Bandgap engineering of zigzag graphene nanoribbons by manipulating edge states via defective boundaries
Aihua Zhang, Yihong Wu, San-Huang Ke, Yuan Ping Feng, Chun Zhang
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Bandgap engineering of zigzag graphene nanoribbons bymanipulating edge states via defective boundaries
Aihua Zhang , Yihong Wu , San-Huang Ke , Yuan Ping Feng , Chun Zhang , ∗ Department of Physics, National University of Singapore,2 Science Drive 3, Singapore 117542 Department of Chemistry, National University of Singapore,3 Science Drive 3, Singapore 117543 Department of Electrical and Computer Engineering,National University of Singapore, 4 Engineering Drive 3, Singapore, 117576 Department of Physics, Tongji University,1239 Siping Road, Shanghai 200092, P. R. China (Dated: November 12, 2018)
Abstract
One of severe limits of graphene nanoribbons (GNRs) in future applications is that zigzag GNRs(ZGNRs) are gapless, so cannot be used in field effect transistors (FETs). In this paper, using tight-binding approach and first principles method, we derived and proved a general edge (boundary)condition for the opening of a significant bandgap in ZGNRs with defective edge structures. Theproposed semiconducting GNRs have some interesting properties including the one that they canbe embedded and integrated in a large piece of graphene without the need of completely cuttingthem out. We also demonstrated a new type of high-performance all-ZGNR FET. . INTRODUCTION Graphene has attracted intensive research efforts due to its unique electronic and me-chanical properties.
A recent experiment demonstrated a beautiful technique in fabri-cating graphene nanoribbons (GNRs) with atomically precise edges, suggesting the greatpotential of GNRs in future applications of graphene-based high performance electronics. Theoretical calculations showed that only two thirds of armchair graphene nanoribbons(AGNRs) with different widths are semiconducting, while zigzag graphene nanoribbons(ZGNRs) are gapless due to localized edge states at the Fermi level.
It was also theoret-ically demonstrated that these bandgap-closing edge states survive in zigzag GNRs witha mixture of zigzag and armchair sites at boundaries.
Another recent theoretical workproved that the confinement by minimal boundaries generally does not produce an insu-lating GNR except for the armchair case. This theoretically predicted edge or orientationand width dependence of bandgap opening in GNRs provide serious limits in real appli-cations of GNR-based electronic devices: First, gapless ZGNRs cannot be used in FETs,and second, the precise control of the width of AGNRs is required.In this paper, using the tight-binding approach and the first principles method based ondensity functional theory (DFT), we derived and proved that when the number of A-sitedefects equals to that of B-site defects at each boundary (A, B denotes two sublatticesof graphene), localized edge states in GNRs will be eliminated, and then a bandgap thatis inversely proportional to the ribbon width will generally be open. We then showedthat ZGNRs with defective boundaries that satisfy the bandgap opening conditions can beembedded and integrated in a large piece of graphene, which may have implications for thefuture design of graphene-based integrated circuits. At last, we demonstrate a new type offield effect transistor completely made of ZGNRS. Note that in all previously theoreticallyproposed GNR-based transistors, the AGNR is indispensable. It is worthy mentioninghere that since the long-range magnetic order is not stable in one-dimensional systems2nder finite temprature, we therefore stick to the non-magnetic case in this study. II. RESULTS AND DISCUSSION
We first focus on a ZGNR with an edge structure with defects as shown in Fig. 1(a). Theedge structure can be specified by a quadruple of segment lengths in unit of the graphenelattice constant ( a = .
46 Å), ( N B , , N A , , N B , , N A , ). So the number of two-coordinatedcarbon atoms at edge belong to A ( B ) sublattice, N A ( N B ), equals to N A , + N A , ( N B , + N B , ). The electronic structure was calculated using the tight-binding approach. Onlythe nearest-neighbor hopping energy (-2.7 eV) was taken into account. It is well knownthat there exist edge states with E = p / < k y a p . The edge state entirely localizes at edge for k y a = p , and otherwisedecays exponentially away from the edge. When two zigzag edges form a ZGNR andthe edge states from both edges interact with each other, the edge states still degenerate at k y a = p , while a small gap that decreases exponentially with the nanoribbon width openselsewhere. The band structure of a perfect ZGNR in a supercell corresponding to (9, 0,0, 0) is reproduced in Fig. 1(b). Due to band folding, there are six bands (marked as red)corresponding to 2 p / < k y a p . We find that these bands originating from edge statesare removed and thus an energy gap opens if N A = N B . An example of the band structurecorresponding to a (3, 3, 3, 3) edge structure is shown in Fig. 1(c), and the squared wavefunctions in the inset clearly indicate they are extended states with the form, sin ( k n x ) ,which has more nodes for larger energies. Therefore, the energy of the valence bandmaximum (VBM), hence the energy gap, is inversely proportional to the nanoribbon width.For the case of either N A < N B or N A > N B , some of the edge states will remain as shownin Figs. 1(d) and 1(e), though the perfectly localized edge state is destroyed. The squaredwave functions in the insets of Figs. 1(d) and 1(e) show the edge states are exponentiallydecaying away from the edge, so the energy gap due to the interaction between the states at3pposite edges also decreases exponentially with respect to the nanoribbon width. Thesetwo distinct behaviors of the bandgap variation as a function of the nanoribbon width canbe seen in Fig. 2(a). The variation of bandgaps with respect to the characteristic length ofthe edge structure is shown in Fig. 2(b). The possibility to tune bandgaps with differentedge structures on the same nanoribbon might provide useful implication in the design ofnanoribbon-based electronic devices.The above mentioned condition for the elimination of edges states that leads to bandgapopening can be understood from the following arguments. Considering a semi-infinitegraphene sheet with N eA ( N eB ) two-coordinated carbon atoms at the edge and N bA ( N bB ) three-coordinated carbon atoms in the bulk belonging to A ( B ) sublattice, we have the followingequation by the conservation of coordinate numbers,2 N eA + N bA = N eB + N bB . Since the total number of carbon atoms in each sublattice is N tA ( B ) = N eA ( B ) + N bA ( B ) , theabove equation can be rewritten as 3 ( N tA − N tB ) = N eA − N eB , which means the relation be-tween N tA and N tB is the same as that between N eA and N eB . The band structure of graphenein the tight-binding approximation is calculated by E y A ( r ) = t [ y B ( r ) + y B ( r − R ) + y B ( r − R )] (1) E y B ( r ) = t [ y A ( r ) + y A ( r + R ) + y A ( r + R )] , (2)where t is the hopping energy, y A ( r ) and y B ( r ) are the wave functions on A and B atomsbelonging to the same unit cell at a discrete coordinate r , and R and R are graphenelattice vectors as shown in Fig. 1(a). For E =
0, Eqs. 1 and 2 are decoupled. There are N tA equations with N tB unknowns for 1 and N tB equations with N tA unknowns for 2. So if N tA > N tB , 1 will have no solution while 2 will have solutions on A sublattice. Similarconclusion will arrive for N tA < N tB . The fact that the wave function will reside on thesublattice with more atoms can be observed by comparison of the insets in Figs. 1(d)4nd 1(e). If N tA = N tB and the break of symmetry leads to no linear dependence amongequations, then Eqs. 1 and 2 will have only zero solution, which is not admissible andresults in the elimination of localized states with E = An interesting property of nanoribbons with a bandgap-opening edge structure is thatif a wide nanoribbon is joined with a narrow nanoribbon, the electronic structure of thewide nanoribbon near the Fermi energy is not altered with electrons still confined in thewide nanoribbon. An example of a nanoribbon with a width of L = √ a and a (3, 3,3, 3) edge structure joined with a nanoribbon with the same edge structure and a widthof L = √ a is shown in Fig. 3(a). The band structure of the compound system near theFermi energy (the conduction and valence band) in Fig. 3(b) is almost the same as that5f the stand-alone nanoribbon shown in Fig. 1(c). The charge distribution of the state atVBM in Fig. 3(a) and the local density of states in Fig. 3(c) clearly indicate that the wavefunction is only localized in the wide nanoribbon. The confinement can be understoodfrom the bandgap difference of two nanoribbons with different widths. Note that the in-tegrated GNRs discussed here can be also regarded as a special type of graphene antidotlattice structures proposed earlier. This property makes it possible to fabricate indi-vidual nanoribbon-based electronic devices by patterning rows of holes in a large pieceof graphene avoiding complete cutout and glued together, which might be beneficial forthe integration of future graphene-based electric circuits. On experimental side, the pat-terned graphene nanostructures discussed here can be obtained experimentally by usingtechniques such as templated self-assembly of block copolymers or direct writing usinga helium ion beam. The periodically patterned structure may be formed by first form-ing resist patterns on the graphene sheet followed by templated self-assembly of blockcopolymers in the region where the resist have been removed and etching of graphene byusing the copolymer patterns as the mask. On the other hand, the random patterns can beformed by direct writing using a helium ion beam. Prior to the lithography processes, analignment mark may be formed on the wafer by using an appropriate graphene edge as areference so as to align the patterns in specific directions with respect to the underlyinggraphene lattice structure.At last, we show a FET completely made of ZGNRs as shown in Fig. 5. In all previoustheoretically proposed GNR-based FETs, the AGNR is an indispensable component dueto the fact that pure ZGNR is metallic. Here, the proposed FET consists of two pureZGNR electrodes (left and right), and a ZGNR with a defective (2, 2, 2, 2) edge structure.The transport calculations were done using a first principles approach combining the non-equilibrium Green’s function’s techniques and DFT via the ATK code.
In the insetof the figure, the current-voltage (I-V) curve is shown for the zero gate voltage. The biasrange of the zero current comes from the bandgap of the defective ZGNR in the center,6onfirming the bandgap opening condition we derived from the tight binding approach.The currents as a function of gate voltage for different bias voltages suggest that the on-offratio of this proposed FET is bigger than 1000. Compared to the previously proposed all-GNR based FET that used two ZGNRs and one AGNR, the FET suggested here has twoobvious advantages: First, the complicated contacts between differently orientated AGNRand ZGNRs are avoided. Second, the precise control of ribbon width is not required. III. CONCLUSION
In conclusion, using the tight-binding approach, we derived a general boundary condi-tion for the band gap opening in the ZGNRs with defective edges: When the number ofA-site defects equals to that of B-site defects, the ZGNRs are semiconducting. We furthershowed that the semiconducting ZGNRs generated this way can be integrated in a largepiece of graphene by correctly patterning holes, which may be useful for the future large-scale integration of GNR-based devices. At last, we demonstrated using first principlescalculations a high-performance FET completely made of ZGNRs. Results presented inthis paper may be used to explain the recent experimental measurements showing that thetransport gap always exists independent of the crystallographic orientations of GNRs. Weexpect these findings to provide impetus for new experiments as well motivations for newideas in designing ZGNR-based electronic devices.
IV. ACKNOWLEDGMENT
We thank Professor A. H. Castro Neto and Dr. V. M. Pereira for stimulating and helpfuldiscussions. This work was supported by NUS Academic Research Fund (Grant Nos: R-144-000-237133 and R-144-000-255-112). Computations were performed at the Centrefor Computational Science and Engineering at NUS.7
IG. 1: (a) The lattice structure of a zigzag graphene nanoribbon with periodic edge structures.The unit cell is indicated by the dashed line. The edge structure is denoted by a quadruple of( N B , , N A , , N B , , N A , ), each number of which corresponds to the segment length in unit of thegraphene lattice constant, a . Other parameters of the system are the nanoribbon width ( L ) and thetranslational vector ( T ). Carbon atoms belonging to different sublattices at edge are designated red( A ) and blue ( B ) colors. (b-e) The band structures of nanoribbons with different edge structuresand the same width L = √ a . In the inset of (c) are shown the squared wave functions along thedash-dotted line in (a) for different states at k y =
0. The squared wave functions corresponding tothe valence band maximums are also plotted in the insets of (d) and (e). The radii of filled discs areproportional to R ( log | y ( r ) | + ) , where R ( x ) is the ramp function, and the color is determinedby the sign of real part of y ( r ) . IG. 2: (a) The variation of bandgaps as a function of the nanoribbon width ( L ) for different edgestructures. (b) The variation of bandgaps as a function of the segment length ( Na ) of a ( N , N , N , N ) edge structure for different widths. IG. 3: (a) The unit cell (dashed line) of a wide nanoribbon ( W ) joined with a narrow nanoribbon( N ) having the same edge structure. The nanoribbon W between two dash-dotted lines is identicalto the nanoribbon shown in Fig. 1(a). (b) The band structure of the system in (a). The squaredwave function plotted in (a) corresponds to the state indicated by an arrow. (c) The correspondinglocal density of states in W and N as shown in (a). IG. 4: The comparison between band structures of graphene nanoribbons with a pure (3, 3, 3, 3)edge structure (dashed line) and with one (3, 3, 2, 3) unit (as shaded) plus ten (3, 3, 3, 3) units in asupercell (solid line). The squared wave function (only part of the supercell is depicted) shown inthe left panel suggests that the corresponding state (as arrowed) is localized around the defective(3, 3, 2, 3) unit. IG. 5: (a) The configuration for the transport calculation. (b) The variation of current as a functionof gate voltage for different bias voltages. The inset shows the bias voltage dependence of currentfor zero gate voltage. ∗ Electronic address: [email protected] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grig-orieva, and A. A. Firsov, Science , 666 (2004). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V.Dubonos, and A. A. Firsov, Nature , 197 (2005). A. K. Geim and K. S. Novoselov, Nat. Mater. , 183 (2007). J. M. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen,M. Saleh, X. L. Feng, et al., Nature , 470 (2010). K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B , 17954 (1996). Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. , 216803 (2006). S. Ihnatsenka, I. V. Zozoulenko, and G. Kirczenow, Phys. Rev. B , 155415 (2009). A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B , 085423 (2008). Q. M. Yan, B. Huang, J. Yu, F. W. Zheng, J. Zang, J. Wu, B. L. Gu, F. Liu, and W. H. Duan,Nano Lett. , 1469 (2007). M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. , 206805 (2007). T. G. Pedersen, C. Flindt, J. Pedersen, N. A. Mortensen, A.-P. Jauho, and K. Pedersen, Phys.Rev. Lett. , 136804 (2008). A. H. Zhang, H. F. Teoh, Z. X. Dai, Y. P. Feng, and C. Zhang, Appl. Phys. Lett. , 023105(2011). J. Y. Cheng, C. A. Ross, H. I. Smith, and E. L. Thomas, Adv. Mater. , 2505 (2006). M. C. Lemme, D. C. Bell, J. R. Williams, L. A. Stern, B. W. H. Baugher, P. Jarillo-Herrero, andC. M. Marcus, ACS Nano , 2674 (2009). J. Taylor, H. Guo, and J. Wang, Phys. Rev. B , 245407 (2001). M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys. Rev. B , 165401(2002). J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal,J. Phys.-Condes. Matter , 2745 (2002)., 2745 (2002).