Exploration of edge-dependent optical selection rules for graphene nanoribbons

doi:10.1364/OE.19.023350

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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 = J^c – J^v = 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.

1. Introduction

The discovery of graphene [1

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

, 2

K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. U. S. A. 102, 10451–10453 (2005). [CrossRef] [PubMed]

] has motivated many studies because of its interesting physical properties and possible applications. Graphene, a two-dimensional carbon-based material [3

R. F. Service, “Materials science carbon sheets an atom thick give rise to graphene dreams,” Science 324, 875–877 (2009). [CrossRef] [PubMed]

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M. I. Katsnelson, “Graphene: carbon in two dimensions,” Mater. Today 10, 20–27 (2007). [CrossRef]

], is a gapless semiconductor with linear energy dispersion near the Dirac point [6

A. K. Geim, “Graphene: Status and prospects,” Science 324, 1530–1534 (2009). [CrossRef] [PubMed]

, 7

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007). [CrossRef] [PubMed]

]. Its cone-like energy spectrum results in particular electronic and transport properties, such as massless Dirac fermions which travel at a extremely high speed [8

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, “Giant intrinsic carrier mobilities in graphene and its bilayer,” Phys. Rev. Lett. 100, 016602 (2008). [CrossRef] [PubMed]

F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, “Detection of individual gas molecules adsorbed on graphene,” Nat. Mater. 6, 652–655 (2007). [CrossRef] [PubMed]

], finite conductivity at zero charge-carrier concentration [10

S. V. Morozov, K. S. Novoselov, and A. K. Geim, “Electron transport in graphene,” Phys. Usp. 51, 744–748 (2008). [CrossRef]

,11

S. Cho and M. S. Fuhrer, “Charge transport and inhomogeneity near the minimum conductivity point in graphene,” Phys. Rev. B 77, 081402 (2008). [CrossRef]

], and an unusual quantum Hall effect [12

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005). [CrossRef] [PubMed]

,13

Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum hall effect and berry’s phase in graphene,” Nature 438, 201–204 (2005). [CrossRef] [PubMed]

]. However, the gapless band structure of graphene limits its applications in electronic devices. In order to induce a tunable band gap for a wide variety of applications, one-dimensional (1D) graphene nanoribbons (GNRs) have been proposed, which can be fabricated using a nanowire etch mask [14

J. W. Bai, X. F. Duan, and Y. Huang, “Rational fabrication of graphene nanoribbons using a nanowire etch mask,” Nano Lett. 9, 2083–2087 (2009). [CrossRef] [PubMed]

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A. Fasoli, A. Colli, A. Lombardo, and A. C. Ferrari, “Fabrication of graphene nanoribbons via nanowire lithography,” Phys. Status Solidi B-Basic Solid State Phys. 246, 2514–2517 (2009). [CrossRef]

], chemical vapor deposition [16

V. L. J. Joly, M. Kiguchi, S. J. Hao, K. Takai, T. Enoki, R. Sumii, K. Amemiya, H. Muramatsu, T. Hayashi, Y. A. Kim, M. Endo, J. Campos-Delgado, F. Lopez-Urias, A. Botello-Mendez, H. Terrones, M. Terrones, and M. S. Dresselhaus, “Observation of magnetic edge state in graphene nanoribbons,” Phys. Rev. B 81, 245428 (2010). [CrossRef]

,17

J. Campos-Delgado, Y. A. Kim, T. Hayashi, A. Morelos-Gomez, M. Hofmann, H. Muramatsu, M. Endo, H. Terrones, R. D. Shull, M. S. Dresselhaus, and M. Terrones, “Thermal stability studies of cvd-grown graphene nanoribbons: Defect annealing and loop formation,” Chem. Phys. Lett. 469, 177–182 (2009). [CrossRef]

], lithographic patterning of graphene [18

L. Tapaszto, G. Dobrik, P. Lambin, and L. P. Biro, “Tailoring the atomic structure of graphene nanoribbons by scanning tunnelling microscope lithography,” Nat. Nanotechnol. 3, 397–401 (2008). [CrossRef] [PubMed]

–20

C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science 312, 1191–1196 (2006). [CrossRef] [PubMed]

], and unzipping of nanotube [21

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature 458, 872–876 (2009). [CrossRef] [PubMed]

–24

A. G. Cano-Marquez, F. J. Rodriguez-Macias, J. Campos-Delgado, C. G. Espinosa-Gonzalez, F. Tristan-Lopez, D. Ramirez-Gonzalez, D. A. Cullen, D. J. Smith, M. Terrones, and Y. I. Vega-Cantu, “Ex-mwnts: Graphene sheets and ribbons produced by lithium intercalation and exfoliation of carbon nanotubes,” Nano Lett. 9, 1527–1533 (2009). [CrossRef] [PubMed]

]. The electronic and transport properties are highly related to the edge structure and ribbon width. Two basic types of GNRs exist, namely, an armchair GNR (AGNR) and a zigzag GNR (ZGNR), which are classified based on their edge structures along the longitudinal direction. A ZGNR is gapless because of the peculiar partial flat bands at the Fermi level (E_F = 0), while an AGNR can be either semiconducting or gapless depending on the ribbon width. Owing to these special electronic properties, GNRs have triggered many studies, e.g., studies of their geometrical and energy band structures [25

R. H. Miwa, R. G. A. Veiga, and G. P. Srivastava, “Structural, electronic, and magnetic properties of pristine and oxygen-adsorbed graphene nanoribbons,” Appl. Surf. Sci. 256, 5776–5782 (2010). [CrossRef]

, 26

S. Dutta and S. K. Pati, “Novel properties of graphene nanoribbons: a review,” J. Mater. Chem. 20, 8207–8223 (2010). [CrossRef]

], as well as their magnetic [16

, 27

H. C. Chung, Y. C. Huang, M. H. Lee, C. C. Chang, and M. F. Lin, “Quasi-landau levels in bilayer zigzag graphene nanoribbons,” Physica E 42, 711–714 (2010). [CrossRef]

–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]

], transport [31

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

–33

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

], and optical properties [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]

–36

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

Optical measurements are very useful in understanding the electronic structure of a material. Unlike the featureless absorption spectrum of graphene at low energy [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]

], the optical spectra of GNRs exhibit many sharp peaks related to van Hove singularities [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]

] in the density of electronic states. These peaks satisfy certain selection rules, which depend on the edge structure. Absorption selection rules of GNRs, which decide the possible transitions, have been studied by Lin et. al. [35

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

] and Hsu et. al. [36

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

]. However, Hsu et. al. only demonstrated the zigzag case and assumed that the momentum matrix elements involved are constant throughout the k-space. Lin et. al. described the selection rules, but have not verified their origin.

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 (N_y) of the dimer lines (armchair or zigzag lines) along the longitudinal direction (x̂). The periodic length is

I_{x} = 3 b (I_{x} = \sqrt{3} b = a)

for AGNRs (ZGNRs). The first Brillouin zone is confined to the region −π/I_x ≤ k_x ≤ π/I_x with 2N_y 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 = \sum_{l, l^{'}} γ_{0} c_{l}^{†} c_{l^{'}}

, where γ₀ is the hopping integral,

c_{l}^{†} (c_{l^{'}})

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 2N_y tight-binding functions |ϕ_l〉’s with the corresponding site amplitudes C_l’s, is expressed as

| Ψ 〉 = \sum_{l = 1}^{2 N_{y}} C_{l} | ϕ_{l} 〉 or | Ψ 〉 = \sum_{m = 1}^{N_{y}} A_{m} | a_{m} 〉 + \sum_{m = 1}^{N_{y}} B_{m} | b_{m} 〉,

(1)

where A_m (B_m) is the site amplitude and |a_m〉 (|b_m〉) is the tight-binding function associated with the periodic A (B) atoms.

Fig. 1 Energy dispersions and geometric structures for N_y = 69 AGNR and ZGNR. The conduction (valence) subbands of the corresponding indices J^c (J^v) 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 2N_y × 2N_y Hermitian matrix. For AGNRs, the upper triangular elements of H are

H_{l, l^{'}} = {\begin{array}{l} γ_{0} exp [i k_{x} (- b)], & if l^{'} = l + 1, & l is odd, \\ γ_{0} exp [i k_{x} (- b / 2)], & if l^{'} = l + 1, & l is even, \\ γ_{0} exp [i k_{x} (b / 2)], & if l^{'} = l + 3, & l is odd, \\ 0, & otherwise . \end{array}

(2)

For ZGNRs, the Hamiltonian elements are

H_{l, l^{'}} = {\begin{array}{l} 2 γ_{0} cos (k_{x} a / 2), & if l^{'} = l + 1, & l is odd, \\ γ_{0}, & if l^{'} = l + 1, & l is even, \\ 0, & otherwise, \end{array}

(3)

where γ₀ = 2.598 eV [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 E^h’s and the eigenstates |Ψ^h〉’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 k_x = 0, and therefore only positive ones are discussed. For the AGNR with N_y = 69, many 1D parabolic subbands are symmetric about the Fermi level [Fig. 1(a)]. The bottoms and tops of these subbands are located at k_x = 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 J^c,v (= 1, 2, 3,...) is decided by the minimum energy spacing between the subbands and the Fermi level. For the ZGNR with N_y = 69, the parabolic subbands are also symmetric about E_F = 0, and their band edges are located at k_x = 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

\begin{array}{l} | Ψ^{h} 〉 = \sum_{m = 1, 2, 3, ...} [A_{3 m}^{h} | a_{3 m} 〉 + A_{3 m + 1}^{h} | a_{3 m + 1} 〉 + A_{3 m + 2}^{h} | a_{3 m + 2} 〉 \\ + B_{3 m}^{h} | b_{3 m} 〉 + B_{3 m + 1}^{h} | b_{3 m + 1} 〉 + B_{3 m + 2}^{h} | b_{3 m + 2} 〉], \end{array}

(4)

where

A {(B)}_{3 m}^{h} ’ s

A {(B)}_{3 m + 1}^{h} ’ s

and

A {(B)}_{3 m + 2}^{h} ’ s

, 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 N_y = 69 AGNR at k_x = 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 J^h − 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 |E^h| 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

A_{m}^{h} (J^{h})

and

B_{m}^{h} (J^{h})

abide by the relations

A_{m}^{h} (J^{h}) = \pm B_{m}^{h} (J^{h}) .

(5)

For example, the site amplitudes of sublattice A of the J^c = 1 subband,

A_{m}^{c} (J^{c} = 1)

[Figs. 2(m)–2(o)], is the negative of

B_{m}^{c} (J^{c} = 1)

[Figs. 2(p)–2(r)]. For the conduction and valence wave-functions of the same index, J^c = J^v = J, the subenvelope functions have the relations

A_{m}^{c} (J) = \pm A_{m}^{v} (J) .

(6)

For instance,

A_{m}^{c} (J = 2) = - A_{m}^{v} (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

A_{3 m}^{h}

and

B_{3 m + 1}^{h}

Fig. 2 Wavefunctions of subband index J^c = 1–3 (red dots) and J^v = 1 and 2 (blue dots) at k_x = 0 for N_y = 69 AGNR.

For ZGNRs, the wavefunction is decomposed as

| Ψ^{h} 〉 = \sum_{odd} [A_{o}^{h} | a_{o} 〉 + B_{o}^{h} | b_{o} 〉] + \sum_{even} [A_{e}^{h} | a_{e} 〉 + B_{e}^{h} | b_{e} 〉],

(7)

where

A_{o}^{h} ’ s

B_{o}^{h} ’ s

A_{e}^{h} ’ s

and

B_{e}^{h} ’ s

are the site amplitudes of sublattices A and B on odd and even zigzag lines, respectively. |a_o〉’s, |b_o〉’s, |a_e〉’s, and |b_e〉’s are the corresponding tight-binding functions. The wavefunctions of the N_y = 69 ZGNR at k_x = 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:

\begin{array}{l} A_{m}^{c} (J^{c}) & = & {(- 1)}^{J^{c} + 1} B_{N_{y} + 1 - m}^{c} (J^{c}), \\ A_{m}^{v} (J^{v}) & = & {(- 1)}^{J^{v}} B_{N_{y} + 1 - m}^{v} (J^{v}), \end{array}

(8)

where the indices (N_y + 1 − m) and m are correlated and the sign factor (−1)^{J^c+1} [(−1)^{J^v}] describes whether the wavefunction is symmetric (positive sign) or anti-symmetric (negative sign). For example, the subband of J^c = 1 is symmetric [compare Figs. 3(i), 3(j) with Figs. 3(k), 3(l)] and anti-symmetric for J^v = 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 J^c = 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

A_{m}^{c} (J^{c})

and

B_{m}^{c} (J^{c})

are very sensitive to whether N_y is odd or even. If N_y is even, the relations become

\begin{array}{l} A_{m}^{c} (J^{c}) & = & {(- 1)}^{J^{c}} B_{N_{y} + 1 - m}^{c} (J^{c}), \\ A_{m}^{v} (J^{v}) & = & {(- 1)}^{J^{v} + 1} B_{N_{y} + 1 - m}^{v} (J^{v}), \end{array}

(9)

which differ from Eq. (8) by a minus sign. However, the alternatively changing symmetry property is retained whether N_y is odd or even.

Fig. 3 Wavefunctions of subband index J^c = 1–3 (red dots) and J^v = 1 and 2 (blue dots) at k_x = 2π/3 for N_y = 69 ZGNR.

4. Absorption spectra

When a GNR is under the influence of an electromagnetic field with polarization ê||x̂, the absorption spectrum A(ω) for direct photon transitions (Δk_x = 0) from the initial state |Ψ^h(k_x, J^h)〉 to the final state |Ψ^h′ (k_x, J^h′)〉 is given by

\begin{array}{l} A (ω) \propto \sum_{J^{h}, J^{h^{'}}} \int_{1 stB . Z .} \frac{d k_{x}}{2 π} {| 〈 Ψ^{h^{'}} (k_{x}, J^{h^{'}}) | \frac{\hat{e} \cdot p}{m_{e}} | Ψ^{h} (k_{x}, J^{h}) 〉 |}^{2} \\ \times Im {\frac{f [E^{h} (k_{x}, J^{h})] - f [E^{h^{'}} (k_{x}, J^{h^{'}})]}{E^{h^{'}} (k_{x}, J^{h^{'}}) - E^{h} (k_{x}, J^{h}) - ω - i Γ}}, \end{array}

(10)

where m_e is the effective mass of an electron, p is the momentum operator, f[E^h(k_x, J^h)] is the Fermi-Dirac distribution function, and Γ is the broadening factor. The low-energy absorption spectra for the N_y = 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 N_y = 69 AGNR and ZGNR.

In the absorption spectrum of the N_y = 69 AGNR [Fig. 4(a)], the first peak, located at ω₁ = 0.055 γ₀, is identified as the excitation from the J^v = 1 to the J^c = 1 subband. The second (ω₂ = 0.105 γ₀) and the third (ω₃ = 0.205 γ₀) ones are the transitions from the J^v = 2 to the J^c = 2 subband and from the J^v = 3 to the J^c = 3 subband, respectively. The allowed transitions originate in the subbands of the same indices, i.e., the selection rule is ΔJ = J^c − J^v = 0. The transition energy is twice the band-edge energy (

ω_{J} = 2 E_{J^{c}}^{edge} = 2 | E_{J^{v}}^{edge} |

, where J^c = J^v = J) because the band structure is symmetric about E_F = 0. Moreover, the peak height increases when the frequency increases as a consequence of the decrease of the subbands’ curvatures.

For ZGNRs, the absorption peaks may be classified into principal peaks and subpeaks based on their heights, as shown in Fig. 4(b). The allowed transitions obey the selection rule |ΔJ| = odd, which is different from the armchair case, ΔJ = 0. The principal peaks become higher with increasing frequency because of the decrease of the bands’ curvatures and the increase of the excitation channels. Due to the symmetry of the energy spectrum, the absorption spectrum of negative ΔJ’s is the same as that of positive ones. Therefore, only the positive ΔJ cases (transitions from the J^v to the J^c = J^v + ΔJ subband of energy

ω = E_{J^{v} + Δ J}^{edge} - E_{J^{v}}^{edge}

) are discussed. The principal peaks,

ω_{J}^{P}

, are contributed by transitions between subbands satisfying the selection rule ΔJ = 1 (J^v → J^c : 1 → 2), ΔJ = 1 (J^v → J^c : 2 → 3), ΔJ = 1,3 (J^v → J^c : 3 → 4, 2 → 5), ΔJ = 1,3,5 (J^v → J^c : 4 → 5, 3 → 6, 2 → 7), and so on. The corresponding numbers of excitation channels are 2, 2, 4, and 6, i.e., double of the allowed positive ΔJ number. On the other hand, there exists one subpeak between two adjacent principal peaks. These subpeaks are indicated by ω_1J, where the subscript 1J denotes the transition from the J^v = 1 valence subband to the J^c conduction subband. The subpeak ω₁₄, for instance, results from the transition between the J^v = 1 and the J^c = 4 subband and follows the optical selection rule ΔJ = 3. Since each subpeak possesses two excitation channels, the increase of subpeak height is not as rapid as the height increase of principal peaks. It is remarkable that only the first principal peak (

ω_{1}^{P}

) and all the subpeaks (ω_1J) are derived from the transitions related to the J^c,v = 1 subbands (E_J^c,v=1 = E_F = 0). The

ω_{1}^{P}

peak is located at

E_{J^{c} = 2}^{edge}

, and the subpeaks ω₁_J are positioned at

E_{J^{c} = J}^{edge}

, which is the energy difference between the subband edge and the Fermi level.

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

M^{h h^{'}} (k_{x}) \equiv 〈 Ψ^{h^{'}} (k_{x}, J^{h^{'}}) | \frac{\hat{e} \cdot p}{m_{e}} | Ψ^{h} (k_{x}, J^{h}) 〉 .

(11)

To evaluate M^hh′, the momentum operator is substituted in terms of the k-space derivative of the Hamiltonian [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

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) = \frac{m_{e}}{\bar{h}} \nabla_{k} H (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

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

M^{h h^{'}} (k_{x}) = \frac{1}{\bar{h}} \sum_{l, l^{'} = 1}^{2 N_{y}} C_{l}^{h^{'} *} (k_{x}) C_{l^{'}}^{h} (k_{x}) \frac{\partial H_{l, l^{'}} (k_{x})}{\partial k_{x}},

(13)

where h̄ 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 k_x = 0, where the excitations occur for AGNRs. Substituting the coefficients of sublattices B with A in Eq. (13) by the relations

B_{m}^{v} = s A_{m}^{v} (s = \pm 1)

and

B_{m}^{c} = t A_{m}^{c} (t = \pm 1)

from Eq. (5), M^vc is

\begin{array}{l} b γ_{0} \sum_{m = 1}^{N_{y}} [- s A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v}) + t A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v})] \\ + \frac{b γ_{0}}{2} \sum_{m = 1}^{N_{y} - 1} [- t A_{m}^{c *} (J^{c}) A_{m + 1}^{v} (J^{v}) + s A_{m}^{c *} (J^{c}) A_{m + 1}^{v} (J^{v})] \\ + \frac{b γ_{0}}{2} \sum_{m = 2}^{N_{y}} [- t A_{m}^{c *} (J^{c}) A_{m - 1}^{v} (J^{v}) + s A_{m}^{c *} (J^{c}) A_{m - 1}^{v} (J^{v})] . \end{array}

(14)

The transfer relations between A_m_±1 and A_m, obtained from H|Ψ〉 = E|Ψ〉 at k_x = 0, are

{\begin{array}{l} A_{2} = Δ_{\pm} A_{1}, \\ A_{m - 1} + A_{m + 1} = Δ_{\pm} A_{m}, & if m = 2, 3, 4, \dots, N_{y} - 1, \\ A_{N_{y} - 1} = Δ_{\pm} A_{N_{y}}, \end{array}

(15)

where Δ_± = ±(E/γ₀) − 1 for A_m = ±B_m. After using Eq. (15) to replace A_m_±1 with A_m, M^vc becomes

\begin{array}{l} b γ_{0} \sum_{m = 1}^{N_{y}} [- s A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v}) + t A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v}) - \frac{Δ_{s}}{2} t A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v}) + \frac{Δ_{s}}{2} s A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v})] \\ = (t - s) (1 - \frac{Δ_{s}}{2}) b γ_{0} \sum_{m = 1}^{N_{y}} A_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v}), \end{array}

(16)

where Δ_s = s(E/γ₀) − 1 for

A_{m}^{c} = s B_{m}^{c}

. Through the relation between the conduction and valence subbands

A_{m}^{v} = u A_{m}^{c} (u = \pm 1)

from Eq. (6) and the normalization of the wavefunction

2 \sum_{m = 1}^{N_{y}} A_{m}^{c *} (J^{c}) A_{m}^{c} (J^{v}) = δ_{J^{c}, J^{v}}

, the velocity matrix element at k_x = 0 is

M^{v c} (k_{x} = 0) = u (t - s) (1 - \frac{Δ_{s}}{2}) b γ_{0} \frac{1}{2} δ_{J^{c}, J^{v}} = \pm (1 - \frac{Δ_{\pm}}{2}) b γ_{0} δ_{J^{c}, J^{v}},

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 k_x = 2π/3, where the transition occur, becomes

h^{'} \sum_{m = 1}^{N_{y}} [A_{m}^{c *} (J^{c}) B_{m}^{v} (J^{v}) + B_{m}^{c *} (J^{c}) A_{m}^{v} (J^{v})],

(17)

where h′ = −aγ₀ sin(πa/3). For an odd N_y, we use Eq. (8) to convert the second term in Eq. (17), which leads to

h^{'} \sum_{m = 1}^{N_{y}} [A_{m}^{c *} (J^{c}) B_{m}^{v} (J^{v}) + {(- 1)}^{J^{c} + 1} A_{N_{y} + 1 - m}^{c *} (J^{c}) {(- 1)}^{J^{v}} B_{N_{y} + 1 - m}^{v} (J^{v})] .

(18)

Then renaming the index of N_y + 1 − m as m gives

[1 + {(- 1)}^{J^{c} + J^{v} + 1}] h^{'} \sum_{m = 1}^{N_{y}} A_{m}^{c *} (J^{c}) B_{m}^{v} (J^{v}) .

(19)

As for an even N_y, the second term in Eq. (18) can also be converted by Eq. (9), and yields the same result as Eq. (19). Hence, the M^vc for ZGNR at k_x = 2π/3 is

M^{v c} (k_{x} = \frac{2 π}{3}) = {\begin{array}{l} 2 h^{'} \sum_{m = 1}^{N_{y}} A_{m}^{c *} (J^{c}) B_{m}^{v} (J^{v}), & if Δ J = odd, \\ 0, & if Δ J = even, \end{array}

(20)

where

\sum_{m = 1}^{N_{y}} A_{m}^{c *} (J^{c}) B_{m}^{v} (J^{v})

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 N_y, the Jth transition energy is equal to

ω_{J} = 2 E_{J^{c}}^{edge} (J^{c} = J)

, i.e., twice of the band-edge state energy of the J^cth subband. For narrower GNRs, there are three groups of transition energies. The ω_J’s of N_y = 3m (full circles) and N_y = 3m + 1 (open circles) groups are very similar, while for the N_y = 3m + 2 (squares) group, the first and second ω_J’s lie close to each other and are sandwiched between the second and third ω_J’s of the N_y ≠ 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 N_y = 3m + 2 and N_y ≠ 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 N_y for AGNR and ZGNR. The notations •, ○ and ▪ correspond to AGNRs of N_y = 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

ω_{1}^{P}

, the first subpeak ω₁₄, the second principal peak

ω_{2}^{P}

, the second subpeak ω₁₆, and the third principal peak

ω_{3}^{P}

, respectively [see Fig. 4(b)]. The predicted corresponding transition energies are

ω_{1}^{P} = E_{J^{c} = 2}^{edge}

ω_{14} = E_{J^{c} = 4}^{edge}

ω_{2}^{P} = E_{J^{c} = 3}^{edge} - E_{J^{v} = 2}^{edge}

ω_{16} = E_{J^{c} = 6}^{edge}

, and

ω_{3}^{P} = E_{J^{c} = 4}^{edge} - E_{J^{v} = 3}^{edge}

. The first subpeak (ω₁₄) and the second principal peak (

ω_{2}^{P}

) are close to each other and merge for a sufficiently large width, as to the second subpeak (ω₁₆) and the third principal peak (

ω_{3}^{P}

). 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 ω^exp. According to the optical selection rule, the measured peak positions

ω_{1}^{P, \exp}

ω_{14}^{\exp}

, and

ω_{16}^{\exp}

can be applied to specify the band-edge states

E_{J^{c} = 2}^{edge}

E_{J^{c} = 4}^{edge}

, and

E_{J^{c} = 6}^{edge}

, respectively, which are the energy differences between the J^c = even subband edges and the Fermi level. Moreover, the band-edge state energies of the J^c = odd subbands are also obtainable, for example,

E_{J^{c} = 3}^{edge} = ω_{2}^{P, \exp} - | E_{J^{v} = 2}^{edge} | = ω_{2}^{P, \exp} - ω_{1}^{P, \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

E_{J^{c}}^{edge} = | E_{J^{v}}^{edge} | = ω_{J}^{\exp} / 2

, where J^c = J^v = 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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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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References

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004). [CrossRef] [PubMed]
K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. U. S. A.102, 10451–10453 (2005). [CrossRef] [PubMed]
R. F. Service, “Materials science carbon sheets an atom thick give rise to graphene dreams,” Science324, 875–877 (2009). [CrossRef] [PubMed]
N. M. R. Peres, “Graphene, new physics in two dimensions,” Europhys. News40, 17–20 (2009). [CrossRef]
M. I. Katsnelson, “Graphene: carbon in two dimensions,” Mater. Today10, 20–27 (2007). [CrossRef]
A. K. Geim, “Graphene: Status and prospects,” Science324, 1530–1534 (2009). [CrossRef] [PubMed]
A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007). [CrossRef] [PubMed]
S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, “Giant intrinsic carrier mobilities in graphene and its bilayer,” Phys. Rev. Lett.100, 016602 (2008). [CrossRef] [PubMed]
F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, “Detection of individual gas molecules adsorbed on graphene,” Nat. Mater.6, 652–655 (2007). [CrossRef] [PubMed]
S. V. Morozov, K. S. Novoselov, and A. K. Geim, “Electron transport in graphene,” Phys. Usp.51, 744–748 (2008). [CrossRef]
S. Cho and M. S. Fuhrer, “Charge transport and inhomogeneity near the minimum conductivity point in graphene,” Phys. Rev. B77, 081402 (2008). [CrossRef]
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature438, 197–200 (2005). [CrossRef] [PubMed]
Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum hall effect and berry’s phase in graphene,” Nature438, 201–204 (2005). [CrossRef] [PubMed]
J. W. Bai, X. F. Duan, and Y. Huang, “Rational fabrication of graphene nanoribbons using a nanowire etch mask,” Nano Lett.9, 2083–2087 (2009). [CrossRef] [PubMed]
A. Fasoli, A. Colli, A. Lombardo, and A. C. Ferrari, “Fabrication of graphene nanoribbons via nanowire lithography,” Phys. Status Solidi B-Basic Solid State Phys.246, 2514–2517 (2009). [CrossRef]
V. L. J. Joly, M. Kiguchi, S. J. Hao, K. Takai, T. Enoki, R. Sumii, K. Amemiya, H. Muramatsu, T. Hayashi, Y. A. Kim, M. Endo, J. Campos-Delgado, F. Lopez-Urias, A. Botello-Mendez, H. Terrones, M. Terrones, and M. S. Dresselhaus, “Observation of magnetic edge state in graphene nanoribbons,” Phys. Rev. B81, 245428 (2010). [CrossRef]
J. Campos-Delgado, Y. A. Kim, T. Hayashi, A. Morelos-Gomez, M. Hofmann, H. Muramatsu, M. Endo, H. Terrones, R. D. Shull, M. S. Dresselhaus, and M. Terrones, “Thermal stability studies of cvd-grown graphene nanoribbons: Defect annealing and loop formation,” Chem. Phys. Lett.469, 177–182 (2009). [CrossRef]
L. Tapaszto, G. Dobrik, P. Lambin, and L. P. Biro, “Tailoring the atomic structure of graphene nanoribbons by scanning tunnelling microscope lithography,” Nat. Nanotechnol.3, 397–401 (2008). [CrossRef] [PubMed]
M. Y. Han, B. Ozyilmaz, Y. B. Zhang, and P. Kim, “Energy band-gap engineering of graphene nanoribbons,” Phys. Rev. Lett.98, 206805 (2007). [CrossRef] [PubMed]
C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science312, 1191–1196 (2006). [CrossRef] [PubMed]
D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature458, 872–876 (2009). [CrossRef] [PubMed]
F. Cataldo, G. Compagnini, G. Patane, O. Ursini, G. Angelini, P. R. Ribic, G. Margaritondo, A. Cricenti, G. Palleschi, and F. Valentini, “Graphene nanoribbons produced by the oxidative unzipping of single-wall carbon nanotubes,” Carbon48, 2596–2602 (2010). [CrossRef]
M. C. Paiva, W. Xu, M. F. Proenca, R. M. Novais, E. Laegsgaard, and F. Besenbacher, “Unzipping of functionalized multiwall carbon nanotubes induced by stm,” Nano Lett.10, 1764–1768 (2010). [CrossRef] [PubMed]
A. G. Cano-Marquez, F. J. Rodriguez-Macias, J. Campos-Delgado, C. G. Espinosa-Gonzalez, F. Tristan-Lopez, D. Ramirez-Gonzalez, D. A. Cullen, D. J. Smith, M. Terrones, and Y. I. Vega-Cantu, “Ex-mwnts: Graphene sheets and ribbons produced by lithium intercalation and exfoliation of carbon nanotubes,” Nano Lett.9, 1527–1533 (2009). [CrossRef] [PubMed]
R. H. Miwa, R. G. A. Veiga, and G. P. Srivastava, “Structural, electronic, and magnetic properties of pristine and oxygen-adsorbed graphene nanoribbons,” Appl. Surf. Sci.256, 5776–5782 (2010). [CrossRef]
S. Dutta and S. K. Pati, “Novel properties of graphene nanoribbons: a review,” J. Mater. Chem.20, 8207–8223 (2010). [CrossRef]
H. C. Chung, Y. C. Huang, M. H. Lee, C. C. Chang, and M. F. Lin, “Quasi-landau levels in bilayer zigzag graphene nanoribbons,” Physica E42, 711–714 (2010). [CrossRef]
T. Nomura, D. Yamamoto, and S. Kurihara, “Electric field effects in zigzag edged graphene nanoribbons,” J. Phys.: Conf. Ser.200, 062015 (2010). [CrossRef]
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]
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]
A. Cresti and S. Roche, “Range and correlation effects in edge disordered graphene nanoribbons,” New. J. Phys.11, 095004 (2009). [CrossRef]
Y. O. Klymenko and O. Shevtsov, “Low-energy electron transport in semimetal graphene ribbon junctions,” Eur. Phys. J. B72, 203–209 (2009). [CrossRef]
E. Perfetto, G. Stefanucci, and M. Cini, “Time-dependent transport in graphene nanoribbons,” Phys. Rev. B82, 035446 (2010). [CrossRef]
J. Jiang, W. Lu, and J. Bernholc, “Edge states and optical transition energies in carbon nanoribbons,” Phys. Rev. Lett.101, 246803 (2008). [CrossRef] [PubMed]
M. F. Lin and F. L. Shyu, “Optical properties of nanographite ribbons,” J. Phys. Soc. Jpn.69, 3529–3532 (2000). [CrossRef]
H. Hsu and L. E. Reichl, “Selection rule for the optical absorption of graphene nanoribbons,” Phys. Rev. B76, 045418 (2007). [CrossRef]
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]
L. Van Hove, “The occurrence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev.89, 1189 (1953). [CrossRef]
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]
J. C. Charlier, X. Gonze, and J. P. Michenaud, “First-principles study of the electronic properties of graphite,” Phys. Rev. B43, 4579–4589 (1991). [CrossRef]
G. Dresselhaus and M. S. Dresselhaus, “Fourier expansion for electronic energy bands in silicon and germanium,” Phys. Rev.160, 649–679 (1967). [CrossRef]
L. G. Johnson and G. Dresselhaus, “Optical properies of graphite,” Phys. Rev. B7, 2275–2285 (1973). [CrossRef]
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]
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]

Amemiya, K.
Angelini, G.
Bai, J. W.
Barros, E. B.
Berger, C.
Bernholc, J.
Besenbacher, F.
Biro, L. P.
Blake, P.
Booth, T. J.
Botello-Mendez, A.
Brown, N.
Campos-Delgado, J.
Cano-Marquez, A. G.
Capaz, R. B.
Cataldo, F.
Chang, C. C.
Chang, C. P.
Charlier, J. C.
Chen, S. C.
Cheng, R.
Chiu, C. W.
Cho, S.
Chung, H. C.
Cini, M.
Colli, A.
Compagnini, G.
Conrad, E. H.
Cresti, A.
Cricenti, A.
Cullen, D. A.
de Heer, W. A.
Dimiev, A.
Dobrik, G.
Dresselhaus, G.
Dresselhaus, M. S.
Duan, X. F.
Dubonos, S. V.
Dutta, S.
Elias, D. C.
Endo, M.
Enoki, T.
Espinosa-Gonzalez, C. G.
Fasoli, A.
Ferrari, A. C.
Firsov, A. A.
First, P. N.
Fuhrer, M. S.
Geim, A. K.
Gonze, X.
Grigorieva, I. V.
Han, M. Y.
Hao, S. J.
Hass, J.
Hayashi, T.
Higginbotham, A. L.
Hill, E. W.
Hofmann, M.
Hsu, H.
Huang, Y.
Huang, Y. C.
Jaszczak, J. A.
Jiang, D.
Jiang, J.
Johnson, L. G.
Joly, V. L. J.
Jorio, A.
Katsnelson, M. I.
Khotkevich, V. V.
Kiguchi, M.
Kim, P.
Kim, Y. A.
Klymenko, Y. O.
Kosynkin, D. V.
Kurihara, S.
Laegsgaard, E.
Lambin, P.
Lee, M. H.
Lee, S. H.
Lew Yan Voon, L. C.
Li, T. B.
Li, X. B.
Liao, L.
Lin, M. F.
Lombardo, A.
Lomeda, J. R.
Lopez-Urias, F.
Lu, W.
Marchenkov, A. N.
Margaritondo, G.
Mayou, D.
Mendes, J.
Michenaud, J. P.
Miwa, R. H.
Morelos-Gomez, A.
Morozov, S. V.
Muramatsu, H.
Naud, C.
Nomura, T.
Novais, R. M.
Novoselov, K. S.
Ozyilmaz, B.
Paiva, M. C.
Palleschi, G.
Patane, G.
Pati, S. K.
Peres, N. M. R.
Perfetto, E.
Price, B. K.
Proenca, M. F.
Ramirez-Gonzalez, D.
Ram-Mohan, L. R.
Reichl, L. E.
Ribic, P. R.
Roche, S.
Rodriguez-Macias, F. J.
Samsonidze, G. G.
Schedin, F.
Service, R. F.
Shailos, A.
Shevtsov, O.
Shull, R. D.
Shyu, F. L.
Sinitskii, A.
Smith, D. J.
Smith, N. V.
Song, Z. M.
Souza, A. G.
Srivastava, G. P.
Stefanucci, G.
Stormer, H. L.
Sumii, R.
Takai, K.
Tan, Y. W.
Tapaszto, L.
Terrones, H.
Terrones, M.
Tour, J. M.
Tristan-Lopez, F.
Ursini, O.
Valentini, F.
Van Hove, L.
Vega-Cantu, Y. I.
Veiga, R. G. A.
Wang, K. L.
Wang, M. S.
Wu, X. S.
Xiu, F. X.
Xu, W.
Yamamoto, D.
Zhang, Y.
Zhang, Y. B.

Appl. Surf. Sci.

R. H. Miwa, R. G. A. Veiga, and G. P. Srivastava, “Structural, electronic, and magnetic properties of pristine and oxygen-adsorbed graphene nanoribbons,” Appl. Surf. Sci.256, 5776–5782 (2010). [CrossRef]

Carbon

F. Cataldo, G. Compagnini, G. Patane, O. Ursini, G. Angelini, P. R. Ribic, G. Margaritondo, A. Cricenti, G. Palleschi, and F. Valentini, “Graphene nanoribbons produced by the oxidative unzipping of single-wall carbon nanotubes,” Carbon48, 2596–2602 (2010). [CrossRef]

Chem. Phys. Lett.

J. Campos-Delgado, Y. A. Kim, T. Hayashi, A. Morelos-Gomez, M. Hofmann, H. Muramatsu, M. Endo, H. Terrones, R. D. Shull, M. S. Dresselhaus, and M. Terrones, “Thermal stability studies of cvd-grown graphene nanoribbons: Defect annealing and loop formation,” Chem. Phys. Lett.469, 177–182 (2009). [CrossRef]

Eur. Phys. J. B

Y. O. Klymenko and O. Shevtsov, “Low-energy electron transport in semimetal graphene ribbon junctions,” Eur. Phys. J. B72, 203–209 (2009). [CrossRef]

Europhys. News

N. M. R. Peres, “Graphene, new physics in two dimensions,” Europhys. News40, 17–20 (2009). [CrossRef]

J. Mater. Chem.

S. Dutta and S. K. Pati, “Novel properties of graphene nanoribbons: a review,” J. Mater. Chem.20, 8207–8223 (2010). [CrossRef]

J. Phys. Soc. Jpn.

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

J. Phys.: Conf. Ser.

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

Mater. Today

M. I. Katsnelson, “Graphene: carbon in two dimensions,” Mater. Today10, 20–27 (2007). [CrossRef]

Nano Lett.

M. C. Paiva, W. Xu, M. F. Proenca, R. M. Novais, E. Laegsgaard, and F. Besenbacher, “Unzipping of functionalized multiwall carbon nanotubes induced by stm,” Nano Lett.10, 1764–1768 (2010). [CrossRef] [PubMed]
A. G. Cano-Marquez, F. J. Rodriguez-Macias, J. Campos-Delgado, C. G. Espinosa-Gonzalez, F. Tristan-Lopez, D. Ramirez-Gonzalez, D. A. Cullen, D. J. Smith, M. Terrones, and Y. I. Vega-Cantu, “Ex-mwnts: Graphene sheets and ribbons produced by lithium intercalation and exfoliation of carbon nanotubes,” Nano Lett.9, 1527–1533 (2009). [CrossRef] [PubMed]
J. W. Bai, X. F. Duan, and Y. Huang, “Rational fabrication of graphene nanoribbons using a nanowire etch mask,” Nano Lett.9, 2083–2087 (2009). [CrossRef] [PubMed]

Nat. Mater.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007). [CrossRef] [PubMed]
F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, “Detection of individual gas molecules adsorbed on graphene,” Nat. Mater.6, 652–655 (2007). [CrossRef] [PubMed]

Nat. Nanotechnol.

L. Tapaszto, G. Dobrik, P. Lambin, and L. P. Biro, “Tailoring the atomic structure of graphene nanoribbons by scanning tunnelling microscope lithography,” Nat. Nanotechnol.3, 397–401 (2008). [CrossRef] [PubMed]
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]

Nature

D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, “Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons,” Nature458, 872–876 (2009). [CrossRef] [PubMed]
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature438, 197–200 (2005). [CrossRef] [PubMed]
Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum hall effect and berry’s phase in graphene,” Nature438, 201–204 (2005). [CrossRef] [PubMed]

New. J. Phys.

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

Phys. Rep.

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]

Phys. Rev.

L. Van Hove, “The occurrence of singularities in the elastic frequency distribution of a crystal,” Phys. Rev.89, 1189 (1953). [CrossRef]
G. Dresselhaus and M. S. Dresselhaus, “Fourier expansion for electronic energy bands in silicon and germanium,” Phys. Rev.160, 649–679 (1967). [CrossRef]

Phys. Rev. B

L. G. Johnson and G. Dresselhaus, “Optical properies of graphite,” Phys. Rev. B7, 2275–2285 (1973). [CrossRef]
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]
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]
J. C. Charlier, X. Gonze, and J. P. Michenaud, “First-principles study of the electronic properties of graphite,” Phys. Rev. B43, 4579–4589 (1991). [CrossRef]
H. Hsu and L. E. Reichl, “Selection rule for the optical absorption of graphene nanoribbons,” Phys. Rev. B76, 045418 (2007). [CrossRef]
E. Perfetto, G. Stefanucci, and M. Cini, “Time-dependent transport in graphene nanoribbons,” Phys. Rev. B82, 035446 (2010). [CrossRef]
V. L. J. Joly, M. Kiguchi, S. J. Hao, K. Takai, T. Enoki, R. Sumii, K. Amemiya, H. Muramatsu, T. Hayashi, Y. A. Kim, M. Endo, J. Campos-Delgado, F. Lopez-Urias, A. Botello-Mendez, H. Terrones, M. Terrones, and M. S. Dresselhaus, “Observation of magnetic edge state in graphene nanoribbons,” Phys. Rev. B81, 245428 (2010). [CrossRef]
S. Cho and M. S. Fuhrer, “Charge transport and inhomogeneity near the minimum conductivity point in graphene,” Phys. Rev. B77, 081402 (2008). [CrossRef]

Phys. Rev. Lett.

M. Y. Han, B. Ozyilmaz, Y. B. Zhang, and P. Kim, “Energy band-gap engineering of graphene nanoribbons,” Phys. Rev. Lett.98, 206805 (2007). [CrossRef] [PubMed]
S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, “Giant intrinsic carrier mobilities in graphene and its bilayer,” Phys. Rev. Lett.100, 016602 (2008). [CrossRef] [PubMed]
J. Jiang, W. Lu, and J. Bernholc, “Edge states and optical transition energies in carbon nanoribbons,” Phys. Rev. Lett.101, 246803 (2008). [CrossRef] [PubMed]

Phys. Status Solidi B-Basic Solid State Phys.

A. Fasoli, A. Colli, A. Lombardo, and A. C. Ferrari, “Fabrication of graphene nanoribbons via nanowire lithography,” Phys. Status Solidi B-Basic Solid State Phys.246, 2514–2517 (2009). [CrossRef]

Phys. Usp.

S. V. Morozov, K. S. Novoselov, and A. K. Geim, “Electron transport in graphene,” Phys. Usp.51, 744–748 (2008). [CrossRef]

Physica E

H. C. Chung, Y. C. Huang, M. H. Lee, C. C. Chang, and M. F. Lin, “Quasi-landau levels in bilayer zigzag graphene nanoribbons,” Physica E42, 711–714 (2010). [CrossRef]

Proc. Natl. Acad. Sci. U. S. A.

K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. U. S. A.102, 10451–10453 (2005). [CrossRef] [PubMed]

Science

R. F. Service, “Materials science carbon sheets an atom thick give rise to graphene dreams,” Science324, 875–877 (2009). [CrossRef] [PubMed]
A. K. Geim, “Graphene: Status and prospects,” Science324, 1530–1534 (2009). [CrossRef] [PubMed]
C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned epitaxial graphene,” Science312, 1191–1196 (2006). [CrossRef] [PubMed]
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004). [CrossRef] [PubMed]

2011, Chung, J. Phys. Soc. Jpn.
2010, Perfetto, Phys. Rev. B
2010, Chiu, New. J. Phys.
2010, Cataldo, Carbon
2010, Paiva, Nano Lett.
2010, Miwa, Appl. Surf. Sci.
2010, Dutta, J. Mater. Chem.
2010, Chung, Physica E
2010, Nomura, J. Phys.: Conf. Ser.
2010, Bai, Nat. Nanotechnol.
2010, Joly, Phys. Rev. B
2009, Campos-Delgado, Chem. Phys. Lett.
2009, Service, Science
2009, Peres, Europhys. News
2009, Geim, Science
2009, Cano-Marquez, Nano Lett.
2009, Bai, Nano Lett.
2009, Fasoli, Phys. Status Solidi B-Basic Solid State Phys.
2009, Kosynkin, Nature
2009, Cresti, New. J. Phys.
2009, Klymenko, Eur. Phys. J. B
2008, Jiang, Phys. Rev. Lett.
2008, Morozov, Phys. Rev. Lett.
2008, Tapaszto, Nat. Nanotechnol.
2008, Morozov, Phys. Usp.
2008, Cho, Phys. Rev. B
2007, Han, Phys. Rev. Lett.
2007, Schedin, Nat. Mater.
2007, Geim, Nat. Mater.
2007, Katsnelson, Mater. Today
2007, Hsu, Phys. Rev. B
2006, Berger, Science
2006, Barros, Phys. Rep.
2005, Novoselov, Nature
2005, Zhang, Nature
2005, Novoselov, Proc. Natl. Acad. Sci. U. S. A.
2004, Novoselov, Science
2000, Lin, J. Phys. Soc. Jpn.
1993, Lew Yan Voon, Phys. Rev. B
1991, Charlier, Phys. Rev. B
1979, Smith, Phys. Rev. B
1973, Johnson, Phys. Rev. B
1967, Dresselhaus, Phys. Rev.
1953, Van Hove, Phys. Rev.

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