## Exploration of edge-dependent optical selection rules for graphene nanoribbons |

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 Δ*J* = *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 |Δ

*J*| =

*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

*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

*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*) of the dimer lines (armchair or zigzag lines) along the longitudinal direction (

_{y}*x̂*). The periodic length is

*π*/

*I*≤

_{x}*k*≤

_{x}*π*/

*I*with 2

_{x}*N*carbon atoms in a primitive unit cell. The Hamiltonian equation of the system is

_{y}*H*|Ψ〉 =

*E*|Ψ〉. With the framework of the tight-binding model considering only the nearest-neighbor interactions, the Hamiltonian operator is

*γ*

_{0}is the hopping integral,

*l*(

*l*′), and the on-site energies are set to zero. The Bloch wavefunction, i.e., the superposition of the 2

*N*tight-binding functions |

_{y}*ϕ*〉’s with the corresponding site amplitudes

_{l}*C*’s, is expressed as where

_{l}*A*(

_{m}*B*) is the site amplitude and |

_{m}*a*〉 (|

_{m}*b*〉) is the tight-binding function associated with the periodic

_{m}*A*(

*B*) atoms.

*H*in the subspace spanned by the tight-binding functions is a 2

*N*× 2

_{y}*N*Hermitian matrix. For AGNRs, the upper triangular elements of

_{y}*H*are

*γ*

_{0}= 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]

*E*’s and the eigenstates |Ψ

^{h}*〉’s are obtained by diagonalizing the Hamiltonian matrix. The superscript*

^{h}*h*can be either

*c*or

*v*, representing the conduction or valence subband, respectively.

*k*= 0, and therefore only positive ones are discussed. For the AGNR with

_{x}*N*= 69, many 1D parabolic subbands are symmetric about the Fermi level [Fig. 1(a)]. The bottoms and tops of these subbands are located at

_{y}*k*= 0, where many band-edge states exist and optical transitions may occur. The energy gap is equal to 0.05

_{x}*γ*

_{0}for this semiconducting nanoribbon. The subband index

*J*(= 1, 2, 3,...) is decided by the minimum energy spacing between the subbands and the Fermi level. For the ZGNR with

^{c,v}*N*= 69, the parabolic subbands are also symmetric about

_{y}*E*= 0, and their band edges are located at

_{F}*k*= 2

_{x}*π*/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 2

*π*/3 to

*π*. Hence, the ZGNR is a gapless semiconductor.

## 3. Features of wavefunctions in the real space

*A*(

*B*) located at the (3

*m*)th, (3

*m*+1)th and (3

*m*+2)th armchair lines, with

*m*being an integer. The wavefunctions of the

*N*= 69 AGNR at

_{y}*k*= 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

_{x}*J*− 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 |

^{h}*E*| 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

^{h}*A*and

*B*, that is, the site amplitudes

*A*of the

*J*= 1 subband,

^{c}*J*=

^{c}*J*=

^{v}*J*, the subenvelope functions have the relations For instance,

*A*and

*B*. There is no relation between subenvelope functions on different armchair lines, for example, the relation between

## 4. Absorption spectra

**ê**||

*x̂*, the absorption spectrum

*A*(

*ω*) for direct photon transitions (Δ

*k*= 0) from the initial state |Ψ

_{x}*(*

^{h}*k*,

_{x}*J*)〉 to the final state |Ψ

^{h}*′ (*

^{h}*k*,

_{x}*J*′)〉 is given by

^{h}*m*is the effective mass of an electron,

_{e}**p**is the momentum operator,

*f*[

*E*(

^{h}*k*,

_{x}*J*)] is the Fermi-Dirac distribution function, and Γ is the broadening factor. The low-energy absorption spectra for the

^{h}*N*= 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.

_{y}*N*= 69 AGNR [Fig. 4(a)], the first peak, located at

_{y}*ω*

_{1}= 0.055

*γ*

_{0}, is identified as the excitation from the

*J*= 1 to the

^{v}*J*= 1 subband. The second (

^{c}*ω*

_{2}= 0.105

*γ*

_{0}) and the third (

*ω*

_{3}= 0.205

*γ*

_{0}) ones are the transitions from the

*J*= 2 to the

^{v}*J*= 2 subband and from the

^{c}*J*= 3 to the

^{v}*J*= 3 subband, respectively. The allowed transitions originate in the subbands of the same indices, i.e., the selection rule is Δ

^{c}*J*=

*J*

^{c}*− J*= 0. The transition energy is twice the band-edge energy (

^{v}*J*=

^{c}*J*=

^{v}*J*) because the band structure is symmetric about

*E*= 0. Moreover, the peak height increases when the frequency increases as a consequence of the decrease of the subbands’ curvatures.

_{F}## 5. Optical absorption selection rules of GNRs

*M*, the momentum operator is substituted in terms of the

^{hh′}*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. 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]

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]

*h̄*is treated as unity in the following derivation.

*k*= 0, where the excitations occur for AGNRs. Substituting the coefficients of sublattices

_{x}*B*with

*A*in Eq. (13) by the relations

*M*is

^{vc}*A*

_{m}_{±1}and

*A*, obtained from

_{m}*H*|Ψ〉 =

*E*|Ψ〉 at

*k*= 0, are where Δ

_{x}_{±}= ±(

*E*/

*γ*

_{0}) − 1 for

*A*= ±

_{m}*B*. After using Eq. (15) to replace

_{m}*A*

_{m}_{±1}with

*A*,

_{m}*M*becomes

^{vc}*=*

_{s}*s*(

*E*/

*γ*

_{0}) − 1 for

*k*= 0 is where Δ

_{x}_{±}= ±(

*E*/

*γ*

_{0}) − 1 and

*u*(

*t − s*) is either 0 or ±2. Eq. (17) yields the selection rule (Δ

*J*= 0) for AGNRs and possible excitation channels in the absorption spectra.

*k*= 2

_{x}*π*/3, where the transition occur, becomes where

*h*′ = −

*aγ*

_{0}sin(

*πa*/3). For an odd

*N*, we use Eq. (8) to convert the second term in Eq. (17), which leads to Then renaming the index of

_{y}*N*+ 1

_{y}*− m*as

*m*gives As for an even

*N*, the second term in Eq. (18) can also be converted by Eq. (9), and yields the same result as Eq. (19). Hence, the

_{y}*M*for ZGNR at

^{vc}*k*= 2

_{x}*π*/3 is where

*J*=

*odd*) for ZGNRs is obtained.

*N*, the

_{y}*J*th transition energy is equal to

*J*th subband. For narrower GNRs, there are three groups of transition energies. The

^{c}*ω*’s of

_{J}*N*= 3

_{y}*m*(full circles) and

*N*= 3

_{y}*m*+ 1 (open circles) groups are very similar, while for the

*N*= 3

_{y}*m*+ 2 (squares) group, the first and second

*ω*’s lie close to each other and are sandwiched between the second and third

_{J}*ω*’s of the

_{J}*N*≠ 3

_{y}*m*+ 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*= 3

_{y}*m*+ 2 and

*N*≠ 3

_{y}*m*+ 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.

*ω*

_{14}, the second principal peak

*ω*

_{16}, and the third principal peak

*ω*

_{14}) and the second principal peak (

*ω*

_{16}) and the third principal peak (

*ω*. According to the optical selection rule, the measured peak positions

^{exp}*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,

*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

*J*= 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 (|Δ

*J*| =

*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

## References and links

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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

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