechnol. 2, 205 共2004兲. 11 T. Noguchi, T. Shimamoto, and K. Watanabe, e-J. Surf. Sci. Nanotechnol. 3, 439 共2005兲. 12 G. F. Bertsch, J.-I. Iwata, A. Rubio, and K. Yabana, Phys. Rev. B 62, 7998 共2000兲. 13 K. Yabana and G. F. Bertsch, Phys. Rev. B 54, 4484 共1996兲. 14 T. Nakatsukasa and K. Yabana, J. Chem. Phys. 114, 2550 共2001兲. 15 A. Grüneis, Ph.D. thesis, Tohoku University, Japan, 2004. 16 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes 共Imperial College Press, London, 1998兲. 17 M.-F. Lin and F.-L. Shyu, J. Phys. Soc. Jpn. 69, 3529 共2000兲. 18 K. Yabana, T. Nakatsukasa, J.-I. Iwata, and G. F. Bertsch, Phys. Status Solidi B 243, 1121 共2006兲. 10

121409-4