Effect of Dimensionality on the Localization Behavior in Hydrogenated

Letter pubs.acs.org/NanoLett

Effect of Dimensionality on the Localization Behavior in Hydrogenated Graphene Systems Duk-Hyun Choe and K. J. Chang* Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea S Supporting Information *

ABSTRACT: Recently, several experiments have shown that graphene exhibits a metal-to-insulator transition by hydrogenation. Here we theoretically study the transport properties of hydrogenated graphene and graphene nanoribbons (GNRs), focusing on the conductance fluctuation behavior in the localized regime. Using a simple model for the conductance distribution in the quasi-localized regime where the conventional theory fails, we derive the modified single parameter scaling (SPS) relations for quasi-one-dimensional (Q1D) GNRs as well as two-dimensional (2D) graphene. We show that, as the dimensional crossover occurs from 2D to Q1D, the shape of the conductance distribution evolves from a positively skewed distribution to a log-normal distribution. We predict that GNRs with widths much larger than the localization lengths do not behave as a Q1D system. Our results provide fundamental insights into the dimensionality change not only in graphene, but also in general mesoscopic systems in the localized regime. KEYWORDS: Graphene, graphene nanoribbons, hydrogenated graphene, electronic transport, localization

T

L are intrinsic conductance and sample size, respectively. The ensemble average denoted by ⟨···⟩ is necessary because the conductance exhibits large sample-to-sample fluctuations. It soon became clear that the SPS theory must be understood in terms of the entire conductance distribution function. For example, in one-dimensional (1D) strongly localized systems, the distribution function of conductance is log-normal, and the variance σ2 of the logarithm of conductance (ln g) is related to the scaling parameter, that is, the mean value (μ = ⟨ln g⟩), by the universal law19,21

he experimental realization of graphene, as a strictly twodimensional (2D) material with intriguing electronic properties,1 has provided great opportunities for investigating the peculiar transport properties of massless Dirac fermions, such as half-integer quantum Hall effect2,3 and absence of backscattering.4 Recently, hydrogenated graphene has attracted much attention due to the experimental observation of a metal−insulator transition for both high and low H concentrations.5−12 While a metal−insulator transition is likely to occur by the formation of band insulators for high H concentrations,6−9 its origin for low H concentrations is attributed to the localization of 2D electronic states.10−12 In graphene nanoribbons (GNRs),13 as quasi-one-dimensional (Q1D) structures, the localization of the low-energy states by hydrogenation was also predicted theoretically.14 Although the underlying conductance behavior of low-energy states in disordered graphene systems, including 2D graphene and Q1D GNRs, has been studied by a number of groups,12,14−16 their behavior in the dimensional crossover between 2D graphene and Q1D GNRs has not been explored. The dimensional crossover of graphene is of great interest for both electronic and thermal transport properties.16,17 Meanwhile, the quantum electronic transport in lowdimensional disordered systems has been the subject of extensive studies for over several decades. One of the most significant findings in such fields is the single parameter scaling (SPS) theory of localization;17−21 the quantum transport feature is completely determined by a single scaling parameter. The original SPS theory of localization was interpreted in terms of the scaling function β(g) = d⟨ln g⟩/d ln L,18,20,22 where g and © 2012 American Chemical Society

σ 2 = −2μ.

(1)

In two-dimensional (2D) systems, their conductance behavior in the localized regime is explained by asymmetric distributions that are deviated from the log-normal. However, despite some efforts to find analytic formulas for the distribution function23 as well as the universal relation24 between σ2 and μ, the understanding of the distribution is so far relatively poor because of the lack of analytical frameworks and the difficulties in calculations. While most studies have been focused on the localization behavior for both 2D and Q1D systems individually,23−28 no attempt has been made to investigate the dimensional transition behavior in the localized regime. In this communication, we investigate such a crossover from 2D to Q1D by examining the localization induced by hydrogenation in graphene systems. We first study the Received: June 12, 2012 Revised: August 17, 2012 Published: September 10, 2012 5175

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properties of the conductance distribution function in hydrogenated Q1D GNRs and 2D graphene separately. We explain the deviation from the conventional SPS theory in the quasilocalized regime by using a cutoff model for the distribution function. We then consider the dimensional transition from 2D to Q1D with adopting two different strategies and discuss the effect of the dimensional crossover on the SPS relation. The transport properties of hydrogenated graphene and GNRs are studied by calculating the conductance based on the tight-binding (TB) model. We use a single-band TB Hamiltonian to describe interactions of GNRs with hydrogen, where parameters are determined by fitting the band structure to that of the density functional calculations.14 For a device model in which a disordered sample is sandwiched between two semi-infinite graphene nanoribbon electrodes, the twoterminal conductance29 is calculated by using the LandauerBüttiker formula,30 gL = Tr(tt†) where t is the transmission matrix and the spin degeneracy is neglected. The details of the calculation method were given elsewhere.12,14 To examine the sample-to-sample fluctuation behavior in the conductance of hydrogenated graphene and GNRs, we analyze the data statistically. Individual data points and the distribution functions are obtained by taking the ensemble average over more than 2000 different configurations in which H adsorbates are randomly distributed. As a first step, we examine the conductance behavior of Q1D GNRs in the localized regime. We consider armchair and zigzag GNRs, which are denoted as Na-aGNR and Nz-zGNR, respectively. Here, Na stands for the numbers of C−C dimer lines across the ribbon in aGNRs, whereas Nz is for the number of zigzag chains in zGNRs. The ribbon lengths (L) range from 10 to 200 nm. Two ribbon widths (W) of 3.6 and 10.0 nm are chosen: 29-aGNR and 17-zGNR correspond to the width of W = 3.6 nm, whereas 83-aGNR and 47-zGNR correspond to W = 10.0 nm. To focus on the disorder effect on conductance and avoid the formation of band insulators,7,8 a relatively low H concentration (nH = 1.5%) is used. For hydrogenated GNRs, the calculated variances of ln gL (σ2) are plotted as a function of the mean value of ln gL (μ) in Figure 1a. We find that all the data points are well fitted to a single curve regardless of the details of the system, such as edge type, length, width, and energy. In the strongly localized limit μ ≪ 0, the results for σ2 and μ are well described by the formula of eq 1, satisfying the conventional 1D SPS hypothesis.19,21 However, as μ approaches to the quasi-localized regime (see Supporting Information), roughly lying between −10 and −2, σ2 does not follow the traditional SPS curve. This discrepancy is attributed to the unique feature of the distribution function. For a Q1D disordered system without time-reversal symmetry, Mutallib and Wölfle showed that in the quasi-localized regime the log-normal distribution exhibits a sharp cutoff on one side.26 Here we recover their results by considering a simple model containing the essence of the cutoff effect. Realizing that there is a cutoff in the log-normal distribution, we modify the conventional SPS theory of localization. This modification is done based on the condition that the shape of the expected lognormal distribution from eq 1 does not change, but, the sharp cutoff emerges at ln gL = 0 2 ⎧ ⎪ Nf (ln gL ; μ′ , σ ′ ) for ln gL ≤ 0 7(ln gL ) = ⎨ ⎪ 0 for ln gL > 0 ⎩

Figure 1. (a) Variance plot for Q1D hydrogenated GNRs with the lengths ranging from 10 to 200 nm. The yellow and cyan curves are derived from our cutoff model for 7 (ln gL) and the model of Muttalib et al. (ref 26), respectively. The channel energies of −0.8 and −0.4 eV are chosen for GNRs with the widths of 3.6 and 10 nm, respectively. The total numbers of channels (Nch) for 17-zGNRs, 29-aGNRs, 47zGNRs, and 83-aGNRs for the chosen channel energies are 3, 3, 5, and 3, respectively (see Supporting Information for more details). (b) Magnified view of the curves and data in (a). (c) Variance plot for 2D hydrogenated graphene systems with the sizes ranging from 10 × 10 to 60 × 60 nm2. For graphene, Nch ranges from 3 to 41 for the considered channel energies (see Supporting Information for more details). The 2D SPS relation in eq 3 (ref 24) is compared with the Q1D SPS curve derived from our cutoff model. The empty symbols denote the numerical results obtained by using the periodic boundary conditions (ref 32).

where N is the normalizing factor and f(x; μ′ , σ′2) is the normal distribution function with μ′ and σ′2 satisfying the relation of eq 1. The prime symbols are used to distinguish between the values from our model and those from eq 1. The existence of the cutoff is verified by the fact that the

(2) 5176


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Figure 2. Conductance distribution functions for various samples with different hydrogen concentrations, channel energies, and sample sizes. Specific details are given inside the figure. The black solid lines are obtained from our cutoff model in eq 2 for 7 (ln gL) with μ = −3.96, −15.8, and −34.3. For Q1D GNRs, the numerically calculated μ values for 3.6 × 60 nm2 aGNR, 10 × 30 nm2 zGNR, 10 × 100 nm2 zGNR, and 4.0 × 40 nm2 zGNR are −3.96, −3.91, −15.8, and −34.3, respectively. For 2D graphene, the numerically calculated μ values for the samples of 10 × 10 nm2, 40 × 40 nm2, and 60 × 60 nm2 with nH = 5% and 60 × 60 nm2 with nH = 10% are −3.96, −4.05, −17.0, and −35.4, respectively.

conductance is governed by only one effective active channel31 in strongly localized and quasi-localized regimes,27 thus, gL cannot exceed one. With only the cutoff correction, we find that the modified relation between μ and σ2 is slightly lowered, as compared to the conventional curve of eq 1, and successfully describes the numerical results in the SPS region, as shown in Figure 1b. Based on our cutoff model, we compare the modified distribution function 7 (ln gL) in eq 2 with the numerical data for 7 (ln gL) in Figure 2, and find good agreement between the two results. The log-normal behavior is apparent for the distributions which have the mean values of μ ≈ − 15.8 and −34.3 in the strongly localized regime, without the cutoff. On the other hand, the sharp cutoff appears in 7 (ln gL) with the mean value of μ ≈ − 3.96 in the quasi-localized regime. Our simple intuitive model provides physical insight that quasilocalized regime can be understood by just a combination of the conventional log-normal distribution in the strongly localized regime [eq 1] and a sharp cutoff at ln gL = 0. Next we turn to study the 2D hydrogenated graphene by using square-shaped zGNRs with the side lengths up to 60 nm. To achieve a certain degree of localization (see Supporting Information), we choose an H concentration of 10% and focus on the channel energies (E) of −0.1, −0.2, −0.4, and −0.6 eV, which satisfy the 2D SPS relation (see Supporting Information). For the 2D system, the calculated σ2 values are drawn as a function of μ in Figure 1c, in the same manner as for Q1D GNRs. The results with and without the periodic boundary condition32 are compared with the approximate 2D SPS relation24 σ 2 ≈ 3.66( −μ)2/3 − 4.4

SPS regime. Unfortunately, the conductance behavior and hence the SPS relation are still not well established in 2D localized systems. Thus, we cannot directly confirm the cutoff effect on the σ2 curve, as done for the Q1D case in Figure 1b. Nevertheless, the cutoff at ln gL = 0 in 7 (ln gL) is also evident for the 2D system,25,33 as shown in Figure 2, and its effect can be clearly seen in the skewness plot. To better understand the conductance distribution behavior, we plot the skewness (γ) as a function of μ for both Q1D and 2D systems in Figure 3. The skewness, which is defined as γ = ⟨[(ln gL − μ)/σ]3⟩, is a measure of the asymmetry in distribution; it shows negative values if the left tail is longer, whereas positive values for the opposite case. In Q1D GNRs, as the localization becomes stronger, γ becomes close to zero due to the perfect symmetry between the left and right tails of the normal distribution of ln gL. On the other hand, γ gradually decreases as μ approaches to zero because the sharp cutoff at ln gL = 0 results in a negative value for γ. This feature is well demonstrated in the comparison of the numerical calculations for the two γ versus μ curves by our cutoff model and the model of Muttalib and Wölfle26 in Figure 3. As compared to the Q1D case, 7 (ln gL) of the 2D localized system is rather asymmetric, thus, γ converges to a certain value in the strongly localized limit. Recently, Somoza et al.23 have derived the analytic form for 7 (ln gL) of the 2D localized system by considering the Anderson model on a square lattice. They showed that for L/lloc ≥ 6, where lloc is the localization length defined as lloc = −2(∂⟨ln gL⟩/∂L)−1, 7 (ln gL) is well fitted to a certain asymmetric function with the asymptotic value of γ = 0.359. However, in disordered graphene, we find that the γ values are somewhat smaller than the asymptotic value of Somoza et al. in the strong localization regime where L/lloc increases to 17.7, as shown in Figure 3. Although it is not clear whether γ will reach the asymptotic value of 0.359 as L/lloc increases further, the difference between the Q1D and 2D

(3)

which was obtained by Prior et al. by fitting the numerical results from the Anderson model. A good overlap between our calculations and eq 3 indicates that the results are within the 2D 5177


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Figure 4. (a) Schematic illustration of a transition from graphene to GNR in the process of reducing the width W, with the length fixed to L. (b) The same transition from graphene to GNR in the process of increasing the length L, with the width fixed to W.

Figure 3. Skewness plot for hydrogenated GNRs and graphene. The Q1D curves are derived from our cutoff model for 7 (ln gL) and the model of Muttalib et al. (ref 26). The orange dashed curve is drawn as guidance for the numerical results in 2D graphene. The triangles denote the numerical results for Q1D GNRs with the lengths ranging from 10 to 200 nm, whereas the stars and squares stand for the results for 2D graphene systems with the sizes ranging from 10 × 10 to 60 × 60 nm2. The four star symbols from right to left correspond to the channel energies of −0.6, −0.4, −0.2, and −0.1 eV, respectively, in graphene with the size of 60 × 60 nm2 and the H concentration of 10%.

while. It is indicated by arrow marks in Figure 5a, in which the sample widths range from 3.8 to 2.8 nm and 3.0 to 1.7 nm for green and blue transition curves, respectively. As the sample width decreases further than 2.8 and 1.7 nm for green and blue transition curves, respectively, the σ2 values deviate from the Q1D SPS curve, turning into the non-SPS regime, although this feature is not shown in the figure. Our results indicate that for a given GNR length, there are some conditions for nH, E, and W to determine whether the GNR system is in the Q1D SPS regime as shown in the arrow marks in Figure 5a. In the second strategy of increasing the length L (Figure 4b), the crossover behavior from 2D to Q1D is not found for both the σ2 and γ plots. Starting from an arbitrary point in the localized regime, the σ2 value deviates from the 2D SPS curve as L increases, as shown in Figure 5a. However, in contrast to the process of reducing W, σ2 does not reach the Q1D curve, with a smaller slope. A similar situation occurs in the γ plot in Figure 5b. The γ value does not converge to zero although L increases from 40 to 320 nm for W = 40 nm, indicating that 7 (ln gL) maintains γ > 0 even if L becomes sufficiently large, making it difficult to induce a transition from 2D to Q1D. On the basis of the results, it is inferred that if the starting W × W graphene is in the localized regime, the localization behavior of W × L GNRs, which are obtained by increasing L, does not follow the Q1D SPS theory because the width is already much larger than the localization length. In conclusion, we have numerically investigated the effect of dimensionality on the localization behaviors of 2D graphene, Q1D GNRs, and in between. We find that a transition from 2D to Q1D is attainable by reducing the sample width, while it is not possible by increasing the length if the 2D system is in the localized regime. In a wider perspective, by taking advantage of the ultimate 2D nature of graphene, we look forward to seeing experimental tests of the localization behavior during the transition from 2D to Q1D. On the other hand, it was reported that for the 3D strongly localized system,34−36 the variance behaves as σ2 = A(−μ)α + B where α = 2/5, as compared to α = 1 and 2/3 for Q1D and 2D systems, respectively, and the skewness converges to the value of about 1.1, while their asymptotic values are 0 and 0.395 for Q1D and 2D systems, respectively. While the dimensional crossover of the localization behavior from 3D to 2D or Q1D is still an open question, the analysis in this study can be similarly applied.

strongly localized systems is apparent, as illustrated in Figures 1c, 2, and 3, and the conductance behavior highly depends on the dimensionality. As can be seen in Figure 2, even though the μ values are similar, 7 (ln gL) is more widely distributed for the Q1D case because the σ2 value is larger (Figure 1c). Moreover, 7 (ln gL) for the 2D case is slightly skewed with the γ values of about 0.20 and 0.28 for μ = −17.0 and −35.4, respectively. On the other hand, in the 2D quasi-localized system, γ decreases and has negative values because of the cutoff effect in 7 (ln gL), similar to the Q1D case. For the mean values of −5 < μ < 0, the γ curves as well as the σ2 curves for the Q1D and 2D systems seem to merge into one (see Figures 1c and 3), indicating that the conductance distributions are almost equal in such a region.28 In fact, this is verified by an excellent match of 7 (ln gL) in the Q1D and 2D systems for μ ≃ −4 in Figure 2. So far we have studied the conductance behaviors of the Q1D and 2D systems separately by considering hydrogenated GNRs and graphene, respectively. Now we examine the crossover behavior of conductance between the Q1D and 2D systems. As shown in Figure 4, two strategies are chosen for the dimensional transition from 2D to Q1D: reducing the width or increasing the length of hydrogenated graphene. In the former (Figure 4a), we start from a certain point on the 2D SPS curve in Figure 5a,b and gradually reduce the width of the sample with fixing all the other parameters such as L, nH, and E. In this transition process from 2D to Q1D, the σ2 values immediately start to deviate from the 2D SPS curve and then meet the Q1D SPS curve for four representative samples chosen. Such immediate deviations from the 2D SPS curve indicate that the conductance feature in the 2D localized system stems from the equality of width and length. Similar evolutions from 2D to Q1D are found for γ (Figure 5b), implying that 7 (ln gL) changes from a positively skewed distribution to a log-normal form as the width decreases. We note that after the transition curve meets the Q1D curve, it follows the Q1D behavior for a 5178


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ACKNOWLEDGMENTS Research supported by Korea Research Foundation under contract no. KRF-2011-0093845 and by Korea Institute of Science and Technology Information through the Supercomputor-Aid Program.

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(1) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, 109. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197. (3) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201. (4) Katsnelson, M. I.; Novoselov, K. S.; Geim, A. K. Nat. Phys. 2006, 2, 620. (5) Sun, Z.; Pint, C. L.; Marcano, D. C.; Zhang, C.; Yao, J.; Ruan, G.; Yan, Z.; Zhu, Y.; Hauge, R. H.; Tour, J. M. Nat. Commun. 2011, 2, 559. (6) Sofo, J. O.; Chaudhari, A. S.; Barber, G. D. Phys. Rev. B 2007, 75, 153401. (7) Elias, D. C.; Nair, R. R.; Mohiuddin, T. M. G.; Morozov, S. V.; Blake, P.; Halsall, M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.; Novoselov, K. S. Science 2009, 323, 610. (8) Balog, R.; Jørgensen, B.; Nilsson, L.; Andersen, M.; Rienks, E.; Bianchi, M.; Fanetti, M.; Lægsgaard, E.; Baraldi, A.; Lizzit, S.; Sljivancanin, Z.; Besenbacher, F.; Hammer, B.; Pedersen, T. G.; Hofmann, P.; Hornekær, L. Nat. Mater. 2010, 9, 315. (9) Grassi, R.; Low, T.; Lundstrom, M. Nano Lett. 2011, 11, 4574. (10) Bostwick, A.; McChesney, J.; Emtsev, K.; Seyller, T.; Horn, K.; Kevan, S.; Rotenberg, E. Phys. Rev. Lett. 2009, 103, 056404. (11) Haberer, D.; Vyalikh, D. V.; Taioli, S.; Dora, B.; Farjam, M.; Fink, J.; Marchenko, D.; Pichler, T.; Ziegler, K.; Simonucci, S.; Dresselhaus, M. S.; Knupfer, M.; Büchner, B.; Grüneis, A. Nano Lett. 2010, 10, 3360. (12) Bang, J.; Chang, K. J. Phys. Rev. B 2010, 81, 193412. (13) Son, Y.-W.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2006, 97, 216803. (14) Choe, D. H.; Bang, J.; Chang, K. J. New J. Phys. 2010, 12, 125005. (15) Lherbier, A.; Biel, B.; Niquet, Y.-M.; Roche, S. Phys. Rev. Lett. 2008, 100, 036803. (16) Adam, S.; Cho, S.; Fuhrer, M. S.; Das Sarma, S. Phys. Rev. Lett. 2008, 101, 046404. (17) Ghosh, S.; Bao, W.; Nika, D. L.; Subrina, S.; Pokatilov, E. P.; Lau, C. N.; Balandin, A. A. Nat. Mater. 2010, 9, 555. (18) Abrahams, E.; Anderson, P. W.; Licciardello, D. C.; Ramakrishnan, T. V. Phys. Rev. Lett. 1979, 42, 673. (19) Anderson, P. W.; Thouless, D. J.; Abrahams, E.; Fisher, D. S. Phys. Rev. B 1980, 22, 3519. (20) Lee, P. A.; Ramakrishnan, T. V. Rev. Mod. Phys. 1985, 57, 287. (21) Beenakker, C. W. J. Rev. Mod. Phys. 1997, 69, 731. (22) Slevin, K.; Markos, Peter.; Ohtsuki, T. Phys. Rev. Lett. 2001, 86, 3594. (23) Somoza, A. M.; Ortuño, M.; Prior, J. Phys. Rev. Lett. 2007, 99, 116602. (24) Prior, J.; Somoza, A. M.; Ortuño, M. Phys. Rev. B 2005, 72, 024206. (25) Prior, J.; Somoza, A. M.; Ortuño, M. Eur. Phys. J. B 2009, 70, 513. (26) Muttalib, K. A.; Wölfle, P. Phys. Rev. Lett. 1999, 83, 3013. (27) Froufe-Pérez, L. S.; García-Mochales, P.; Serena, P. A.; Mello, P. A.; Sáenz, J. J. Phys. Rev. Lett. 2002, 89, 246403. (28) Qiao, Z.; Xing, Y.; Wang, J. Phys. Rev. B 2010, 81, 085114. (29) Note the difference between two-terminal conductance gL, which corresponds to the conductance of the total system including the disordered region and the leads, and four-terminal (or intrinsic) conductance g = (1/gL − 1/Nch)−1, where Nch is the number of channels. To examine the conductance distribution behavior in the

Figure 5. (a) Variance and skewness as a function of μ in the dimensional crossover process. Specific details for samples are given in (b). The sample widths are reduced from 40 to 2.8 nm, 60 to 1.7 nm, 60 to 1.9 nm, and 60 to 2.1 nm for green, blue, red, and gray transition curves, respectively. The sample length is increased from 40 to 320 nm for black transition curve. In (a), the orange curve denotes the 2D SPS relation in eq 3, whereas the yellow curve represents the Q1D SPS curve derived from our cutoff model for 7 (ln gL). In (b), the Q1D γ curve is derived from our cutoff model for 7 (ln gL), whereas the orange dashed curve for the 2D skewness is the same as that in Figure 3.

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ASSOCIATED CONTENT

S Supporting Information *

Justification of using two-terminal conductance; conductance behavior in strongly localized, quasi-localized, and crossover regimes; degree of localization in hydrogenated graphene; validity of the single parameter scaling; gL versus E plot for graphene and GNRs. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: 82-42-350-2531. Fax: 82-42-350-2510. Notes

The authors declare no competing financial interest. 5179


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strongly localized as well as quasi-localized regime, we have used twoterminal conductance gL rather than g throughout this paper. The difference between gL and g is negligible in strongly localized regime. See Supporting Information for more details. (30) Datta, S. Electronic Transport in Mesoscopic Systems, 1st ed.; Cambridge University Press: Cambridge, 1995; Vol. 1. (31) Imry, Y. Europhys. Lett. 1986, 1, 249. (32) In 2D graphene, the periodic boundary condition imposed in the transverse direction slightly affects μ and σ2. However, all the data with and without the periodic boundary condition are well fitted to the curve given by equation 3, as shown in Figure 1c. For consistency with the Q1D case, we deal with 2D graphene systems without the periodic boundary condition throughout the paper. (33) Rühländer, M.; Soukoulis, C. M. Physica B 2001, 296, 32. (34) Somoza, A. M.; Prior, J.; Ortuño, M. Phys. Rev. B 2006, 73, 184201. (35) Douglas, A.; Muttalib, K. A. Phys. Rev. B 2010, 82, 035121. (36) Douglas, A.; Muttalib, K. A. Phys. Rev. B 2009, 80, 161102(R).

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