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Conductance of bilayer graphenenanoribbons with different widthsT.S. Li a , Y.C. Huang b , M.F. Lin c & S.C. Chang aa Department of Electrical Engineering, Kun Shan University,Tainan, Taiwan, Republic of Chinab Center for General Education, Kao Yuan University, Kaoshiung,Taiwan, Republic of Chinac Department of Physics, National Cheng Kung University, Tainan,Taiwan, Republic of China

Available online: 23 Jun 2010

To cite this article: T.S. Li, Y.C. Huang, M.F. Lin & S.C. Chang (2010): Conductance of bilayergraphene nanoribbons with different widths, Philosophical Magazine, 90:23, 3177-3187

To link to this article: http://dx.doi.org/10.1080/14786435.2010.482914

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Philosophical MagazineVol. 90, No. 23, 14 August 2010, 3177–3187

Conductance of bilayer graphene nanoribbons with different widths

T.S. Lia*, Y.C. Huangb, M.F. Linc and S.C. Changa

aDepartment of Electrical Engineering, Kun Shan University, Tainan, Taiwan, Republic ofChina; bCenter for General Education, Kao Yuan University, Kaoshiung, Taiwan,

Republic of China; cDepartment of Physics, National Cheng Kung University, Tainan,Taiwan, Republic of China

(Received 14 December 2009; final version received 29 March 2010)

The electronic and transport properties of bilayer graphene nanoribbonswith different width are investigated theoretically by using the tight-bindingmodel. The energy dispersion relations are found to exhibit significantdependence on the interlayer interactions and the geometry of the bilayergraphene nanoribbons. The energy gaps are oscillatory with the upperribbon displacement. For all four types of bilayer graphene nanoribbons,the bandgaps touch the zero value and exhibit semiconductor–metaltransitions. Variations in the electronic structures with the upper ribbondisplacement will be reflected in the electrical and thermal conductance.The chemical-potential-dependent electrical and thermal conductancesexhibit a stepwise increase and spike behavior. These conductances can betuned by varying the upper ribbon displacement. The peak and trenchstructures of the conductance will be stretched out as the temperature rises.In addition, quantum conductance behavior in bilayer graphene nano-ribbons can be observed experimentally at temperature below 10 K.

Keywords: transport properties; electronic properties; graphenenanoribbon

1. Introduction

Graphene is a monolayer of carbon atoms tightly packed into a two-dimensional(2D) honeycomb lattice. Its unusual electronic properties were first predictedtheoretically by Wallace [1] in 1947 and the subsequent studies of its magneticproperties [2]. It took 50 years for the experimental realization of single-layergraphene [3]. Single-layer graphene demonstrates exceptional electronic properties,such as the half-integer quantum Hall effect [4–7] and temperature-induced plasmons[8]. These fascinating properties are due to its unique band structure, that is, thevalence and conduction bands in graphene touch conically at the Dirac points. Forbilayer graphenes, an additional degeneracy of the Landau levels and a Berry phaseof 2� were predicted to cause the anomalous quantum Hall effect [9], which differsfrom either conventional quantum well devices or single-layer graphene. Electronmobilities as high as 104 cm2/Vs are observed in single and bilayer graphenes [10].

*Corresponding author. Email: tsli@mail.ksu.edu.tw

ISSN 1478–6435 print/ISSN 1478–6443 online

� 2010 Taylor & Francis

DOI: 10.1080/14786435.2010.482914

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Very recently, mobilities approaching 2� 105 cm2/Vs have been reported for

suspended graphene devices [11]. High carrier mobility implies ballistic transport

and long electronic phase coherence length. Graphene nanoribbons (GNRs), 1D

graphene with finite width, can be obtained by cutting mechanically exfoliated

graphenes [4,10], or by patterning graphenes with lithographic techniques [7]. The

presence of edges can have a strong influence on the properties of the �-electrons.There are two basic shapes of edges, namely zigzag and armchair. The ribbons with

zigzag edges (zigzag GNRs) possess partial flat bands at the Fermi level, which is

caused by strongly localized electronic states at the edges [12]. On the other hand,

there is no partial flat band in armchair GNRs.In 1957, Landauer proposed one of the most important predictions of modern

condensed-matter physics: the electrical conductance in 1D channels is quantized in

steps of e2/h [13–15]. His theory is intuitive, simple and practically useful. Motivated

by possible device applications, electron transport through various graphene-based

nanostructures was extensively studied. Experiments showed that the conductance of

a single-walled carbon nanotube, which can be regarded as a rolled-up graphene

sheet, is quantized and shows Fabry–Perot interference patterns [16]. These results

can be explained within a generalized Landauer formalism [16]. Magnetotransport

experiments performed on exfoliated or epitaxial graphene have reported weak

localization effects [17,18]. Quantum transport in GNRs has been studied previously

for ideal [19] and defective cases [20]. Mucciolo and collaborators studied

numerically the effects of edge disorder on the conductance of GNRs [21]. Apart

from their fascinating transport properties, the spin flip length in graphene-based

materials is long [22], making them possible candidates for spin-polarized electronics.

In this paper, the electrical and thermal transport properties of bilayer graphene

nanoribbons (BGNRs) with different widths are studied within the ballistic regime.

They can be calculated by using the Landauer–Buttiker formalism, which relates the

electrical conduction to an independent electron scattering problem. Progress in the

fabrication of graphene-based nanostructures that permits us to study transport in

graphene nanoribbons motivates this work. Such research has not yet been explored,

to the best of our knowledge.

2. Theory

The electronic properties of single-layer graphene nanoribbons are briefly reviewed

here. The ribbon width is defined by Ny, the number of armchair lines or zigzag lines

in the transverse (y) direction. Zigzag GNRs are always gapless, whereas armchair

GNRs with Ny¼ 3Iþ 2 are also gapless, but those with Ny 6¼ 3Iþ 2 possess a

bandgap and are semiconductors. For semiconducting armchair GNRs, their

bandgaps scale inversely with Ny. Details of the electronic structure of GNRs are

given in [12,23,24]. In this study, we consider the bilayer GNRs consisting of an

armchair GNR with width NL lying below another one with width NU. A schematic

plot of such a system is depicted in Figure 1. The displacement of the upper GNR is

specified by yd. There are 2(NLþNU) carbon atoms in the primitive unit cell. The

first Brillouin zone is confined within jkj ��/3b. The Hamiltonian operator can be

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written as

H ¼ �Xhi, j i

t intrai, j cþi cj �Xhhi, j ii

t interi, j cþi cj, ð1Þ

where i¼ 1, 2, . . . , 2(NLþNU), and ti, j is the transfer integral. cþi and cj are thecreation and annihilation operators at sites i and j, respectively. The intralayerhopping is fixed at tintrai, j ¼ �0¼ 2.66 eV, whereas the interlayer hopping is assumed todecay exponentially with interatom distance dij according to the model byNemec [25],

tinteri, j ¼ � expa� dij�

� �,

with �¼ �0/8, a¼ 3.34 A, and �¼ 0.45 A. A cutoff is chosen at dcutoff¼ aþ 5�. Thedangling bonds on the edge sites are assumed to be terminated by hydrogen atoms,

and they will not contribute to the electronic states near the Fermi level [12]. Aftersolving the state energy E, one may proceed to calculate the transport properties.

We consider a BGNR suspended between two reservoirs or macroscopic leads.

The left and right reservoirs are assumed to have the chemical potentials and thetemperatures of (�þ eV,T ) and (�,TþDT ), respectively. Making use of theLandauer–Buttiker formula, the electrical conductance G and thermal conductance �in the ballistic regime are L0 and ðL2 � L

21L�10 Þ=T, respectively. The details are given

in our previous work [26]. L� is defined as

L� ¼ 2e2��To

h

Zk40

dk@E

@k

��������ðE� �Þ��@f

oðE Þ

@E: ð2Þ

Figure 1. Schematic plot of a bilayer GNR consisting of an armchair GNR with width NL

lying below another one with width NU. The displacement of the upper GNR is specified by yd.

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In Equation (2), the factor of 2 in front accounts for the two spin states, and To is thetransmission probability for electrons in the reservoirs to enter the BGNR. f o is theFermi–Dirac distribution function. At low temperature, the main contributions to L�(�¼ 0, 1, 2) come from electronic states very close to the Fermi level.

3. Results and discussion

Four types of BGNRs: (I) NL¼ 150 and NU¼ 29 armchair GNRs; (II) NL¼ 150 andNU¼ 30 armchair GNRs; (III) NL¼ 152 and NU¼ 29 armchair GNRs; and (IV)NL¼ 152 and NU¼ 30 armchair GNRs; are chosen to be part of a model study. TheNL¼ 150 and 152 GNRs have roughly the same width, but the former issemiconducting and the latter is gapless. In addition, NU¼ 30 GNR has a bandgapand NU¼ 29 GNR has zero bandgap. When the lower and upper GNRs aredecoupled (independent subsystems), the band structures are symmetric about theFermi level EF¼ 0 for all four BGNRs (Figure 2). NL¼ 152 or NU¼ 29 armchairGNR have linear subbands intersecting at EF. With the inclusion of coupling, theband structures are considerably altered, such as the modification of the subbandcurvature, the creation of additional band-edge states, and the change of subbandspacing or energy gap. The energy bands are no longer symmetric about EF

(Figure 3); the energy band distortion varies with yd (not shown).For type-I BGNR, the linear dispersions originating from the NU¼ 29 upper

GNR become parabolic with the interlayer coupling, and new band-edge states arecreated (Figure 3a). A small bandgap �0.001�0 is opened by such coupling. Fortype-II BGNR, the first, second, and third subbands closest to EF all belong to theNL¼ 150 lower GNR. The subbands of the NU¼ 30 are farther away from EF due toits smaller width (Figure 2b). Nevertheless, the interlayer hoppings will modify thelowest subbands considerably and reduce the energy gap (Figure 3b). For type-IIIBGNR, both NL¼ 152 and NU¼ 29 GNRs have linear subbands intersecting at EF

(Figure 2c). They change to parabolic subbands, and a small bandgap �0.001�0 isopened. The interlayer-coupling-induced bandgap is relatively large �0.013�0 intype-IV BGNR (Figure 3d). The electronic structures of BGNRs vary sensitivelywith yd, and are demonstrated in the yd dependence of the energy gap (Figure 4). Ingeneral, the energy gap is oscillatory with yd. For all four BGNRs, the bandgapstouch the zero value and exhibit semiconductor–metal transitions at certain values ofyd. Additionally, bandgap closing occurs more frequently for the type-IV BGNR.

The electrical conductance of BGNR at low temperature is determined by thenumber of conducting channels at the Fermi level. As a result, the chemical-potential-dependent conductance demonstrates a ladder behavior (Figure 5a). Suchladders appear whenever � intersects a subband. The stepwise increases with �are caused by the increasing number of conducting channels. In Figure 5b,�o ¼ �k

2BT=6�h is the quantum of the thermal conductance. Apart from the step

behavior, the conductance of type-III BGNR demonstrates sharp peak structures.The thermal conductance also exhibits a stepwise increase behavior (Figure 5b)similar to that of the electrical conductance. However, there is a small difference inthe shape and extent of the ladder structures or the peak structures between G and �due to the different integrands in Equation (2).

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It is of interest to study the effect of the displacement yd on the conductance.The electrical conductances of all four BGNRs demonstrate prominent peak structureat certain values of yd. In addition to prominent peak behavior, the conductanceof type-II BGNR also displays small-plateau structures. Explanations for this

(a) (b)

(c) (d)

NL=152NU=29

NL=150NU=29

NL=150NU=30

NL=152NU=30

Figure 2. The low-energy subbands of BGNRs without interlayer interactions (independentsubsystems) for (a) NL¼ 150, NU¼ 29, (b) NL¼ 150, NU¼ 30, (c) NL¼ 152, NU¼ 29, and (d)NL¼ 152, NU¼ 30. The subbands of the upper GNR are shown as green (online) solid lines.b¼ 1.42 A is the C–C bond length.

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spiking behavior are as follows. The @f o(E )/@E function is a sharp Lorentzian functionat the Fermi level. In additional, the electron state energies are very sensitive to yd.Therefore, whether the state closest to the Fermi level may contact EF and thuscontribute to the electrical conductance, or may not contact EF and thus notcontribute, depends very strongly on yd. The positions of the peak correspond to the ydvalues that lead to semiconductor–metal transitions, and their heights are related

(a) (b)

(c) (d)

NL=152NU=29

NL=150NU=29

yd=0 NL=150NU=30

NL=152NU=30

Figure 3. The low-energy subbands of BGNRs with interlayer interactions for (a) NL¼ 150,NU¼ 29, (b) NL¼ 150, NU¼ 30, (c) NL¼ 152, NU¼ 29, and (d) NL¼ 152, NU¼ 30, at yd¼ 0.

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to the number of subbands at the Fermi energy. Type-IV BGNR has the largestnumber of conductance peaks corresponding to its largest number of bandgap closingoccurrances. The thermal conductances closely resemble the electrical conductances(Figure 5b). The slight difference between them is caused by the different integrands inEquation (2).

We have explored the dependence of the electrical conductance on the GNRwidth NL (Figure 6c). The general trend is that the conductance oscillates with NL,except for that at NU¼ 29 and �¼ 0.005 �0. With fixed NU, adjacent values of NL

may correspond to different types of BGNR. For example, with NU fixed at 30,NL¼ 150 and 151 are type-II BGNRs, whereas NL¼ 152 is type-IV BGNR. Theelectrical conductance at NU¼ 29 and �¼ 0.005 �0 is almost a constant at 8 T0e

2/hwith a small fluctuation at large NL, indicating that there are eight conductionchannels for all values of NL. Although varying NL certainly modifies the bandstructures accordingly, it does not change the number of conducting subbands at EF,which is the only factor that determines the conductance. The thermal conductancecurves are similar to the electrical curves (Figure 6d).

Figure 7 illustrates the effects of temperature on the conductance. The electricalconductance has a single-peak structure adjacent to a trench structure, whereas thethermal conductance has a double-peak adjacent to the trench. The difference arisesfrom the different integrands in Equation (2). The @f o(E )/@E Lorentzian function isstretched with a width kBT by the finite temperature. As a result, the number ofconducting channels is also changed. The peak structures or the trench structures ofthe electrical conductance and thermal conductance are gradually extended by theincreasing temperature. At T¼ 10K, the single-peak structure of G is stillrecognizable (Figure 7a), whereas the corresponding double-peak structure of �disappears (Figure 7b). At the same temperature, the integrand of L2 (Equation (2)),

NL=150; NU=29NL=150; NU=30NL=152; NU=29NL=152; NU=30

yd (b)

Figure 4. Energy gap dependence on yd for different BGNRs. The small discontinuities arecaused by the cutoff dcutoff.

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which is quadratic in E��, is more stretched out in the energy domain than that ofL0. Therefore, � is more sensitive to the temperature change than G. According toour computation results, quantum conductance behavior in BGNRs can be observedexperimentally at temperatures below 10 K by conductance measurements [27].

4. Conclusions

In conclusion, we have presented a theoretical investigation on the electronic andtransport properties of BGNRs. A detailed analysis of the role played by the

(a)

(b)

NL=150; NU=29

T=2 K; yd =0

NL=150; NU=30NL=152; NU=29NL=152; NU=30

Figure 5. Dependence of (a) the electrical conductance, and (b) the thermal conductance onthe chemical potential � for different BGNRs at T¼ 2K and yd¼ 0. �o ¼ �k

2BT=6�h is the

quantum of the thermal conductance.

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interlayer interactions and the geometry of the BGNRs on the energy dispersions wasperformed. The interlayer couplings modify the subband curvature, createadditional band-edge states, change the subband spacing and open or close bandgaps.The energy gaps are found to be oscillatory with the upper ribbon displacement.

(a) (c)

(b) (d)

NL=150; NU=29

T=2 K; m=0

NL=150; NU=30NL=152; NU=29NL=152; NU=30

Nu =29; m=0.005 g0

Nu =29; m=0.01 g0

Nu =30; m=0.005 g0

Nu =30; m=0.01 g0

yd =0

yd (b) NL

Figure 6. Dependence of (a) the electrical conductance, and (b) the thermal conductance on ydfor different BGNRs at T¼ 2K. Dependence of (c) the electrical conductance, and (d) thethermal conductance on NL for different BGNRs at T¼ 2K and yd¼ 0.

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For all four bilayer graphene nanoribbons, the bandgaps touch the zero value andexhibit semiconductor–metal transitions. Variations in the electronic structures withthe upper ribbon displacement will be reflected in the electrical and thermalconductance. The chemical-potential-dependent electrical and thermal conductancesare found to exhibit a stepwise increase and spike behavior. These conductances can betuned by varying the upper ribbon displacement. The peak and trench structures of theconductance will be stretched as the temperature rises. Moreover, quantumconductance behavior in BGNRs can be observed experimentally at temperaturesbelow 10K. These theoretical predictions can be validated by conductancemeasurements.

(a)

(b)

NL=152; NU=29

yd

=0

Figure 7. Dependence of (a) the electrical conductance, and (b) the thermal conductance, onthe chemical potential � for type-III BGNR at various temperatures.

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Acknowledgements

This work was supported in part by the National Science Council of Taiwan, the Republic ofChina under Grant No. NSC 98-2112-M-168-001-MY2.

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