Current-voltage (I-V) characteristics of armchair graphene nanoribbons under uniaxial strain
CCurrent-voltage (I-V) characteristics of armchair graphene nanoribbons underuniaxial strain
M. Topsakal, V. M. K. Bagci, and S. Ciraci
1, 3, ∗ UNAM-Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan Department of Physics, Bilkent University, Ankara 06800, Turkey (Dated: June 7, 2010)The current-voltage (I-V) characteristics of armchair graphene nanoribbons under a local uniaxialtension are investigated by using first principles quantum transport calculations. It is shown thatfor a given value of bias-voltage, the resulting current depends strongly on the applied tension. Theobserved trends are explained by means of changes in the band gaps of the nanoribbons due to theapplied uniaxial tension. In the course of plastic deformation, the irreversible structural changesand derivation of carbon monatomic chains from graphene pieces can be monitored by two-probetransport measurements.
PACS numbers: 72.80.Vp, 62.25.-g, 77.80.bn
I. INTRODUCTION
Graphene, as a (2D) monolayer honeycomb structureof carbon, has attracted a great deal of interest since itssuccessful preparation in 2004. Due to its unique me-chanical, structural and electronic properties, graphenehave been realised as an important material for nu-merous theoretical investigations and promising applica-tions. Among these are charge carriers behaving as mass-less Dirac fermions, Klein tunneling, ballistic trans-port at room temperature, and anomalous quantumHall effects. From experimental points of view, field-effect transistors , micromechanical resonators , gassensors of graphene have already been proposed. Mostof these are directly related with its transport properties.Earlier transport studies predict that spin-valve de-vices based on graphene nanoribbons can exhibit mag-netoresistance values that are thousands of times higherthan previously reported experimental values. Unusualeffects of dopings on the transport properties of graphenenanoribbons were also reported.
Nevertheless, thetransport properties of graphene nanoribbons under uni-axial tension have not been fully explored even from thetheoretical points of view. While the effect of strainon the electronic properties of graphene is becoming anactive field of study, the transport properties and I-V characteristics of nanoribbons under local or uniformstrain is of crucial interest for development of future de-vice applications.In this study, based on state-of-the-art first-principlesquantum transport calculations, we investigate the ef-fects of uniaxial strain on the current-voltage (I-V) char-acteristics of graphene nanoribbons. We showed thatelastic strain can alter the electron transport propertiesdramatically. In some cases, under a 10% strain, thecurrent can change as much as 400-500 %. However, thevariation of current with strain is sample specific. Evenmore remarkable is that the chain formation of carbonatoms from the graphene nanoribbons undergoing a plastic deformation can be monitored through I-V char-acteristics showing negative differential resistance. II. MODEL AND METHODOLOGY
Geometry relaxations and electronic structures arecalculated by using SIESTA package, which uses nu-merical atomic orbitals as basis sets and Troullier-Martin type norm-conserving pseudopotentials. Theexchange-correlation functional of the generalized gradi-ent approximation is represented by the Perdew-Burke-Ernzerhof approximation. A 300 Ryd mesh cut-off ischosen and the self-consistent calculations are performedwith a mixing rate of 0.1. The convergence criterion forthe density matrix is taken as 10 − . Brillouin zone (BZ)sampling of the calculations have been determined afterextensive convergence analysis. The conjugate gradientmethod is used to relax all the atoms until the maximumabsolute force was less than 0.05 (eV/˚A). Interactionsbetween adjacent graphene layers is hindered by a largespacing of ∼
10 ˚A.The electronic transport properties are studied by thenon-equilibrium Green’s function (NEGF) techniques,within the Keldysh formalism, based on density func-tional theory (DFT) as implemented in the TranSIESTA(Ref. 22) module within the SIESTA (Ref. 19) pack-age. A single ζ -plus-polarization basis set is used. Testcalculations with larger basis set and mesh cut-off werealso performed, which give almost identical results. Thecurrent through the contact region was calculated usingLandauer-Buttiker formula, I ( V b ) = G (cid:90) µ L µ R T ( E, V b ) dE, (1)where G = 2( e /h ) is the unit of quantum conductanceand T ( E, V b ) is the transmission probability of electronsincident at an energy E through the device under the po-tential bias V b . The electrochemical potential difference a r X i v : . [ c ond - m a t . m e s - h a ll ] J un between the left and right electrodes is eV b = µ L − µ R . III. I-V CHARACTERISTICS UNDER ELASTICSTRAIN
The band gaps of armchair graphene nanoribbons(AGNRs), which we consider in this study, depend ontheir widths, which are conventionally specified accord-ing to the number of dimer lines, N in their primitiveunit cell. AGNR( N )’s are grouped into three families,namely N = 3 m − N = 3 m family having medium band gaps, and N = 3 m +1 family having largest gaps, where m is an in-teger. Band gaps of each family decrease with increasing m and eventually goes to zero as m → ∞ . AGNRs arenonmagnetic direct band gap semiconductors, which can,however, be modified by vacancies and impurities. The NEGF technique used to study the electronictransport employs a two-probe system; semi-infinite left-and right-electrode regions are in contact with a confinedcentral scattering region. A two-probe system, specificto AGNR with N = 8, but representative of any N , isshown in Fig. 1 (a). Both electrodes and the centralregion are made from AGNR(8). Periodic boundary con-ditions were imposed on the plane perpendicular to theaxis of the nanoribbon. The carbon atoms at the edgesare saturated with H atoms. The central region con-tains 5 primitive unit cells, with a total length of 21.76˚A (= 5 c ). The length of the central region is sufficientenough to avoid an abrupt change in electronic struc-ture while progressing from the electrode region to thestrained region of interest.We first consider the electronic transport properties ofthe unstrained two-probe system presented in Fig. 1 (a).To provide for an intuitive understanding of the trans-port phenomena, the band structure of the electrodesor the scattering region in their primitive unit cell areshown in Fig. 1 (b). The lowest conduction and highestvalance bands originate from π ∗ - and π -states, respec-tively. Unlike the perfect 2D graphene, where π - and π ∗ -bands cross at the K -corners of BZ, AGNR(8) is adirect band gap material having 0.20 eV band gap value.The calculated zero-bias transmission spectrum is givenin Fig. 1 (c), which apparently mimics the band struc-ture of AGNR(8). There is a region of zero-transmissionwith a width of 0 .
20 eV and located around the Fermilevel, coinciding with the band gap of AGNR(8). Like-wise, the step-like behavior of the spectrum is relatedwith the available conductance channels due to bands.The current as a function of the applied bias voltage V b is presented in Fig. 1 (d). For this type of calculations,we increased V b in steps of 0.1 V and used the convergeddensity matrix of the previous state as an initial guess forthe next step. Applying a bias voltage shifts the Fermilevel of the left-electrode with respect to the Fermi levelof the right-electrode. The current starts flowing oncethe top of the valence band of the left-electrode matches Figure 1: (Color Online) (a) Schematic view of two-probearmchair graphene nanoribbon system AGNR(8), having N =8 dimers along the z -axis. Central scattering region, leftand right electrodes are indicated. Carbons atoms are repre-sented by large (brown) and H atoms by small (light) balls.Primitive unit cell of electrodes and scattering region are thesame and represented by dashed lines. The lattice constant ofthe primitive unit cell is c , and that of the central scatteringregion is 5 c . (b) Band structure of AGNR(8) in its primitiveunit cell. (c) Transmission spectrum of the system shown in(a) under zero-bias voltage. (d) I-V plot of AGNR(8) for abias voltage from 0 to 2.5 V. (e) The transmission spectrumsfor 4 different bias voltage calculated for the system shown in(a). in energy with the bottom of the conduction band of theright-electrode, as expected from the T(E, V b = 0 ) forlow-bias values. The T(E, V b ) does not alter much withthe bias, since it is a uniform system and no significantpermanent charge migrations should occur. This is evi-dent from the linear response of the current to the biasvoltage for values of V b > . V b ) contributing to the current always keep near 1 G , and the higher values in transmission values movefurther away from the Fermi energy. This is due to thefact that as the bands of the leads move up or down in en-ergy scale with the varying bias, only a single conductionchannel is open, or in other words, only one band crossesthe energy of interest, at either one or both of the leads.This holds true for the bias voltages we consider in thisstudy and as a result, we see a linear current response tovoltage for zero-strain.Earlier, we have investigated the elastic and plastic de-formation of graphene and its nanoribbons under uniaxialtension. Mechanical properties were revealed from thestrain energy, E S = E T ( (cid:15) ) − E T ( (cid:15) = 0); namely, the totalenergy at a given uniaxial strain (cid:15) minus the total energyat zero-strain. Here, the uniaxial strain is (cid:15) = ∆ c/c ,where c and c = c + ∆ c are equilibrium and stretchedlattice constants of the nanoribbon, respectively. Thetension force, F T = − ∂E S ( (cid:15) ) /∂c and the force constant κ = ∂ E S /∂c are obtained from the strain energy. Cal-culated elastic constants were in good agreement withavailable experimental data obtained from graphene. Here we consider the I-V characteristics of AGNR(8) un-der a uniform uniaxial tension of the central scatteringregion for 0 ≤ (cid:15) ≤ .
18. The strain is introduced asfollows: The electrode atoms are fixed in their equilib-rium positions while the length of the central region isincreased uniformly by ∆ c . Subsequently, the structureof the central region is fully optimized in a larger super-cell containing also unstrained electrode regions. In thisrespect, our study reveals the effect of a local strain ina long unstrained nanoribbon. The total energy of thesystem is recalculated. The strain energy E S , is obtainedaccording to above definition. E S versus the elongation∆ c , as well as (cid:15) = ∆ c/ c plot for AGNR(8) system isgiven in Fig. 2 (a). The segment of AGNR(8) in the cen-tral scattering region undergoes an elastic deformationup to strain values (cid:15) (cid:39) .
18, where the honeycomb-likestructure is maintained, and the system returns to itsoriginal configuration if the tension is released. However,for higher values of strain, the system deforms plastically,where irreversible structural changes occur and the strainenergy suddenly drops. Further information for this typeof elastic and plastic deformation can be found in Ref.18.In Fig. 2 (b)-(d), we present the I-V plots of stretchednanoribbons. The electrode regions are identical to theno-strain case, but the central region under strain causesthe changes. Once again, due to the band gap of theelectrodes of 0.2 eV in AGNR(8), no current is observedup to a bias voltage of 0.2 V. The current response tobias voltage is linear for low strain, but becomes in-creasingly non-linear for higher strain. It is importantto notice that higher strain in the central region inducesstronger non-uniformity on its geometry and thus on its
Figure 2: (Color Online) (a) The strain energy E S , as a func-tion of elongation ∆ c , or strain (cid:15) = ∆ c/c for AGNR(8) sys-tem. (b) I-V characteristics of AGNR(8) for a bias voltagefrom 0 to 2.5 V for different values of strain. The increaseand decrease trends are shown by arrows. (c) and (d) sameas (b) for different systems, AGNR(6) and AGNR (10), re-spectively. electronic structure as compared to the electrodes. Equi-librium charge transfer may occur and alter the systemsresponse to the non-equilibrium perturbation. This willresult in a varying T(E, V b ) for different values of V b . Itis informative to compare the current values for systemsunder different strain at a given bias voltage. For exam-ple, at V b =2 V, the current is around 62 µA at (cid:15) = 0.It increases to 65 µA at (cid:15) = 0 .
02, but steadily decreasesfor higher strain, having values of 28 µA at (cid:15) = 0 .
12 and
Figure 3: (Color Online) The values of band gaps and currentsof (a) AGNR(8) and (b) AGNR(10) systems calculated as afunction of strain, (cid:15) . The currents are calculated for 2.0 Vbias-voltage. (c) The band structures of AGNR(10) underdifferent strains. The band gaps are shaded and Fermi energyis set to zero. µA at (cid:15) = 0 .
18. One also notes that the I-V curve inFig. 1 (d), which is almost linear for (cid:15) = 0, starts to loseits linearity for higher values of strain as seen from Fig. 2(b).Other ribbons such as AGNR(6) and AGNR(10) whoseI-V characteristics are given in Fig. 2 (c) and (d).AGNR(6) belongs to the N = 3 m family and it has alarger band gap ( (cid:39) (cid:15) = 0 .
10, then starts to increase as seen from Fig. 2(c). AGNR(10) is another system which has a band gapvalue around 1.00 eV and its I-V characteristics are givenin Fig. 2 (d). In contrast to AGNR(6), the current firstincreases until (cid:15) = 0 .
08 and then starts to decrease forhigher strain values. All these results in Fig. 2 (b),(c),and (d) show that the current passing through nanorib-bons is very sensitive to the strain values and the behav-iors of I-V curves are sample specific.The increase and decrease of the currents given inFig. 2 due to the changes in the strain is directly relatedwith the electronic structure of the central scattering re-gion, which is modified as a result of changes in atomicstructure under tension.
In Fig. 3 (a) we show thevariation of current and band gap of AGNR(8) with ap-plied uniaxial strain. Here current values are extractedfrom Fig. 2 (b) for 2 V bias voltage. As seen from theplots, there is an inverse relationship between the cur-rent and band gap values. Any increase in the band gapdecreases the current and vice versa. The same analy-sis performed for AGNR(10) in Fig. 3 (b) also confirmsthis relationship. The changes in the band structures ofAGNR(10) can also be followed from Fig. 3 (c), where the lowest conduction and highest valence bands approach toeach other until (cid:15) (cid:39) .
09 and then move away for highervalues of strain. The band gap variations occur due todifferent nature of bands around the conduction and va-lence band edges exhibiting different shifts with strain.In particular, note that π ∗ - and π -bands of AGNR(10)cross linearly at Γ-point by closing the band gap. This isthe realization of massless Dirac Fermion behavior in ananoribbon, which is semiconductor under zero-strain. In a simple model, an electron is ejected from the leftelectrode at an energy value lower than the shifted va-lence band maximum, for available ranges within the biasvoltage. It is incident upon the central region, with lowerchemical potential, and tunnels through to the right elec-trode, still lower in energy. The smaller the band gapvalue for the central region, the larger the number ofpossible states that participate in the tunneling, thus thelarger is the value of the current.
IV. TRANSPORT PROPERTIES OF AGNR(8)UNDER PLASTIC DEFORMATION
While the elastic deformation imposes changes in theband gap and current values, the onset of plastic defor-mation results in dramatic changes in the structure. Af-ter yielding, the modification of honeycomb structure issomehow stochastic and sample specific. It depends onthe conditions, such as the defects in the sample, thetemperature effects and the rate of stretching. However,it has been shown theoretically that under certain cir-cumstances a long carbon atomic chain (identified ascumulene having double bonds and polyyne with alter-nating triple and single bonds) can form in the courseof plastic deformation of graphene, unless the edges ofAGNR is not terminated with hydrogen. Upon furtherstretching, each carbon atom of graphene implementedto chain results in a stepwise elongation of the chain be-tween two graphene pieces. Monatomic carbon chain,which was derived experimentally from graphene, canbe a potential nanostructure for various future applica-tions. The important issue we address here is how thesesequential structural changes reflects the transport prop-erties.In Fig. 4 (a) we present the atomic structure of a two-probe graphene nanoribbon system which is formed afterthe plastic deformation of AGNR(8) nanoribbon. A shortchain containing 4 carbon atoms between the grapheneflakes is formed in the scattering region and its zero-biastransmission spectrum is presented in Fig. 4 (b). Thisspectrum is composed of peaks rather than step-like lev-els as in Fig. 1 (c). The calculated I-V plot in Fig. 4(c) also contains some peaks, which may lead to neg-ative differential resistance. Similar situation also ex-ists for a longer chain in Fig. 4 (d), which occurred ata more advanced stage of plastic deformation wherebythe nanoribbon in the central region is more stretchedthan in Fig. 4 (a). At the end, two more carbon atoms
Figure 4: (Color Online) (a) Schematic view of AGNR(8) system which is deformed plastically due to the high values of strain( (cid:15) > . are included to the chain. The differences between zero-bias transmission curves in Fig. 4(b) and (e) occur be-cause of the energy level diagram and their positions rel-ative to Fermi levels are different. Also, the I-V curvecorresponding to two carbon chains of different lengthsoccurring in subsequent stages of stretching are ratherdifferent. We note that the conductance of the longercarbon chain in Fig. 4(d) and the corresponding currentvalues (I) of a given bias voltage (V) can be higher thanthe shorter chain in Fig. 4(a). This paradoxical situa-tion is related with the fact that some energies of thechannels can be closer to the Fermi level as the lengthof the chain increases. Further stretching of the sys-tem shown in Fig. 4(d) can produce longer carbon chainstructures. The length of these chains can be as long as10 carbon chains. As found for the structures in Fig. 4(a)and Fig. 4(d), the I-V characteristics of the longer car-bon chains will be different and will allow one to monitorthe structural changes. Finally the plastic deformationterminates upon breaking of the chain.
V. CONCLUSION
We have shown that the transport properties of thesegment of an armchair graphene nanoribbon in a two- probe system can be modified with uniaxial strain. Thecurrent under a fixed bias can change several times withapplied uniaxial strain. However, these changes are sam-ple specific and related with strain induced changes in theelectronic structure near the band gap. Irreversible struc-tural changes and the formation of monatomic carbonchain between graphene pieces in the advanced stages ofplastic deformation can be monitored through two-probetransport experiments. We believe that our findings areof crucial importance for recent active studies aiming toreveal the effects of strain on the electronic properties ofgraphene. Also our results suggest that these systemscan be used as nanoscale strain gauge devices.
VI. ACKNOWLEDGEMENT
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