Negative Differential Resistance and Steep Switching in Chevron Graphene Nanoribbon Field Effect Transistors
Samuel Smith, Juan-Pablo Llinás, Jeffrey Bokor, Sayeef Salahuddin
NNegative Differential Resistance and Steep Switching in Chevron GrapheneNanoribbon Field Effect Transistors
Samuel Smith, Juan-Pablo Llin´as, Jeffrey Bokor, Sayeef Salahuddin University of California, Berkeley, Department of Electrical Engineering and Computer Sciences, Berkeley,CA 94720
Ballistic quantum transport calculations based on the non-equilbrium Green’s function formalism show thatfield-effect transistor devices made from chevron-type graphene nanoribbons (CGNRs) could exhibit negativedifferential resistance with peak-to-valley ratios in excess of 4800 at room temperature as well as steep-slope switching with 6 mV/decade subtheshold swing over five orders of magnitude and ON-currents of88 µ A µ m − . This is enabled by the superlattice-like structure of these ribbons that have large periodic unitcells with regions of different effective bandgap, resulting in minibands and gaps in the density of states abovethe conduction band edge. The CGNR ribbon used in our proposed device has been previously fabricatedwith bottom-up chemical synthesis techniques and could be incorporated into an experimentally-realizablestructure.In 1970, L.Esaki and R. Tsu predicted that in an ap-propriately made superlattice, it should be possible toobtain very narrow width bands, which could then leadto negative differential resistance. The remarkable prop-erty of these superlattices is in the fact that, unlike theEsaki diodes, this negative differential resistance does notneed any tunneling, rather it comes from the direct con-duction of electrons. Nonetheless, significant difficultyin synthesizing atomically precise, eptiaxial heterostruc-tures has made it very challenging to realize such su-perlattice structures . Much work has been done onmodeling graphene nanoribbon heterostructures and su-perlattices which could exhibit NDR . Other workhas been done on steep slope devices based on GNR andCNT heterojunctions . Gnani et al. showed how su-perlattices could be used in a III-V nanowire FET toachieve steep slope behavior by using the superlatticegap to filter high energy electrons in the OFF state .Here, we show that the recently synthesized chevronnanoribbons provides a natural, monolithic materialsystem where narrow-width energy bands and negativedifferential resistance (NDR) can be achieved. Our atom-istic calculations predict that the NDR behavior shouldmanifest at room temperature along with sub-thermalsteepness ( <
60 mV/decade at room temperature). SuchNDR behavior could lead to completely novel devices fornext generation electronics.Unlike a graphene sheet, a narrow strip etched out ofgraphene, often called a graphene nanoribbon (GNR),can provide a sizeable bandgap. As a result, GNRscould lead to devices with good ON/OFF ratio at thenanoscale. However, a number of studies have also shownthe deleterious effect of edge roughness on the deviceperformance . Recent breakthroughs in bottom-upchemical synthesis can produce GNRs with atomisticallypristine edge states and overcome this shortcoming .In fact, a recent experimental work demonstrated work-ing transistors with 9- and 13-AGNRs made with thesetechniques . The methods used to synthesize theseribbons can also be used to generate complex pe-riodic structures beyond simply armchair and zigzagnanoribbons . In this work, we will consider one of those structures, the chevron graphene nanoribbon(CGNR).Fig. 1 shows both the atomic structure of the 6-9CGNR originally fabricated by Cai et al. and the elec-tronic structure calculated through a p z orbital-basedtight-binding method . A key feature of the band struc-ture is the presence of minibands with regions of forbid-den energy above the conduction band edge, such as thoseseen in superlattices of III-V semiconductors. Analogousto III-V superlattices, the CGNR contains regions of dif-ferent effective bandgap. When we look at the CGNRin Fig. 1, we see that its narrowest segment is 6 car-bon atoms across and its widest segment is 9 carbonatoms across, with both segments having armchair-typeedges. Using a pz-basis set (GW ), the bandgap, E g , ofa 6-AGNR is 1.33 eV (2.7 eV) and the bandgap of a 9-AGNR is 0.95 eV (2.0 eV). However, given the very shortlength scale over which the width changes in our struc-ture ( ∼ E g for the isolated AGNRs. In fact,our chevron structure has an overall bandgap of 1.59 eV.This value is consistent with the 1 .
62 eV bandgap fromLDA DFT calculations, but significantly smaller than the3 .
74 eV value from calculations incorporating the GWcorrection . Both LDA and GW calculations show thepresence of minibands and gaps above the conductionband edge .The structure of the simulated device is shown in inFig. 2. Like a typical MOSFET, our superlattice field-effect transistor (SLFET) can be turned ON and OFFwith a gate voltage at low drain biases. Operation dif-fers from a MOSFET in two key ways. The first is thatthe device shows NDR with respect to the drain volt-age. At some value of V ds determined by the width ofthe first miniband, I d decreases substantially when theconduction band at the source becomes aligned with thesuperlattice gap at the drain. At higher drain bias, cur-rent increases again when the conduction band at thesource is aligned with the second miniband at the drain.The second feature of the SLFET is that the superlat-tice gap at the drain filters out higher energy electrons a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r from the first miniband at the source when a source-drainbias is applied. This cuts off the higher energy portion ofthe thermionic tail at the source, which would contributeto leakage current in a traditional MOSFET. This filter-ing does not, however, affect the low-energy electrons,which carry most of the ON state current as they are inthe window where the first minibands at the source anddrain overlap. Transport in an SLFET is entirely intra-band like a MOSFET, whereas a TFET relies on band-to-band tunneling. This could possibly allow higher ONcurrent than a TFET.The CGNR used in our simulation has a width of 1.9nm. The simulation domain is approximately 70 nm long,and the gate has a length of 15 nm. The source and drainare doped with N D = 1 . × cm − donors. An ef-fective oxide thickness of 1 . . A simple p z -basisis used for the Hamiltonian for the chevron graphenenanoribbon with the hopping parameter set to t =2 . , and contact self-energyis computed with the Sancho-Rubio iteration scheme .The NEGF equations are solved self-consistently with
1. 0 0. 5 0. 0 0. 5 1. 0 k (normalized) E ( e V ) FIG. 1. Band structure of a chevron graphene nanoribbonbased on a p z orbital basis set. The width (and thus quantumconfinement) varies across the unit cell, giving a superlattice-like band structure. Forbidden energies are highlighted inred. The bandgap of the ribbon is 1.59 eV, the first con-duction band has a bandwidth of 0.272 eV, and the first gapbetween minibands is 0.178 eV. Inset: Molecular structure ofthe chevron nanoribbon. E n e r g y Transmission (V s = 0) Transmission (V ds = 0)
21 21
Transmission (V ds > qE mb1 ) Current at low V ds Reflection at higher V ds Source DrainE mb1
FIG. 2. Artistic rendering of double-gate CGNR on insulatorSLFET. Parts of the top gate and oxide region have been cutaway so that the channel is visible. When a gate voltage isapplied to turn the device ON, current conduction occurs atlow values of V ds where the first miniband at the source isaligned with the first miniband at the drain. As the drainvoltage is increased beyond qE mb , the bandwidth of the firstminiband, transmission from source to drain is cut off and thedevice exhibits negative differential resistance. the Poisson equation for electrostatics in three dimen-sions. Our simulator solves the nonlinear Poisson equa-tion using the predictor-corrector scheme described byTrellakis et al . using a semiclassical approximation forthe charge density . The geometry of the system is mod-eled using a tetrahedral finite element mesh generatedwith the SALOME package . The solution of the finalsparse matrix form of the Poisson equation discretizedthe with finite element method is performed using theconjugate gradient solver from the Eigen library .The local density of states for the CGNR MOSFETis shown in Fig. 3 for several biasing conditions. Fig.3b shows the case for peak current for the device whena large enough drain bias has been applied to generateenough splitting between the source and drain Fermi lev-els to allow significant current to flow, but not a highenough bias to move the the first miniband outside ofthe current conduction window. For higher bias as in Fig.3c, intraband conduction from the first miniband is com-pletely cut off. As the drain bias is further increased, cur-rent can only flow due to a band-to-band tunneling fromthe first miniband at the source to the second minibandat the drain. Note that, due to the minibands, there willbe regions of operation for both gate and drain voltageswhere current flow is abruptly turned on or off, as theoverlap between source minibands and drain minibandsis modified. This leads to a steep subthreshold swing( <
60 mV/decade at room temperature) in the I d − V g x (nm)
0. 40. 20. 00. 20. 40. 60. 8 E ( e V ) (a) V gs = 0 .
55 V , V ds = 0 .
10 V x (nm)
0. 40. 20. 00. 20. 40. 60. 8 E ( e V ) (b) V gs = 0 .
70 V , V ds = 0 .
10 V x (nm)
0. 40. 20. 00. 20. 40. 60. 8 E ( e V ) (c) V gs = 0 .
70 V , V ds = 0 . FIG. 3. Local density of states for several different biasing conditions. Fig. (a) shows the OFF state, in which leakage currentis substantially reduced because the superlattice gap in the drain region filters higher energy electrons, which could otherwisetravel over the source-side barrier. Fig. (b) shows the ON state, in which current is primarily carried by lower energy carriers,which are not blocked through the density of states filtering at the drain. Fig. (c) shows the ON state for a higher value of V ds . Significant ballistic transport from source to drain is no longer possible when the drain voltage is greater than the widthof the first miniband minus the height of the source-side barrier. The colormap is based on a logarithmic scale.
0. 5 0. 6 0. 7V gs (V)10 -12 -9 -6 -3 I d ( n A ) V ds =0.10 VV ds =0.20 VV ds =0.30 VV ds =0.40 V FIG. 4. I d − V gs plot for different values of V ds . Steep-slopebehavior is observed with a subtheshold swing of around 6mV/decade over five orders of magnitude around V gs = 0 . V ds = 0 . I d is lower for higher valuesof V ds . characteristic and a negative differential resistance in the I d − V d characteristic.We shall first discuss the current vs gate voltage( I d − V g ) characteristics. Fig. 4 shows I d vs. V gs forseveral values of V ds . While steep slope behavior is ex-hibited at some point for all values of V ds , the highest ONcurrent is obverved for V ds = 0 . µ A µ m − ) is achieved at agate bias of V gs = 0 .
75 V. In the steep slope region of thiscurve, the subthreshold swing is 6 mV/decade when aver-aged over five orders of magnitude of I d . With gate workfunction engineering and additional device optimization, it should be possible to achieve reasonable ON currentwith a low supply voltage in devices of this type. Theorigin of the steep-slope behavior can be understood fromFig. 3. In the OFF state shown in Fig. 3a, the super-lattice gap at the drain prevents leakage current fromflowing over the source-side injection barrier. The statesnear the top of the barrier are seen to decay rapidly inthe drain region. Fig. 3b shows the ON state, in whichlow-energy electrons, which make up virtually all of theON current, can flow unimpeded from source to drain.The I d − V ds curves from the results of our simulationare shown in Fig. 5. Considering the case when V gs =0 . V ds = 0 .
10 V.As the drain bias is further increased, we see a decreasein current as the drain miniband goes out of alignmentwith the source miniband. The current beigins to pick upagain as the second miniband at the drain starts to comein alignment with the source miniband again. The peak-to-valley ratio (PVR) at this gate voltage is 4 . × .At V gs = 0 .
60 V, the calculated PVR is 1 . × . Note,however, that this value is expected to much smaller ina practical device due to the electron-phonon scatteringmechanisms that were not taken into account in ourballistic simulations.In summary, we have shown that chevron graphenenanoribbon devices can exhibit both steep-slope sub-threshold behavior and negative differential resistance.Both properties are the result of the superlattice-like elec-tronic structure of the ribbon. CGNR SLFETs couldbe promising for a number of applications ranging fromlow-power logic transistors to high speed oscillators. Amajor obstacle to building a real device is making con-tacts with appropriate Schottky barrier heights to beable to match the band alignment conditions achievedin this work through a simple doping model. The per-formance of a real device would also likely be impactedby scattering mechanisms we have not considered here, FIG. 5. I d − V ds plot for different values of V gs . For thecase, when V gs = 0 . . × isachieved. though the ability to synthesize ribbons with virtuallyno defects may minimize these effects. Additional op-timization will also likely be necessary to make a func-tioning device. DFT+GW calculations predict a muchhigher bandgap for the CGNR in vacuum than the tight-binding model used in this work. While surface screeningmay reduce the bangap somewhat, a wider ribbon with anarrower bandgap may be required. Co-optimization ofthe bandgap with the bandwidths of the minibands andthe gaps between minibands is also a necessary topic forfuture work. ACKNOWLEDGEMENTS
This work was supported by NSF CAREER grantCISE-1149804. Work by JPL and JB was supportedin part by the Office of Naval Research BRC programunder Grant N00014-16-1-2229. The authors would liketo acknowledge useful discussions with Felix Fischer andMichael Crommie. L. Esaki and R. Tsu, IBM Journal of Research and Development , 61 (1970). L. Esaki and L. Chang, Physical Review Letters , 495 (1974). K. Choi, B. Levine, R. Malik, J. Walker, and C. Bethea, PhysicalReview B , 4172 (1987). K. Ismail, W. Chu, A. Yen, D. Antoniadis, and H. I. Smith,Applied physics letters , 460 (1989). G. Bernstein and D. Ferry, Journal of Vacuum Science & Tech-nology B: Microelectronics Processing and Phenomena , 964(1987). H. Grahn, R. Haug, W. M¨uller, and K. Ploog, Physical reviewletters , 1618 (1991). J. Kastrup, H. Grahn, K. Ploog, F. Prengel, A. Wacker, andE. Sch¨oll, Applied physics letters , 1808 (1994). A. Warren, D. Antoniadis, H. I. Smith, and J. Melngailis, IEEEelectron device letters , 294 (1985). G. J. Ferreira, M. N. Leuenberger, D. Loss, and J. C. Egues,Physical Review B , 125453 (2011). S. Li, C. K. Gan, Y.-W. Son, Y. P. Feng, and S. Y. Quek, AppliedPhysics Letters , 013302 (2015). H. Teong, K.-T. Lam, S. B. Khalid, and G. Liang, Journal ofApplied Physics , 084317 (2009). H. Sevin¸cli, M. Topsakal, and S. Ciraci, Physical Review B ,245402 (2008). M. Sharifi, E. Akhoundi, and H. Esmaili, Journal of Computa-tional Electronics , 1 (2016). G. Saha, A. K. Saha, and A. H.-u. Rashid, in
Nanotechnology(IEEE-NANO), 2015 IEEE 15th International Conference on (IEEE, 2015) pp. 440–443. E. C. Girao, E. Cruz-Silva, and V. Meunier, ACS nano , 6483(2012). S. Kim, M. Luisier, T. B. Boykin, and G. Klimeck, AppliedPhysics Letters , 243113 (2014). Y. Yoon and S. Salahuddin, Applied Physics Letters , 033102(2010). E. Gnani, P. Maiorano, S. Reggiani, A. Gnudi, and G. Baccarani,in
Electron Devices Meeting (IEDM), 2011 IEEE International (IEEE, 2011) pp. 5–1. J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blanken-burg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. M¨ullen,and R. Fasel, Nature , 470 (2010). G. Fiori and G. Iannaccone, IEEE Electron Device Letters ,760 (2007). Y. Yoon and J. Guo, Applied Physics Letters , 073103 (2007). J. P. Llin´as, A. Fairbrother, G. Barin, P. Ruffieux, W. Shi,K. Lee, B. Y. Choi, R. Braganza, N. Kau, W. Choi, Z. Pedram-razi, T. Dumslaff, A. Narita, X. Feng, K. M¨ullen, F. Fischer,A. Zettl, P. Ruffieux, E. Yablonovitch, M. Crommie, R. Fasel,and J. Bokor, arXiv preprint arXiv:1605.06730 (2016). J. Cai, C. A. Pignedoli, L. Talirz, P. Ruffieux, H. S¨ode, L. Liang,V. Meunier, R. Berger, R. Li, X. Feng, K. M¨ullen, and R. Fasel,Nature nanotechnology , 896 (2014). Y.-C. Chen, T. Cao, C. Chen, Z. Pedramrazi, D. Haberer, D. G.de Oteyza, F. R. Fischer, S. G. Louie, and M. F. Crommie,Nature nanotechnology , 156 (2015). L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, and S. G. Louie,Physical Review Letters , 186801 (2007). S. Wang and J. Wang, The Journal of Physical Chemistry C ,10193 (2012). S. Datta,
Quantum transport: atom to transistor (CambridgeUniversity Press, 2005). R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, Journalof Applied Physics , 7845 (1997). M. L. Sancho, J. L. Sancho, J. L. Sancho, and J. Rubio, Journalof Physics F: Metal Physics , 851 (1985). A. Trellakis, A. Galick, A. Pacelli, and U. Ravaioli, Journal ofApplied Physics , 7880 (1997). A. Ribes and C. Caremoli, in
Computer Software and Applica-tions Conference, 2007. COMPSAC 2007. 31st Annual Interna-tional , Vol. 2 (IEEE, 2007) pp. 553–564.32