Lack of a bandgap is one of the significant challenges for application of graphene as the active element of an electronic device. A bandgap can be induced in bilayer graphene by application of a potential difference between the two layers. The simplest geometry for creating such a potential difference is two overlayed graphene nanoribbons independently contacted. Calculations, based on density functional theory and the nonequilibrium Green's function formalism, show that transmission through such a structure is a strong function of applied bias. The simulated current voltage characteristics mimic the characteristics of resonant tunneling diode featuring negative differential resistance. © 2011 American Institute of Physics.published_or_final_versio

Habib, KMM

Lake, RK

Zahid, F

Applied Physics Letters

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Title Negative differential resistance in bilayer graphene nanoribbonsAuthor(s) Habib, KMM; Zahid, F; Lake, RKCitation Applied Physics Letters, 2011, v. 98 n. 19Issued Date 2011URL http://hdl.handle.net/10722/135375Rights Creative Commons: Attribution 3.0 Hong Kong LicenseNegative differential resistance in bilayer graphene nanoribbonsK. M. Masum Habib,a Ferdows Zahid, and Roger K. LakebDepartment of Electrical Engineering, University of California, Riverside, California 92521-0204, USAReceived 1 January 2011; accepted 24 April 2011; published online 13 May 2011Lack of a bandgap is one of the significant challenges for application of graphene as the activeelement of an electronic device. A bandgap can be induced in bilayer graphene by application of apotential difference between the two layers. The simplest geometry for creating such a potentialdifference is two overlayed graphene nanoribbons independently contacted. Calculations, basedon density functional theory and the nonequilibrium Green’s function formalism, show thattransmission through such a structure is a strong function of applied bias. The simulated currentvoltage characteristics mimic the characteristics of resonant tunneling diode featuring negativedifferential resistance. © 2011 American Institute of Physics. doi:10.1063/1.3590772Graphene has fascinating electronic properties featuringthe Dirac fermion1 with high mobility2 and a long coherencelength. However, lack of a bandgap in two-dimensional 2Dgraphene3 reduces its utility for conventional electronic de-vice applications. A bandgap can be introduced by patterninga 2D graphene sheet into a narrow 10 nm nanoribbon,known as a graphene nanoribbon GNR.4,5 Another way tomodify the band structure of graphene is to stack two mono-layers to form a bilayer in which the bandgap can be tunedby creating a potential difference between the two layers.6,7The possibility of field effect transistors FETs usingbilayer graphene as the channel material was recentlystudied.8 It was shown that such a FET had a poor on-offcurrent ratio, Ion / Ioff, due to strong band-to-band tunneling.However, a tunnel FET using bilayer graphene showedpromising performance.9 Other proposed devices includea nanoelectromechanical FET based on interlayer distancemodulation,10,11 a FET utilizing a bilayer excitoncondensate,12 and GNR junction diodes featuring negativedifferential resistance NDR based on chemical13 and fieldeffect14 doping.Many proposed FET type graphene based devices havemultiple gates making them relatively complex device struc-tures. We consider the simplest possible geometry by whicha potential can be applied between two GNR layers. Such ageometry consists of two single layer GNRs with one placedon top of the other. Each GNR is independently contactedsuch that one GNR is held at ground while the other has abias applied to it. Such a geometry and biasing schemewould occur, for example, in a cross-bar architecture. Inde-pendently contacting the top and bottom GNR maximizes thevoltage drop between them. Assuming that the majority ofthe potential drop occurs between the two nanoribbons, thepotential difference between the two nanoribbons is the ap-plied bias. Since the bandgap increases with applied source-drain bias, we hypothesized that NDR would occur.To test this hypothesis, we performed numerical simula-tions of a model GNR geometry using ab initio density func-tional theory DFT to simulate the electronic structure and anonequilibrium Green’s Function NEGF approach to deter-mine the electron transport. The model structure of the over-lapping GNRs is shown in Fig. 1. It consists of a left and aright semi-infinite, armchair, H-passivated GNR which over-lap in the central region. Two well known bilayer stackingsequences, AB and AA, are considered. The widths of thearmchair GNRs AGNRs are chosen to be 14 atomic C lay-ers 3n+2 1.8 nm to minimize the bandgap resultingfrom the finite width. The bandgap of the 14-AGNR calcu-lated from DFT code, Fireball,15,16 is 130 meV which is ingood agreement with Son et al.17 When one GNR is stackedon top of another to form AB or AA bilayer GNRs, the band-gap is reduced further consistent with the results of Lam andLiang.5 For AA GNR, the bandgap is removed completely,and for AB GNR, the bandgap is reduced to 20 meV. Thelengths of the overlap regions for AB and AA stacking are1.7 nm and 1.6 nm, respectively. The total simulated lengthbetween the two ideal leads indicated by the self-energies inFig. 1 is 6.8 nm. Transmission through similar systems inequilibrium and with gate bias was recently studied in detailwith -band and k ·p models,18,19 and strong resonant andantiresonant features were observed in the transmission inagreement with our results below.Both the AA and AB GNR bilayers are either metallic orhave a bandgap less than kBT at room temperature. Creatinga potential difference between the two layers creates a band-gap with a maximum of 0.25 eV for the AB GNR and 130meV for AA GNR as shown in Fig. 2. Understanding theband structure of the bilayer GNRs and the effect of bias, weare now ready to investigate the current-voltage response ofthe structure shown in Fig. 1. Before doing so, we provide abrief description of the theoretical models.The electronic structures of the single and bilayerGNRs are modeled with the quantum molecular dynamics,aElectronic mail: khabib@ee.ucr.edu.bElectronic mail: rlake@ee.ucr.edu. FIG. 1. Atomic geometry of modeled AB–stacked device.APPLIED PHYSICS LETTERS 98, 192112 20110003-6951/2011/9819/192112/3/$30.00 © 2011 American Institute of Physics98, 192112-1Downloaded 15 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsDFT code Fireball using separable, nonlocal Troullier–Martins pseudopotentials,20 the BLYP exchange correlationfunctional,21,22 a self-consistent generalization of the Harris–Foulkes energy functional23,24 known as DOGS after theoriginal authors,25,26 and a minimal sp3 Fireball basis set.The radial cutoffs of the localized pseudoatomic orbitalsforming the basis are rc1s=4.10 Å for hydrogen and rc2s=4.4 Å and rc2p=4.8 Å for carbon.27A supercell of hydrogen passivated single layer AGNRwith periodic boundary conditions is relaxed quantum-mechanically with Fireball. The relaxed single layer super-cell is then repeated to construct single-layer GNRs. Thesesingle-layer GNRs are then placed one above the other at theexperimental separation distance of 3.35 Å and aligned toform the AB stacked structure shown in Fig. 1. The sameprocedure is followed to form the AA stacked structure witha separation distance of 3.55 Å. No further relaxation is per-formed on the structure. The region between the verticallines in Fig. 1 is used as the supercell for bilayer GNR withlattice vector, a. A single point self-consistent calculation isperformed with Fireball to generate the Hamiltonian matrixelements of this supercell. The matrix elements within 16atomic layers of the end overlap regions are discarded andreplaced with the matrix elements for the relaxed single-layerGNR.The applied bias is modeled by applying a rigid shift tothe energy of the lower GNR by the amount of the appliedbias, U=−eV. The matrix elements of U are calculated asi ,Uj ,	=Si,jUri+Ur j /2 where, the indices i andj label the atoms, the indices  and  label the basis orbitals,and Si,j is the overlap matrix i ,  j ,	. Uri=U for atomson the lower GNR and zero for atoms on the upper GNR.This approach in which the matrix elements have the sameform as in an extended Huckel model has been used byothers.28 The approach captures the Stark effect, but not non-equilibrium self-consistency. These and the Fireball Hamil-tonian matrix elements are used in the NEGF algorithm tocalculate the surface self-energies, Green’s function of thedevice, the spectral function, the transmission, and the cur-rent as described in Ref. 29.The simulated I-V characteristics for the AB- and AA-stacked GNRs corresponding to Fig. 1 are shown in Fig. 3.Both the AB and AA structure exhibit NDR. The peak cur-rent of the AB structure occurs at 0.4 V and the valley mini-mum occurs at 0.7 V. The peak and valley voltages for theAA structure are approximately twice those of the AB struc-ture. Thus, the model structure does exhibit NDR confirmingthe initial hypothesis.As a check, we repeated the I-V calculation of the ABstacked structure using a -bond model with tight-bindingparameters for the intralayer coupling 2.569 eV and theinterlayer coupling 0.361 eV taken from Ref. 30. Thepeak and valley currents resulting from the -bond modelwere, respectively, 24.5 A and 2.1 A, occurring at thepeak and valley voltages of 0.5 V and 1.0 V. Thus, the twomodels, DFT and -bond, give qualitatively the same I-Vwith the -bond model giving approximately twice the peakcurrent and four times the peak-to-valley ratio as the DFTmodel.To understand the I-V characteristics shown in Fig. 3, thetransmission coefficients are plotted as a function of electronenergy. The transmission plots for the AB structure areshown in Figs. 4a and 4b at the peak and valley biasvoltages, V=0.4 V and V=0.7 V, respectively. In both fig-ures, the unbiased transmission and the biased quasi-Fermilevels of the left and right contacts are shown for reference.In agreement with and as discussed in Refs. 18 and 19the transmission shows a Fabry–Perot resonant feature at lowenergy and both resonances and antiresonances at more ex-cited energies. The ends of the GNRs result in potential dis-continuities at both ends of the overlap region giving rise toa resonant cavity in which multiple reflections can occur. Athigher and lower energies multiple subbands allow multiplepaths which can constructively or destructively interfere.Edge states also occur on the cut ends, and these states resultin transmission peaks similar to those observed from the cutends of carbon nanotubes.27At V=0.4 V, the energy of the bottom GNR has beenshifted down by 0.4 eV, and the low transmission regionsnear E=0 and E=−0.4 eV are the result of the small 130meV bandgaps of the GNR leads. The region in betweencorresponds to transmission from hole states of the top leadto electron states of the bottom lead. As the bias of the bot-tom layer is increased to 0.7 V, the dip in transmission nearBandgap,Eg(eV)00.10.2Applied Bias, V (V)0 0.5 1 1.5AAABFIG. 2. Color online Bandgap as a function of applied bias for infinite AA-and AB-stacked bilayer GNRs.Current,I(μA)01020Applied Bias, V (V)0 0.5 1 1.5AAABFIG. 3. Color online Simulated current voltage I-V characteristics ofAA- and AB-stacked devices. The valley current minimums occur at 0.7 Vand 1.4 V for AB- and AA-stacked devices, respectively.FIG. 4. Color online Transmission as a function of energy for AB-stackeddevice: a at bias, V=0.4 V and b at bias, V=0.7 V, superimposed ontransmission at no bias. The vertical lines at the lower and upper energiesrepresent the quasi-Fermi levels of right and left contacts, respectively. Thequasi-Fermi level of the left contact is set at 0.192112-2 Habib, Zahid, and Lake Appl. Phys. Lett. 98, 192112 2011Downloaded 15 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions0.4 eV rigidly shifts down to −0.7 eV, and the transmis-sion from hole states to electron states between 0 and 0.7eV is strongly suppressed due to the large wave vectormismatch14,19 of the states inside the contacts and the bilayerregion as illustrated in Fig. 5. The resonant feature at0.3 eV results from an edge state on the cut end of the topGNR.The coherent current at any bias is proportional to thearea under the transmission curve bounded by the Fermi lev-els of the contacts. Beyond 0.7 V bias, the transmission be-tween the Fermi levels in Fig. 4b begins to increase as thefirst excited subbands of the top and bottom GNR leads arepulled into the energy window, and the current begins toincrease.The dependence of transmission of the AA device onbias follows similar trends. However, the peak current istwice as large, and the strong suppression of transmissionoccurs at approximately twice the bias of the AB device.This can be understood by noting that the wave vector mis-match for the AA case is less, since at any energy, E, be-tween the quasi-Fermi levels of the left and right contacts,the AA bilayer has two states whereas the AB bilayer hasonly one state for right moving electrons as illustrated in Fig.5. Therefore, more voltage is required to generate the sameamount of wave vector mismatch for the AA case. This alsoexplains the doubling of the magnitude of the peak currentfor the AA structure compared to that of the AB structure.This is consistent with the fact that there are twice as manynearest-neighbor matrix elements in AA stacking comparedto those in AB stacking; i.e., in AA stacking, every atom inthe lower GNR is directly below a corresponding atom in theupper GNR, whereas, in AB stacking, every second atom inthe lower GNR is directly beneath an atom in the upperGNR.In summary, we have performed ab initio DFT, -bond,and NEGF based calculations to study the I-V characteristicsof a bilayer GNR structure where bias is applied between theGNRs by independently contacting each layer. The simula-tions of the model structures with both AB and AA stackingprovide proof-of-principle that NDR can occur in such struc-tures.This work is supported by the Microelectronics Ad-vanced Research Corporation Focus Center on Nano Materi-als FENA.1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson,I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature London 438,197 2005.2K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V.Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 2004.3T. Ando, Physica E 40, 213 2007.4X. Li, X. Wang, L. Zhang, S. Lee, and H. Dai, Science 319, 1229 2008.5K.-T. Lam and G. Liang, Appl. Phys. Lett. 92, 223106 2008.6H. Min, B. Sahu, S. K. Banerjee, and A. H. MacDonald, Phys. Rev. B 75,155115 2007.7Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F.Crommie, Y. R. Shen, and F. Wang, Nature London 459, 820 2009.8G. Fiori and G. Iannaccone, IEEE Electron Device Lett. 30, 261 2009.9G. Fiori and G. 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At a given energy between thequasi-Fermi levels of the left and right contacts, the number of right movingstates available for carrying current is a one in the AB device and b twoin the AA device.192112-3 Habib, Zahid, and Lake Appl. Phys. Lett. 98, 192112 2011Downloaded 15 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

Negative differential resistance in bilayer graphene nanoribbons

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