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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23350–23363
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Exploration of edge-dependent optical selection rules for graphene nanoribbons

H. C. Chung, M. H. Lee, C. P. Chang, and M. F. Lin  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23350-23363 (2011)
http://dx.doi.org/10.1364/OE.19.023350


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Abstract

Optical selection rules for one-dimensional graphene nanoribbons are explored based on the tight-binding model. A theoretical explanation, through analyzing the velocity matrix elements and the features of the wavefunctions, can account for the selection rules, which depend on the edge structure of the nanoribbon, i.e., armchair or zigzag edges. The selection rule of armchair nanoribbons is ΔJ = Jc – Jv = 0, and the optical transitions occur from the conduction to the valence subbands of the same index. Such a selection rule originates in the relationships between two sublattices and between the conduction and valence subbands. On the other hand, zigzag nanoribbons exhibit the selection rule |ΔJ| = odd, which results from the alternatively changing symmetry property as the subband index increases. Furthermore, an efficient theoretical prediction on transition energies is obtained by the application of selection rules, and the energies of the band-edge states become experimentally attainable via optical measurements.

© 2011 OSA

1. Introduction

In this work, the selection rules for the two typical types of GNR are analytically studied through the tight-binding model and a theoretical explanation of why they are different is given. The optical selection rule is significantly determined by the velocity matrix element, which is sandwiched by the wavefunctions of the initial and final states. The velocity matrix elements here are evaluated directly from the k-space derivatives of the Hamiltonian. On the other hand, the characteristics of state wavefunctions, which strongly depends on the width and edge structure of GNR, are intentionally obtained. Based on these characteristics of wavefunctions, the optical selection rules are then analytically explored.

2. Tight-binding model and electronic properties

The geometric structures of AGNR and ZGNR are shown in Fig. 1. Atoms A (dots) and B (circles) are the nearest neighbors to each other with a C–C bond length of b = 1.42 Å. The ribbon width is characterized by the number (Ny) of the dimer lines (armchair or zigzag lines) along the longitudinal direction (). The periodic length is Ix=3b(Ix=3b=a) for AGNRs (ZGNRs). The first Brillouin zone is confined to the region −π/Ixkxπ/Ix with 2Ny carbon atoms in a primitive unit cell. The Hamiltonian equation of the system is H|Ψ〉 = E|Ψ〉. With the framework of the tight-binding model considering only the nearest-neighbor interactions, the Hamiltonian operator is H=l,lγ0clcl, where γ0 is the hopping integral, cl(cl) is the creation (annihilation) operator on the site l (l′), and the on-site energies are set to zero. The Bloch wavefunction, i.e., the superposition of the 2Ny tight-binding functions |ϕl〉’s with the corresponding site amplitudes Cl’s, is expressed as
|Ψ=l=12NyCl|ϕlor|Ψ=m=1NyAm|am+m=1NyBm|bm,
(1)
where Am (Bm) is the site amplitude and |am〉 (|bm〉) is the tight-binding function associated with the periodic A (B) atoms.

Fig. 1 Energy dispersions and geometric structures for Ny = 69 AGNR and ZGNR. The conduction (valence) subbands of the corresponding indices Jc (Jv) are depicted in red (blue) color. The dashed-line rectangles represent primitive unit cells. The numbers of the mth dimer line are on the right side of the hexagonal lattice. The notations in the rectangles denote the A and B atoms on the mth dimer line.

The Hamiltonian H in the subspace spanned by the tight-binding functions is a 2Ny × 2Ny Hermitian matrix. For AGNRs, the upper triangular elements of H are
Hl,l={γ0exp[ikx(b)],ifl=l+1,lisodd,γ0exp[ikx(b/2)],ifl=l+1,liseven,γ0exp[ikx(b/2)],ifl=l+3,lisodd,0,otherwise.
(2)
For ZGNRs, the Hamiltonian elements are
Hl,l={2γ0cos(kxa/2),ifl=l+1,lisodd,γ0,ifl=l+1,liseven,0,otherwise,
(3)
where γ0 = 2.598 eV [40

40. J. C. Charlier, X. Gonze, and J. P. Michenaud, “First-principles study of the electronic properties of graphite,” Phys. Rev. B 43, 4579–4589 (1991). [CrossRef]

]. The eigenvalues Eh’s and the eigenstates |Ψh〉’s are obtained by diagonalizing the Hamiltonian matrix. The superscript h can be either c or v, representing the conduction or valence subband, respectively.

The energy dispersions of GNRs are symmetric with respect to kx = 0, and therefore only positive ones are discussed. For the AGNR with Ny = 69, many 1D parabolic subbands are symmetric about the Fermi level [Fig. 1(a)]. The bottoms and tops of these subbands are located at kx = 0, where many band-edge states exist and optical transitions may occur. The energy gap is equal to 0.05 γ0 for this semiconducting nanoribbon. The subband index Jc,v (= 1, 2, 3,...) is decided by the minimum energy spacing between the subbands and the Fermi level. For the ZGNR with Ny = 69, the parabolic subbands are also symmetric about EF = 0, and their band edges are located at kx = 2π/3 [Fig. 1(b)]. Furthermore, there are degenerate partial flat bands lying on the Fermi level due to the zigzag edge structure, and the dispersionless region extends from 2π/3 to π. Hence, the ZGNR is a gapless semiconductor.

3. Features of wavefunctions in the real space

Wavefunctions give information on the charge density distribution, and their features are very important in understanding optical selection rules. They are strongly dependent on the edge structure of nanoribbon. For AGNRs, the wavefunction is a combination of six subenvelope functions
|Ψh=m=1,2,3,...[A3mh|a3m+A3m+1h|a3m+1+A3m+2h|a3m+2+B3mh|b3m+B3m+1h|b3m+1+B3m+2h|b3m+2],
(4)
where A(B)3mhs, A(B)3m+1hs and A(B)3m+2hs, are the site amplitudes of sublattice A (B) located at the (3m)th, (3m+1)th and (3m+2)th armchair lines, with m being an integer. The wavefunctions of the Ny = 69 AGNR at kx = 0, where the optical transitions occur, are shown in Fig. 2. They have an oscillating pattern, and the number of nodes (zero points) is given by Jh − 1. The square of the wavefunction represents the charge density distribution. The local maxima and minima of the wavefunctions are of a high charge density and the nodes are zero. As the state energy |Eh| increases, the oscillations become more severe. More importantly, each of the wavefunctions exhibits two kinds of relations. For a certain wavefunction, there is a relation between the subenvelope functions of sublattices A and B, that is, the site amplitudes Amh(Jh) and Bmh(Jh) abide by the relations
Amh(Jh)=±Bmh(Jh).
(5)
For example, the site amplitudes of sublattice A of the Jc = 1 subband, Amc(Jc=1) [Figs. 2(m)–2(o)], is the negative of Bmc(Jc=1) [Figs. 2(p)–2(r)]. For the conduction and valence wave-functions of the same index, Jc = Jv = J, the subenvelope functions have the relations
Amc(J)=±Amv(J).
(6)
For instance, Amc(J=2)=Amv(J=2) [compare Figs. 2(g)–2(i) with Figs. 2(y)–2(aa)]. The relations between the subenvelope functions of the same subband index are either in phase or out of phase. It is marked that Eqs. (5) and (6) are in terms of the sublattices A and B. There is no relation between subenvelope functions on different armchair lines, for example, the relation between A3mh and B3m+1h.

Fig. 2 Wavefunctions of subband index Jc = 1–3 (red dots) and Jv = 1 and 2 (blue dots) at kx = 0 for Ny = 69 AGNR.

For ZGNRs, the wavefunction is decomposed as
|Ψh=odd[Aoh|ao+Boh|bo]+even[Aeh|ae+Beh|be],
(7)
where Aohs, Bohs, Aehs and Behs are the site amplitudes of sublattices A and B on odd and even zigzag lines, respectively. |ao〉’s, |bo〉’s, |ae〉’s, and |be〉’s are the corresponding tight-binding functions. The wavefunctions of the Ny = 69 ZGNR at kx = 2π/3, shown in Fig. 3, reveal profiles different from the armchair ones. The subenvelope functions of the conduction and valence subbands follow the relations:
Amc(Jc)=(1)Jc+1BNy+1mc(Jc),Amv(Jv)=(1)JvBNy+1mv(Jv),
(8)
where the indices (Ny + 1 − m) and m are correlated and the sign factor (−1)Jc+1 [(−1)Jv] describes whether the wavefunction is symmetric (positive sign) or anti-symmetric (negative sign). For example, the subband of Jc = 1 is symmetric [compare Figs. 3(i), 3(j) with Figs. 3(k), 3(l)] and anti-symmetric for Jv = 1 [compare Figs. 3(m), 3(n) with Figs. 3(o), 3(p)]. It is noteworthy that the symmetry property is associated with the subband index. For instance, considering the conduction subbands, the Jc = 1, 2, and 3 wavefunctions are respectively symmetric, anti-symmetric, and symmetric. In other words, the symmetry property changes alternatively from symmetry to anti-symmetry with increasing subband index. Moreover, the results of this study show that the characteristics of the subenvelope functions Amc(Jc) and Bmc(Jc) are very sensitive to whether Ny is odd or even. If Ny is even, the relations become
Amc(Jc)=(1)JcBNy+1mc(Jc),Amv(Jv)=(1)Jv+1BNy+1mv(Jv),
(9)
which differ from Eq. (8) by a minus sign. However, the alternatively changing symmetry property is retained whether Ny is odd or even.

Fig. 3 Wavefunctions of subband index Jc = 1–3 (red dots) and Jv = 1 and 2 (blue dots) at kx = 2π/3 for Ny = 69 ZGNR.

4. Absorption spectra

When a GNR is under the influence of an electromagnetic field with polarization ê||, the absorption spectrum A(ω) for direct photon transitions (Δkx = 0) from the initial state |Ψh(kx, Jh)〉 to the final state |Ψh′ (kx, Jh′)〉 is given by
A(ω)Jh,Jh1stB.Z.dkx2π|Ψh(kx,Jh)|e^pme|Ψh(kx,Jh)|2×Im{f[Eh(kx,Jh)]f[Eh(kx,Jh)]Eh(kx,Jh)Eh(kx,Jh)ωiΓ},
(10)
where me is the effective mass of an electron, p is the momentum operator, f[Eh(kx, Jh)] is the Fermi-Dirac distribution function, and Γ is the broadening factor. The low-energy absorption spectra for the Ny = 69 AGNR and ZGNR at zero temperature are illustrated in Fig. 4. The spectra show a lot of 1D peaks caused by the inter-band transitions of the 1D subbands, and they are greatly affected by the geometric structure.

Fig. 4 Absorption spectra for the Ny = 69 AGNR and ZGNR.

In the absorption spectrum of the Ny = 69 AGNR [Fig. 4(a)], the first peak, located at ω1 = 0.055 γ0, is identified as the excitation from the Jv = 1 to the Jc = 1 subband. The second (ω2 = 0.105 γ0) and the third (ω3 = 0.205 γ0) ones are the transitions from the Jv = 2 to the Jc = 2 subband and from the Jv = 3 to the Jc = 3 subband, respectively. The allowed transitions originate in the subbands of the same indices, i.e., the selection rule is ΔJ = Jc − Jv = 0. The transition energy is twice the band-edge energy ( ωJ=2EJcedge=2|EJvedge|, where Jc = Jv = J) because the band structure is symmetric about EF = 0. Moreover, the peak height increases when the frequency increases as a consequence of the decrease of the subbands’ curvatures.

5. Optical absorption selection rules of GNRs

Based on the above-discussed features of wavefunctions, the optical selection rules are also determined by the velocity matrix elements [the former term in Eq. (10)], which are defined as
Mhh(kx)Ψh(kx,Jh)|e^pme|Ψh(kx,Jh).
(11)

To evaluate Mhh′, the momentum operator is substituted in terms of the k-space derivative of the Hamiltonian [41

41. G. Dresselhaus and M. S. Dresselhaus, “Fourier expansion for electronic energy bands in silicon and germanium,” Phys. Rev. 160, 649–679 (1967). [CrossRef]

44

44. L. C. Lew Yan Voon and L. R. Ram-Mohan, “Tight-binding representation of the optical matrix-elements - theory and applications,” Phys. Rev. B 47, 15500–15508 (1993). [CrossRef]

]:
p(k)=meh¯kH(k).
(12)
This substitution is valid independent of the band-structure model. Thus, it is equally applicable and straightforward to use for any empirical approach for the electronic dispersion relation [44

44. L. C. Lew Yan Voon and L. R. Ram-Mohan, “Tight-binding representation of the optical matrix-elements - theory and applications,” Phys. Rev. B 47, 15500–15508 (1993). [CrossRef]

], so that
Mhh(kx)=1h¯l,l=12NyClh*(kx)Clh(kx)Hl,l(kx)kx,
(13)
where is treated as unity in the following derivation.

At zero temperature, only the inter-band transitions from the valence to the conduction sub-bands are valid. To investigate the optical properties, we focus on kx = 0, where the excitations occur for AGNRs. Substituting the coefficients of sublattices B with A in Eq. (13) by the relations Bmv=sAmv(s=±1) and Bmc=tAmc(t=±1) from Eq. (5), Mvc is
bγ0m=1Ny[sAmc*(Jc)Amv(Jv)+tAmc*(Jc)Amv(Jv)]+bγ02m=1Ny1[tAmc*(Jc)Am+1v(Jv)+sAmc*(Jc)Am+1v(Jv)]+bγ02m=2Ny[tAmc*(Jc)Am1v(Jv)+sAmc*(Jc)Am1v(Jv)].
(14)
The transfer relations between Am±1 and Am, obtained from H|Ψ〉 = E|Ψ〉 at kx = 0, are
{A2=Δ±A1,Am1+Am+1=Δ±Am,ifm=2,3,4,,Ny1,ANy1=Δ±ANy,
(15)
where Δ± = ±(E/γ0) − 1 for Am = ±Bm. After using Eq. (15) to replace Am±1 with Am, Mvc becomes
bγ0m=1Ny[sAmc*(Jc)Amv(Jv)+tAmc*(Jc)Amv(Jv)Δs2tAmc*(Jc)Amv(Jv)+Δs2sAmc*(Jc)Amv(Jv)]=(ts)(1Δs2)bγ0m=1NyAmc*(Jc)Amv(Jv),
(16)
where Δs = s(E/γ0) − 1 for Amc=sBmc. Through the relation between the conduction and valence subbands Amv=uAmc(u=±1) from Eq. (6) and the normalization of the wavefunction 2m=1NyAmc*(Jc)Amc(Jv)=δJc,Jv, the velocity matrix element at kx = 0 is
Mvc(kx=0)=u(ts)(1Δs2)bγ012δJc,Jv=±(1Δ±2)bγ0δJc,Jv,
where Δ± = ±(E/γ0) − 1 and u(t − s) is either 0 or ±2. Eq. (17) yields the selection rule (ΔJ = 0) for AGNRs and possible excitation channels in the absorption spectra.

For ZGNRs, the velocity matrix element in Eq. (13) at kx = 2π/3, where the transition occur, becomes
hm=1Ny[Amc*(Jc)Bmv(Jv)+Bmc*(Jc)Amv(Jv)],
(17)
where h′ = −0 sin(πa/3). For an odd Ny, we use Eq. (8) to convert the second term in Eq. (17), which leads to
hm=1Ny[Amc*(Jc)Bmv(Jv)+(1)Jc+1ANy+1mc*(Jc)(1)JvBNy+1mv(Jv)].
(18)
Then renaming the index of Ny + 1 − m as m gives
[1+(1)Jc+Jv+1]hm=1NyAmc*(Jc)Bmv(Jv).
(19)
As for an even Ny, the second term in Eq. (18) can also be converted by Eq. (9), and yields the same result as Eq. (19). Hence, the Mvc for ZGNR at kx = 2π/3 is
Mvc(kx=2π3)={2hm=1NyAmc*(Jc)Bmv(Jv),ifΔJ=odd,0,ifΔJ=even,
(20)
where m=1NyAmc*(Jc)Bmv(Jv) has a non-zero value, and the selection rule (ΔJ = odd) for ZGNRs is obtained.

By using the selection rules, the transition energies (peak positions) can be efficiently obtained from the energy dispersion. For AGNRs, the dependence of the first five predicted transition energies on the ribbon width is shown in Fig. 5(a). For a certain width Ny, the Jth transition energy is equal to ωJ=2EJcedge(Jc=J), i.e., twice of the band-edge state energy of the Jcth subband. For narrower GNRs, there are three groups of transition energies. The ωJ’s of Ny = 3m (full circles) and Ny = 3m + 1 (open circles) groups are very similar, while for the Ny = 3m + 2 (squares) group, the first and second ωJ’s lie close to each other and are sandwiched between the second and third ωJ’s of the Ny ≠ 3m + 2 nanoribbons. For wider GNRs, these transition energies make a red-shift and merge. The GNRs may therefore be sorted into the two groups Ny = 3m + 2 and Ny ≠ 3m + 2. The former is gapless due to the linear bands crossing at the Fermi level, while the latter is semiconducting with a band gap energy corresponding to the first transition energy.

Fig. 5 The first five consecutive transition energies with respect to the ribbon width Ny for AGNR and ZGNR. The notations •, ○ and ▪ correspond to AGNRs of Ny = 3m, 3m + 1, and 3m + 2, respectively.

The first five ribbon-width-dependent transition energies of ZGNRs are presented in Fig. 5(b). They are associated with the first principal peak ω1P, the first subpeak ω14, the second principal peak ω2P, the second subpeak ω16, and the third principal peak ω3P, respectively [see Fig. 4(b)]. The predicted corresponding transition energies are ω1P=EJc=2edge, ω14=EJc=4edge, ω2P=EJc=3edgeEJv=2edge, ω16=EJc=6edge, and ω3P=EJc=4edgeEJv=3edge. The first subpeak (ω14) and the second principal peak ( ω2P) are close to each other and merge for a sufficiently large width, as to the second subpeak (ω16) and the third principal peak ( ω3P). In short, the peak frequencies can be predicted by a combination of the band structures and selection rules. This is an efficient way to obtain wide-ranging information on the transition energies without extrapolation.

It is worth mentioning that the exact energies of the band-edge states for ZGNRs can be specified by the experimentally measured transition energy ωexp. According to the optical selection rule, the measured peak positions ω1P,exp, ω14exp, and ω16exp can be applied to specify the band-edge states EJc=2edge, EJc=4edge, and EJc=6edge, respectively, which are the energy differences between the Jc = even subband edges and the Fermi level. Moreover, the band-edge state energies of the Jc = odd subbands are also obtainable, for example, EJc=3edge=ω2P,exp|EJv=2edge|=ω2P,expω1P,exp. A similar backtracking strategy is valid for AGNRs based on the band symmetry about the Fermi energy. The band-edge state energy, which is half of the experimentally measured peak position, is given by EJcedge=|EJvedge|=ωJexp/2, where Jc = Jv = J. Hence, the selection rule provides a way to overcome the disadvantages of optical measurements, which only yield information about the energy difference between two subbands.

6. Conclusion

We employ the tight-binding model to study the absorption spectra of GNRs. To our knowledge, this is the first time that the corresponding optical selection rules are analytically specified. The main results of this work are summarized as follows. First, the optical transition channels of absorption spectra for AGNRs and ZGNRs are exactly identified. Then, the characteristics of the absorption peaks, such as positions and heights relating to the energy dispersions, are discussed in detail. Most importantly, this work provides a theoretical explanation for the optical selection rules through analysis of the velocity matrix elements and the wavefunction features. The optical selection rules depend on the edge structure, i.e., armchair or zigzag edges. The selection rule of AGNRs (ΔJ = 0) has its origin in the relation between sublattices A and B within a certain state and the relation between the conduction and valence subenvelope functions of the same index, whereas the selection rule for ZGNRs (|ΔJ| = odd) is derived from the alternatively changing symmetry property with an increasing index. The quite different features of the wavefunctions for AGNRs and ZGNRs result in different selection rules. In addition, according to the selection rules, we can efficiently predict the peak positions and identify the corresponding transition channels. This gives us a way to overcome the limitations of optical measurements, and allows us to obtain exact energies of band-edge states.

Acknowledgments

One of us (H. C. Chung) wishes to thank M. H. Chung and S. M. Chen for financial support. This work was supported in part by the National Science Council of Taiwan under grant numbers NSC 99-2112-M-165-001-MY3 and 98-2112-M-006-013-MY4 and the National Center for Theoretical Sciences south (NCTS south).

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28.

T. Nomura, D. Yamamoto, and S. Kurihara, “Electric field effects in zigzag edged graphene nanoribbons,” J. Phys.: Conf. Ser. 200, 062015 (2010). [CrossRef]

29.

J. W. Bai, R. Cheng, F. X. Xiu, L. Liao, M. S. Wang, A. Shailos, K. L. Wang, Y. Huang, and X. F. Duan, “Very large magnetoresistance in graphene nanoribbons,” Nat. Nanotechnol. 5, 655–659 (2010). [CrossRef] [PubMed]

30.

H. C. Chung, M. H. Lee, C. P. Chang, Y. C. Huang, and M. F. Lin, “Effects of transverse electric fields on quasi-landau levels in zigzag graphene nanoribbons,” J. Phys. Soc. Jpn. 80, 044602 (2011). [CrossRef]

31.

A. Cresti and S. Roche, “Range and correlation effects in edge disordered graphene nanoribbons,” New. J. Phys. 11, 095004 (2009). [CrossRef]

32.

Y. O. Klymenko and O. Shevtsov, “Low-energy electron transport in semimetal graphene ribbon junctions,” Eur. Phys. J. B 72, 203–209 (2009). [CrossRef]

33.

E. Perfetto, G. Stefanucci, and M. Cini, “Time-dependent transport in graphene nanoribbons,” Phys. Rev. B 82, 035446 (2010). [CrossRef]

34.

J. Jiang, W. Lu, and J. Bernholc, “Edge states and optical transition energies in carbon nanoribbons,” Phys. Rev. Lett. 101, 246803 (2008). [CrossRef] [PubMed]

35.

M. F. Lin and F. L. Shyu, “Optical properties of nanographite ribbons,” J. Phys. Soc. Jpn. 69, 3529–3532 (2000). [CrossRef]

36.

H. Hsu and L. E. Reichl, “Selection rule for the optical absorption of graphene nanoribbons,” Phys. Rev. B 76, 045418 (2007). [CrossRef]

37.

C. W. Chiu, S. H. Lee, S. C. Chen, F. L. Shyu, and M. F. Lin, “Absorption spectra of aa-stacked graphite,” New. J. Phys. 12, 083060 (2010). [CrossRef]

38.

L. Van Hove, “The occurrence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev. 89, 1189 (1953). [CrossRef]

39.

E. B. Barros, A. Jorio, G. G. Samsonidze, R. B. Capaz, A. G. Souza, J. Mendes, G. Dresselhaus, and M. S. Dresselhaus, “Review on the symmetry-related properties of carbon nanotubes,” Phys. Rep. 431, 261–302 (2006). [CrossRef]

40.

J. C. Charlier, X. Gonze, and J. P. Michenaud, “First-principles study of the electronic properties of graphite,” Phys. Rev. B 43, 4579–4589 (1991). [CrossRef]

41.

G. Dresselhaus and M. S. Dresselhaus, “Fourier expansion for electronic energy bands in silicon and germanium,” Phys. Rev. 160, 649–679 (1967). [CrossRef]

42.

L. G. Johnson and G. Dresselhaus, “Optical properies of graphite,” Phys. Rev. B 7, 2275–2285 (1973). [CrossRef]

43.

N. V. Smith, “Photoemission spectra and band structures of d-band metals .7. extensions of the combined interpolation scheme,” Phys. Rev. B 19, 5019–5027 (1979). [CrossRef]

44.

L. C. Lew Yan Voon and L. R. Ram-Mohan, “Tight-binding representation of the optical matrix-elements - theory and applications,” Phys. Rev. B 47, 15500–15508 (1993). [CrossRef]

OCIS Codes
(160.4760) Materials : Optical properties
(300.1030) Spectroscopy : Absorption
(300.6170) Spectroscopy : Spectra
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Materials

History
Original Manuscript: August 22, 2011
Revised Manuscript: October 8, 2011
Manuscript Accepted: October 8, 2011
Published: November 1, 2011

Citation
H. C. Chung, M. H. Lee, C. P. Chang, and M. F. Lin, "Exploration of edge-dependent optical selection rules for graphene nanoribbons," Opt. Express 19, 23350-23363 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23350


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  30. H. C. Chung, M. H. Lee, C. P. Chang, Y. C. Huang, and M. F. Lin, “Effects of transverse electric fields on quasi-landau levels in zigzag graphene nanoribbons,” J. Phys. Soc. Jpn.80, 044602 (2011). [CrossRef]
  31. A. Cresti and S. Roche, “Range and correlation effects in edge disordered graphene nanoribbons,” New. J. Phys.11, 095004 (2009). [CrossRef]
  32. Y. O. Klymenko and O. Shevtsov, “Low-energy electron transport in semimetal graphene ribbon junctions,” Eur. Phys. J. B72, 203–209 (2009). [CrossRef]
  33. E. Perfetto, G. Stefanucci, and M. Cini, “Time-dependent transport in graphene nanoribbons,” Phys. Rev. B82, 035446 (2010). [CrossRef]
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  35. M. F. Lin and F. L. Shyu, “Optical properties of nanographite ribbons,” J. Phys. Soc. Jpn.69, 3529–3532 (2000). [CrossRef]
  36. H. Hsu and L. E. Reichl, “Selection rule for the optical absorption of graphene nanoribbons,” Phys. Rev. B76, 045418 (2007). [CrossRef]
  37. C. W. Chiu, S. H. Lee, S. C. Chen, F. L. Shyu, and M. F. Lin, “Absorption spectra of aa-stacked graphite,” New. J. Phys.12, 083060 (2010). [CrossRef]
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  42. L. G. Johnson and G. Dresselhaus, “Optical properies of graphite,” Phys. Rev. B7, 2275–2285 (1973). [CrossRef]
  43. N. V. Smith, “Photoemission spectra and band structures of d-band metals .7. extensions of the combined interpolation scheme,” Phys. Rev. B19, 5019–5027 (1979). [CrossRef]
  44. L. C. Lew Yan Voon and L. R. Ram-Mohan, “Tight-binding representation of the optical matrix-elements - theory and applications,” Phys. Rev. B47, 15500–15508 (1993). [CrossRef]

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