Mechanical Manipulations on Electronic Transport of Graphene Nanoribbons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Mechanical Manipulations on Electronic Transport of Graphene Nanoribbons
Jing Wang , Guiping Zhang , ∗ Fei Ye , and Xiaoqun Wang , , , † Department of Physics, Renmin University of China, Beijing 100872, China Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China Beijing Laboratory of Optoelectronics Functional Materials and Micronano Device,Renmin University of China, Beijing 100872, China Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China (Dated: September 13, 2018)We study the effects of uniaxial strains on the transport properties of the graphene nanorib-bons(GNRs) connected with two metallic leads in heterojunctions, using the transfer matrix method.Two typical GNRs with zigzag and armchair boundaries are considered, and the tension is appliedeither parallel or perpendicular to the ribbon axis. It turns out that the electron-hole symmetry ismissing in the gate voltage dependence of the conductance data of the armchair GNRs, while it per-sists in the zigzag ribbons under any strains. For an armchair GNR with a vertical tension applied,a sharp drop of conductance is found near the critical value of the strain inducing a quantum phasetransition, which allows to determine the critical strain accurately via measuring the conductance.In the zigzag ribbon, there exists a range of gate voltage around zero, where the conductance isinsensitive to the small horizontal strains. The band structures and low-energy properties are cal-culated to elucidate the mechanism on the strain effects in GNRs. We expect that our results canbe useful in developing graphene-based strain sensors.
PACS numbers: 72.80.Vp; 73.22.Pr; 74.25.F-; 73.40.Sx
I. INTRODUCTION
Graphene has attracted widespread interests boththeoretically and experimentally since its discovery in2004, because of its unique properties and promisingapplications[1–3]. Recently, there are many studies onits mechanical deformation[4, 5] and the correspondingeffects on the Raman spectroscopy[6], since the strain isinevitable for the fabrication of graphene on substrate.Unlike the conventional materials, graphene has a toughmechanical property and could sustain elastic deforma-tion up to 15 ∼ µ m to 1.5 µ m. When a free-standing monolayer graphene is transferred onto it, anonlinear strain-stress relation is observed by nanoin-dentation in the atomic force microscope[7], which hasbeen verified theoretically[8]. The other one is by exert-ing tension on the subtrate to control the strain on thegraphene[6, 9, 10]. Graphene ripples on polydimethyl-siloxane(PDMS) substrate can afford a reversible struc-tural deformation under tensile strains as large as 20 ∼ ∗ Electronic address: bugubird˙[email protected] † Electronic address: [email protected]
In contrast, a much smaller uniaxial strain can con-trol (close or open) the band gap in the narrow arm-chair graphene nanoribbons(AGNRs), while the zigzaggraphene nanoribbons(ZGNRs) are quite robust againstgap opening for small strain [13–15].In recent years, the field of graphene-based strain sen-sors develops rapidly, since it is feasible to mediate elec-tronic properties of graphene by applying tensions. In asample of graphene from chemical vapor deposition, theresistance remains around 7.5 K Ω under the strain lessthan 2.47% applied along the electronic transport direc-tion, while increases rapidly to 25 K Ω under 5% strain[9].This is because that the ripples in graphene do not dis-appear until the strain exceeds 2.47%. The strain depen-dent transport properties enable graphene to have po-tential applications in the fields of the displays, robotics,fatigue detection, body monitoring, and so forth. Forinstance, the graphene-based strain sensors on the trans-parent gloves can measure the magnitudes and directionsof the principal strains on the glove induced by the mo-tion of fingers[10].The previous theoretical investigations on the trans-port properties of the strained graphene nanoribbons[16–22] mostly deal with the small-scale GNRs with a widthof about several nanometers using homojunction con-tacts, while one may encounter more complicated situa-tions in practice, e.g., heterojunction contacts and widerGNRs in the fabrication of the GNR-based nanodevices.In this paper, we utilize a transfer matrix method[23] tostudy the transport properties of both narrow and wideGNRs under the strain, which are in particular connectedto two metallic leads with heterojunctions. The width ofgraphene can reach the order of microns by means of thetransfer matrix method[23]. A tight binding model istaken to describe the low energy physics for both the π -electrons of graphene and metallic electrons in two leads.The effects of strains on the hoping integral of C-C bondsin graphene are elucidated in Sec.II. The band structuresof AGNRs and ZGNRs for various sizes under differentstrains are presented in Sec.III. In Sec. IV, we showthe effects of strains on the transport properties of bothAGNRs and ZGNRs. The edge effects are discussed inSec. V. We note that strain only affects the band struc-ture of graphene and the electronic transport of strainedgraphene we present embodies the combined effects ofstrains and the heterojunctions composed of grapheneand metallic contacts. II. MODEL AND METHODA. Strained Graphenes
To investigate the influence of uniaxial strain on theelectronic transport properties of GNRs, we connect itwith semi-infinite quantum wires, which are character-ized by the square lattices, as illustrated in Fig. 1.Each interface between the GNR and a lead is a het-erojunction. For AGNRs in Fig. 1(a), the interface isa ring consisting of five atoms, which eventually breaksthe electron-hole(e-h) symmetry of the system, while forZGNRs in Fig. 1(b), each ring at two interfaces containsfour or six atoms which retains the e-h symmetry sincethe tight binding model involves only the nearest neigh-bor hopping in this paper. This is revealed by the de-pendence of conductance on gate voltages as shown later.The uniaxial tension is only applied to the GNRs leadingto the deformation of C-C bonds in an anisotropic way.The strain-stress relation for graphene is given in Ref.[12]. We quote those relevant results here for our furtherdiscussions. The tension is applied along the directioncos θ~e x + sin θ~e y , and the corresponding tensile strainsparallel and perpendicular to this direction are S and − νS , respectively, with the Poisson’s ratio ν = 0 . ǫ = S (cid:18) cos θ − ν sin θ (1 + ν ) cos θ sin θ (1 + ν ) cos θ sin θ sin θ − ν cos θ (cid:19) . (1)For any vector ~l in the undeformed graphene plane, itis straightforward to obtain its deformed counterpart tothe leading order by the transformation ~l = (1 + ǫ ) ~l . (2)The hopping amplitude t i with i = 1 , , δ i via the following formula[12] t i = t e − . δia − , (3)with t = 2 . eV and a = 1 . A for the undeformedgraphene. The bond length δ i under the strain can be FIG. 1: (Color online.) The schematic illustration of AGNRs(a) and ZGNRs (b), connected to two semi-infinite quantumwires. There are N and M carbon atoms in x and y directions,respectively. (c) AGNRs with the tension along x -axis, i.e. θ = 0, and (d) along y -axis with θ = π/ calculated by Eq. (1) and Eq. (2). Without loss of gen-erality, we focus on two cases with θ = 0 and π/ • For θ = 0 shown in Fig. 1(c), δ = δ = (1 + 14 S − νS ) a, δ = (1 + S ) a. All the three bond lengths increase as S increases,and the t i ’s subsequently decrease for all i = 1 , , δ increases faster than δ and δ . There-fore, we have t = t > t as long as S > • For θ = π/ δ = δ = (1 + 34 S − νS ) a, δ = (1 − νS ) a. In this case, we also have δ = δ and t = t . As S increases, δ , increase, while δ decreases. It turnsout that t and t decrease and t increases withincreasing S > π/ θ = 0 (or π/
2) is identical to that for AGNRs with θ = π/ t i ’s as functions of S are plottedfor AGNRs in Fig. 2(a), in the unit of t , which is set asone in the following discussions. B. Tight-binding Model and Transfer MatrixMethod
The π -electrons of carbon atoms are responsible for thelow energy physics of graphene which can be describedby the tight binding model on the honeycomb latticeˆ H = X h ij,i ′ j ′ i t ij,i ′ j ′ ˆ c † ij ˆ c i ′ j ′ + V g X ij ˆ c † ij ˆ c ij , (4)where a pair of integers ij indicates the lattice position ~R ij = x i ~e x + y j ~e y , and ˆ c ij (ˆ c † ij ) is the corresponding elec-tron annihilation(creation) operator. The summation isover the nearest neighbors indicated by h· · · i , and t ij,i ′ j ′ is the hopping amplitude which takes the value of t , t or t depending on the relative position ~R i ′ j ′ − ~R ij . Thespin indices of electrons are omitted simply for conve-nience. V g is the gate voltage which is applied onlyto the GNRs, not on the leads. In our simulation, weconsider a simplified case with V g changing abruptly atthe interfaces between the GNR and the leads. In fact,this simplification is reasonable for small V g . For large V g , there may exist a junction between the leads andGNR with finite width of several atoms. This situationwould not be considered here, since it only incurs furtherunnecessary complexities as far as the strain effects areconcerned.The left and right electrodes are also described by theHamiltonian in Eq. (4) with V g = 0, but the latticevectors ~R ij describe a rectangular lattice instead of thehexagonal one. All the hopping integrals in the leads arefixed as t , despite the vertical lattice constants may notbe uniform in the leads connected to ZGNRs as shown inFig. 1b. We also assume the leads are unaffected by thestrain in our numerical simulation. This idealized setupmimics a normal-metal/GNR heterojunction, by whichwe shall demonstrate the strain effects on the transportthrough GNRs.The single-particle eigenstate with energy E canbe expressed as ˆ ψ † ( E ) = P ij α ij ˆ c † ij , which satisfies[ ˆ ψ ( E ) , ˆ H ] = E ˆ ψ ( E ), leading to( E − V g ) α ij = X h ij,i ′ j ′ i t ij,i ′ j ′ α i ′ j ′ . (5)The wavefunctions of the electrodes can be representedin terms of two numbers k x and k y , where k x describesthe plane wave traveling along the x direction and k y isquantized as k y,n = nπ/ ( M + 1) with n = 1 , , · · · , M due to the open boundary condition imposed in the y direction, to characterize different channels. The corre-sponding eigenenergy reads E = 2 t (cos k y,n + cos k x,n ) , (6)which determines the wave number k x,n in n -th channelfor the given Fermi energy E . Note that, since the hop-ping amplitudes of all the bonds are given, the lattice constant is not needed anymore and one can simply usedimensionaless wave numbers k x,n and k y,n to label thequantum states.If we assume the electrons are incident from the left,the wavefunctions in the left and right electrodes can bewritten as[23, 25] α Lij = X n ′ ( δ n ′ n e ik x,n ′ x i + r n ′ n e − ik x,n ′ x i ) sin( k y,n ′ y j ) ,α Rij = X n ′ t n ′ n e ik x,n ′ x i sin( k y,n ′ y j ) , (7)where t n ′ n and r n ′ n are the transmission and reflectionamplitudes from n -th to n ′ -th channel, respectively. Cur-rent conservation requires P n ′ η n,n ′ [ | t n,n ′ | + | r n,n ′ | ] =1 for each n with η n,n ′ ≡ | sin( k x,n ′ ) | / | sin( k x,n ) | . In or-der to calculate the transmission coefficients t n,n ′ , weadopt the transfer matrix method developed in Ref. 23,and then we can designate α j for the M coefficients withcolumn index j , which satisfies the matrix equation (cid:18) α j α j +1 (cid:19) = χ j (cid:18) α j − α j (cid:19) , (8)where χ j is the 2 M × M transfer matrix as a function ofFermi energy E , gate voltage V g and the hopping ampli-tudes t i ’s. By acting the transfer matrices consecutively,the coefficients in the left and right interfaces are con-nected in the following form (cid:18) α N α N +1 (cid:19) = χ N χ N − . . . χ χ (cid:18) α α (cid:19) . (9)Combining Eq. (7) and Eq. (9) one can obtain thetransmission and reflection coefficients t n,n ′ and r n,n ′ .In order to investigate the transport properties of thelarge scale GNRs, we actually utilize the renormalizedtransfer matrix method as described in Ref. 26. It isstraightforward to calculate the conductance by employ-ing Landauer-B¨uttiker formula G = 2 e h M X n,n ′ =1 η n,n ′ | t n,n ′ | , (10)where the factor 2 is a consequence of the spin degener-acy. III. BAND STRUCTURE OF STRAINEDGRAPHENE NANORIBBONS
In this section, we study the band structure and lowenergy excitations of GNRs under strains by solving thetight binding model Eq. (4) with the hopping amplitudesgiven by Eq. (3). For convenience, we impose the periodicboundary condition along x -axis and the open boundarycondition along y -axis, and the energy E is taken in unitof t in the following discussions. We also assume thehorizontal lattice spacing to be unit so that k x is alwaysin the interval [0 , π ), although it changes as the tensionis applied. A. Strained AGNRs
The spectra of AGNRs are plotted in Figs. 2(b)-(f)as functions of k x with N = 100 and M = 100 for dif-ferent strains. The unstrained data is given in Fig.2(b)which is precisely gapless at k x = 0. In the presence ofstrains, the spectrum changes upon the direction of theapplied tension. When the tension is applied horizon-tally to AGNRs, i.e. θ = 0, the spectra in Figs. 2(c)and (e) are similar to the unstrained case, except thatthe uniaxial strain may open a small gap at k x = 0. Thisgap is proportional to M − and becomes almost invisiblefor M = 100, which results from the combined effect ofthe finite ribbon widths and the strains. When the ten-sion is applied vertically, i.e. θ = π/
2, the spectrum inFig. 2(d) with strain S = 0 .
15 shows a tiny gap, which isalso proportional to M − with the same origin of that for θ = 0. For S = 0 .
3, another type of gap opens at k x = 0as shown in Fig. 2(f), which is induced entirely by thestrain [12] and can survive the thermodynamic limit un-like the previous gaps. In fact there is a critical strain S separating the two different gaps as to be discussed indetails later.Figures 2(g) and (h) show the density of states (DOS) ρ ( E ) for AGNRs under different strains for θ = 0 and π/
2, respectively. The band width is D = 2 t + t plus anegligible dependence on the ribbon width, which obvi-ously shrinks as S increases for both θ = 0 and θ = π/ t ( t = t ) and t behave very differently asfunctions of S in Fig. 2(a). When θ = 0, t and t aredecreasing function of S so is the band width. Whenthe strain increases for θ = π/ t increases, but it isthe decreasing t that dominates the strain dependenceof the band width. Besides the shrinking band width,there are no other common features in the DOS for bothcases with θ = 0 and θ = π/
2. For the unstrained rib-bons, there are two peaks of DOS located at ± t . Eachof them splits into double peaks if the tension is appliedhorizontally, which locate at ± t and ± (2 t − t ) as seenin Fig. 2(g). When the tension is applied vertically, thepeaks at E = ± t in Fig. 2(h) move slightly outwardsinstead of splitting. However no peaks are observed at E = ± (2 t − t ), except two shoulders emerging at ± . t for S = 0 . S = 0 .
3, the DOS vanishes in the energy range[ − . , . . t ) opens.Now we turn to the dependence of energy gaps at k x =0 on the ribbon widths and strains. For the unstrainedAGNRs, the gap is zero for mod( M,
3) = 1 and inverselyproportional to M for mod( M,
3) = 0 ,
2, which coincideswith previous studies using the first-principle calculation[27] and the tight binding model[23, 28]. This feature ismanifested in Figs. 3(a)-(d) for M = 10, 11, 12, 49 and100 with a fixed ribbon length N = 100.When the tension is applied horizontally with differentribbon widths, the band gaps oscillate with the strains inthe similar zigzag patterns, but with different “phases” -3-2-101230.0 0.5 1.0 1.5 2.0-3-2-10123 0.0 0.5 1.0 1.5 2.0-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 30.000.010.020.03 0.0 0.5 1.0 1.5 2.0-3-2-101234 = /2, t (t )= /2, t (a) =0,t (t ) =0,t (c) AGNRs, =0N=100,M=100,S=0.15 (d)
AGNRs, = /2N=100,M=100,S=0.15 t i E (e) E S k x / AGNRs, =0N=100,M=100,S=0.3 (f) k x / AGNRs, = /2N=100,M=100,S=0.3 (h) E M=100, = /2 S=0 S=0.1 S=0.2 S=0.3 (g)
E(E)
M=100, =0 S=0 S=0.1 S=0.2 S=0.3 (b)
AGNRsN=100,M=100,S=0.0
FIG. 2: (Color online.) (a) shows the strain dependence of thethree hopping amplitudes. (b)-(f) are the band structures ofAGRNs with N = M = 100 under various strains. (g) and (h)are the density of states ρ ( E ) of AGNRs with M = 100 and N = 1600 for the tensions along x and y axis, respectively. according to different values of mod( M,
3) as shown inFig. 3(a). The oscillatory amplitude is inversely propor-tional to M and barely changes with S , and the oscilla-tory frequency increases with M , but decreases with S as shown in Fig. 3(c).For θ = π/ θ = 0.The oscillation only happens for S < S c . In this re-gion, the oscillatory amplitude decreases with both M and S , while the frequency increases with both M and S . The band gaps for both θ = 0 and θ = π/ M as large as 1000. The essential difference occursfor S > S c , where a gap opens for θ = π/ S − S c in the thermodynamiclimit. This is demonstrated with the finite- M scaling fordifferent values of S in Fig. 3(f).In fact, for AGNRs with vertical strains, the gapopening implies a quantum phase transition occurringat S = S c from a Fermi liquid to a dimerized solid phase. (e) (f) N=100,M=1000 =0 = /2 (d)(c) (b)(a) =0,N=100 M=10 M=49 M=100 = /2,N=100 M=10 M=49 M=100
N=100, = /2 S=0 S=0.05 S=0.1 S=0.2 S=0.25 S=0.3 S =0,N=100 M=10 M=11 M=12 SS = /2,N=100 M=10 M=11 M=12 FIG. 3: (Color online) Band gaps of AGNRs as functions ofstrains with various sizes. The tension is applied horizontallyin (a) and (c), and vertically in (b) and (d). A very largeribbon width M = 1000 is taken in (e) for both θ = 0 and π/
2, where the finite size effect is too small to be observable.(f) shows the extrapolation of the band gap to the infinitewidth limit under various strains in the case of θ = π/ As shown in Fig. 2(a), t increases and t , decrease withincreasing S for θ = π/
2, which eventually leads to thedimerized t -bonds with an energy gap ∆ = 2 | t − t | opening. The critical strain S c = 0 .
235 can be deter-mined by solving the equation ∆( S ) = 0, or equivalently t ( S ) = 2 t ( S )[12]. It is then understandable that thegaps of AGNRs with finite widths diminish to zero as S approaching S c from the left, since a quantum phasetransition occurs there. For θ = 0, all t i ’s decrease with S monotonically, and t decreases even faster, hence thereis no phase transition at all. B. Strained ZGNRs
In this subsection, we discuss the band structure ofZGNRs with the periodical boundary condition in x di-rection and open boundary condition in the y direction.Fig. 4(a) is the band structure of unstrained ZGNRs,which shows a midgap flat band corresponding to theedge states[29] localized in the upper and lower zigzagboundaries. The flat band exists in a finite region of mo-mentum [ k s , π − k s ] with k s = 2 π/ | k x / | ≤ θ = 0as shown in Figs. 4(c) and (e), k s moves towards zero ZGNRsN=60,M=100,S=0 (b)(c) (h)(g) (f)(e)(a) (d)
ZGNRs, = /2N=60,M=100,S=0.15 k x / ZGNRs, = /2N=60,M=100,S=0.3 E k x / ZGNRs, =0N=60,M=100,S=0.3
ZGNRs, =0N=60,M=100,S=0.15 =0 k s = /2 k s S k S / t /t t /t t / t (E) E M=100, =0 S=0 S=0.1 S=0.2 S=0.3 E M=100, = /2 S=0 S=0.1 S=0.2 S=0.3
FIG. 4: (Color online.) Band structures of ZGNRs with N =60 and M = 100 for various sizes and strains in (a) and (c)-(f). (g) and (h) are the density of states ρ ( E ) of strainedZGNRs with M = 100 and N = 1600 for θ = 0 and θ = π/
2, respectively. (b) shows the strain dependence of k s and t /t (see text for details). with increasing S until S = S c , after that k s = 0 andthe flat band with zero energy extends over the wholeBrillouin zone accompanied by the conduction and va-lence bands detached from each other. S c is the sameas that defined for AGNRs in previous subsection whichsignals the dimerization of the t -bonds. In contrast, forthe vertical strain of θ = π/ k s moves toward π withincreasing S and the flat band shrinks into a single pointwith k s = π in the large S limit as shown in Figs. 4(d)and (f).In fact, the range of the momentum for the flat bandis given by the convergent condition on the wave func-tion, which requests | t /t cos( k x / | ≤ k s = 2 cos − ( t / t ) [13]. We plot k s and t /t as func-tions of strain S in Fig. 4(b). When θ = 0, t / (2 t ) ≤ S ≤ S c = 0 . k s has a solu-tion between 0 and π . If S > S c , t / (2 t ) > k x , therefore the flat band extends throughout the wholeBrillouin zone. When θ = π/ t / (2 t ) ≤ S . In this case, k s always has a solutionbetween 0 and π . We note that the flat band can shrinkinto a point with k s = π if t = 0, which corresponds tothe horizontal t -bonds broken and the ribbon becomes M independent carbon chains connecting the left andright electrodes.The edge states are also revealed by the zero energypeak in the DOS shown in Figs. 4(g) and (h). As thetensile strain increases, the peak intensity is enhancedfor θ = 0, while it is suppressed for θ = π/
2. This co-incides with previous analysis for the region of momen-tum allowed for the edge states. Similar to the AGNRs,the band width is D = 2 t + t with a minor correctionproportional to M − , which also shrinks with increasing S . In fact, except for the additional zero energy peaks,the characteristics of ρ ( E ) for ZGNRs under the uniaxialstrains with θ = 0(or θ = π/
2) are quite similar to thosefor AGNRs with θ = π/ θ = 0), including the posi-tions of the shoulders for S = 0 . π/ IV. TRANSPORT PROPERTIES OFGRAPHENE NANORIBBONS
The interplay between the strain and the finite sizeeffect leads to the fine tuning of the band structures ofGNRs as presented in the previous section. This allowsGNRs to be considered as a promising candidate for me-chanically controllable electronic nano-devices. In thissection, we use the transfer matrix method described inSec. II B to explore in details the transport propertiesof strained GNRs with various sizes and gate voltagesas well. The transport properties essentially depend onboth the band structures of GNRs and two leads. Trans-port results discussed in this section are expected to bringsome insights into the designation of GNR-based nano-devices for experimentalists.
A. Conductance of strained AGNRs
We first discuss the conductance G of neutral GNRs,i.e., the gate voltage V g = 0. As discussed inSec. III A, the band gaps oscillate with the strains andribbon widths, which signals one sort of the metal-semiconductor transition[16, 18]. This results in that theconductances in Fig. 5 also oscillate accordingly, wherethe conductance peaks are precisely located at the gap-less points of Fig. 3.Figures 5(a) and (b) show the conductances for thenarrow ribbons with M ≤ N . When M is small enough,say M = 10, all the maxima of G equal 2 e /h , which im-plies that only one effective conducting channel is max-imally opened due to the strong confinement in the y direction. As the ribbon width increases from 10 to 400as shown in Figs. 5(a-d), more and more channels are G ( / h ) N=100,Vg=0, =0 M=10 M=49 M=100 (g)(e)(a)(c) =0N=100,Vg=0 M=100 M=200 M=400 k y / =0,Vg=0N=100,M=400 S=0.0 S=0.1 S=0.2 S=0.3 =0N=100,Vg=0 M=100 M=200 M=400 G / M S S = /2N=100,Vg=0 M=10 M=49 M=100 (f)(d)(h)(b) = /2N=100,Vg=0 M=100 M=200 M=400 = /2N=100,Vg=0 M=100 M=200 M=400 T k k y / = /2,Vg=0N=100,M=400 S=0.0 S=0.1 S=0.2 S=0.23 S=0.2363 S=0.2365 S=0.2370 S=0.3 FIG. 5: (Color online.) The strain dependence of the con-ductance in charge neutral AGNRs (i.e. E = V g = 0) fordifferent ribbon widths M . The tension is along x -axis in (a),(c) and (e), and along y axis in (b), (d) and (f). (e) and (f)show the universal behavior of the scaled conductance G/M with M ≥ N particularly. (g) and (h) display the transmis-sion probabilities in different channels for θ = 0 and θ = π/ involved in the electronic transport, leading to the en-hancement of the conductance. In fact G is almost linearin M for AGNRs with fixed lengths, which is reflectedby the universal strain dependence of G/M in Fig. 5(e,f).Besides the ribbon width, the hopping amplitude t alsohas a positive correlation with the conductance. As onecan see in an extreme situation as indicated by the ge-ometry of AGNRs in Fig. 1, that if t = 0, the electronictransport would be completely shut down. At the sametime t is controlled by the strain, which monotonicallydecreases for θ = 0 and increases for θ = π/ S increases as shown in Fig. 2(a). This explains the strik-ingly different strain dependence of the conductance inFig. 5(c) for θ = 0 and in Fig. 5(d) for θ = π/ S < S c ), respectively. Figs. 5(a)-(f) also indicate thatthe conductance oscillation is greatly suppressed in widerribbons. For the ribbons with the same size, the oscil-lation is obviously violently under the horizontal strainthan that under the vertical one.It is interesting to note that when θ = π/
2, the con-ductance of AGNRs vanishes completely in the region
S > S c for any widths. This is because a gap is openedin this region mainly by the uniaxial strain, on which theribbon width has little effect. In fact there is a quan-tum phase transition occurring at the critical strain S c as we have discussed in Sec. III A. Correspondingly, wefind a λ -like in the strain dependence of the conductancein Figs. 5(d,f). The sudden drop of the conductance isexpected useful in the identification of the tension-drivenphase transition, as well as the determination of the crit-ical strain accurately via electronic measurements.To further understand the electronic transport fea-tures of AGNRs, we plot the transmission probability T k ( k y,n ) ≡ P n ′ η nn ′ | t nn ′ | [23] under different circum-stances as functions of k y,n in Fig. 5(g) for θ = 0 andFig. 5(h) for θ = π/
2. One can see then T k has a spike atthe momentum k s , which is exactly the onset momentumof the flat band in the spectra of ZGNRs with the samestrain as seen in Figs. 4. In fact, the interface betweeneach lead and the AGNR has a zigzag pattern, wherelocalized states might exist similar to the edge states inZGNRs. As long as k y is close enough to k s , the local-ization length is comparable to the ribbon length[29–31].Therefore the corresponding quantum states extend fromthe left lead to the right one, giving the major contribu-tions to the conductances. For θ = 0, the peak position k s moves from 2 π/ π and the peak height de-creases with increasing S . However, when θ = π/ k s moves towards zero and the height increases slightly as S increases, until S = S c . After that, the peak posi-tion shifts backward and the height drops rapidly when S > S c . For S = 0, the analytic T k obtained in Ref.23gives rise to GN/M = 4 e / h at V g = 0 as N, M → ∞ and
M/N ≫
1. This finite value is the maximal valuefor θ = 0 and all S , but the minimal value for θ = π/ S < S c . -3 -2 -1 0 1 2 304080120 -3 -2 -1 0 1 2 304080120-1.0 -0.5 0.0 0.5 1.002004006008001000 -1.0 -0.5 0.0 0.5 1.002004006008001000 M=1000N=100, =0 S=0.0 S=0.1 S=0.2 S=0.3
M=100N=100, =0 S=0.0 S=0.1 S=0.2 S=0.3 (d)(c) (b)(a)
VgVgVg
M=100N=100, = /2 S=0.0 S=0.1 S=0.2 S=0.3 G ( / h ) Vg M=1000N=100, = /2 S=0.0 S=0.1 S=0.2 S=0.3
FIG. 6: (Color online.) The conductance of AGNRs as afunction of the gate voltage V g : (a) and (c) for θ = 0, (b)and (d) for θ = π/
2. The size parameters: (a) and (b) with N = 100 and M = 100, (c) and (d) with N = 100, M = 1000. Figure 6 with M = N = 100 shows the overall fea-tures of the conductance as a function of the gate volt- age for AGNRs. One can see that the conductance isnot symmetric with respect to V g = 0, revealing anelectron-hole asymmetry, which has been observed inmany experiments[1–3]. This is a direct consequence ofusing ordinary metallic leads[23]. It is well-known that atight binding model on a bipartite lattice with only near-est neighbor hopping is e-h symmetric. In the presentsystem, the interface between each lead and the AGNRconsists of five-atom rings, which cannot be bipartite andbreaks the e-h symmetry eventually[30]. One can also seethat the conductance fluctuates with V g , which is dueto the scattering of electrons off the lead-ribbon inter-faces and the armchair edges, since there is no impurityand disorder in the present system. The edges reflec-tion of AGNRs can be suppressed relatively by increas-ing the width as demonstrated in Figs. 6(c) and (d) with M = 1000, especially for small gate voltage and strains.The remaining fluctuations in Figs. 6(c,d) should be at-tributed essentially to the scattering on the lead-ribboninterfaces.In Fig. 6(a) for θ = 0, the conductance curves arerather smooth for V g < V g =2 t − t > ρ ( E ) in Fig. 2(g) and moves outwards as S increases.However, the lower energy peak of ρ ( E ) at E = t showsno evidence in the conductance curves. When θ = 0,the conductances are suppressed by increasing S for all V g as observed in previous studies[9, 10]. As a contrast,when θ = π/
2, the conductance in Fig. 6(b) shows amore complicated V g -dependence, which reflects the sig-nificant differences between the strain dependences of theband structures in the two cases. When S increases butis still smaller than S c , we find the conductance domein the negative energy region, an abrupt increase for0 < V g < t − t , a gentle slop for 2 t − t < V g < t ,and finally a decreasing region for V g > t . Although thehopping amplitudes t i ’s shift with S , we can still claimthat, G is an increasing function of S for small V g , and adecreasing function for large V g . When S ≥ S c , a gap ∆opens and G vanishes for V g < ∆ and is suppressed byincreasing S for any V g > ∆ as can be seen in Figs. 6(b)and (d). -0.4 -0.2 0.0 0.2 0.4010203040 -0.4 -0.2 0.0 0.2 0.4050100150 -0.4 -0.2 0.0 0.2 0.40.00.10.20.3 (a) (b) G ( / h ) = /2M=100,S=0.3 N=400 N=800 N=1600 G/M Vg Vg = /2N=500,S=0.3 M=100 M=300 M=500 FIG. 7: (Color online.) The conductance of AGNRs for thevertical strain S = 0 . > S c with various lengths N andwidths M . Inset in (b): the conductance scaled by the width M . Figures 7(a) and (b) show more details on G for S =0 . − . t < V g < . t . The conductance is zero for − . t < V g < . t , which indicates ∆ = 0 . t for S =0 . N ≥ N due tothe ballistic transport. Fig. 7(b) shows the dependence ofthe conductance on M and its inset reveals the universalbehavior of renormalized conductance G/M . B. Conductance of strained ZGNRs
Figures 8(a)-(d) display the conductance of the ZGNRsas a function of V g with various S and M for fixed N = 100. In particular, Figs. 8(a) and (b) with M = 101show the overall features of G in the full range of the bandwidth, while Figs. 8(c) and (d) with a larger M = 901 isshown to demonstrate less conductance fluctuations for | V g | ≤ .
0. The conductance data obviously shows thee-h symmetry unlike the AGNRs case. This is becausethe whole system is still bipartite since those rings onthe interfaces between the leads and ZGNR contain ei-ther four or six atoms as seen in Fig. 1(b), in contrastto the non-bipartite five-atom rings on the interfaces inthe system of AGNRs. It thus seems that all the fea-tures of the conductance are essentially consistent withthe DOS for ZGNRs in Fig. 4. It is also remarkable thatthe conductance is a constant around G ( G ≡ e /h )or vanishingly small at zero gate voltage in the large N and M limit and the flat band is apparently not involvedin electronic transport even at V g = 0. This feature isunchanged under the strains and we will discuss it latter. -3 -2 -1 0 1 2 301530456075 -3 -2 -1 0 1 2 301530456075-1.0 -0.5 0.0 0.5 1.00200400600 -1.0 -0.5 0.0 0.5 1.00200400600 M=901 =0,N=100 S=0.0 S=0.1 S=0.2 S=0.3
VgVg Vg Vg G ( / h ) M=101 N=100, =0 S=0.0 S=0.1 S=0.2 S=0.3 (b)(a)(c) (d)
M=901 N=100, = /2 S=0.0 S=0.1 S=0.2 S=0.3
M=101 N=100, = /2 S=0.0 S=0.1 S=0.2 S=0.3
FIG. 8: (Color online.) The conductance of ZGNRs as afunction of the gate voltage V g : (a) and (c) for θ = 0; (b) and(d) for θ = π/
2. The size parameters: (a) and (b) for N = 100and M = 101; (c) and (d) for N = 100 and M = 901. In Fig. 8(a) for θ = 0, two sharp peaks of the conduc-tance for each different strain correspond to the peaksof ρ ( E ) at E = ± t in Fig. 4(g), which move outwardswhen the strain increases. It is interesting to note that inFigs. 8(a) and (c) the conductance for S = 0 . S = 0 in the region | V g | < . t .In fact this phenomena emerges for any given strain S < S c = 0 . G ( S ) and G (0) is given by | V g | < t − t which is ob-viously strain dependent. In other word, given a small V g , the relation | V g | = 2 t − t gives rise to a thresholdof strain, below which the measured conductance barelychanges with respect to S . Despite of this identical re-gion, Fig. 8(a) for θ = 0 also indicates that the conduc-tance is reduced with increasing S if the gate voltageis fixed. Figs. 8(b) and (d) show the conductance for θ = π/
2, where we find a cross point of the conduc-tance curves under different strains. For convenience wedenote the corresponding gate voltage as V Cg which isaround 1 . t . The strain enhances the conductance for | V g | < V Cg and suppresses it otherwise.To interpret the difference of the conductances between θ = 0 and θ = π/
2, we recall that the hopping inte-grals t i ’s have different strain dependence as seen in Fig.2(a). When the strain increases for θ = 0, t increases,while t , decreases. However, t , effectively favors thehorizontal electronic transport, while t may cause theformation of the dimers for vertical bonds which hindersthe electrons from moving freely. Therefore, the conduc-tance is reduced by increasing the strain for given V g .However, for θ = π/
2, both t , and t decrease, but t drops faster. As a consequence, the ribbon tends toform M metallic chains with a weak interchain coupling.The conductance is then enhanced with an upper limit M G as t → V g . Since more and more chan-nels below V Cg are fully filled due to the reduction ofthe hopping amplitudes t , , they do not contribute tothe conductance for large gate voltage. It turns out thatthere are two turning points ± V Cg in Fig. 8(b) and oppo-site strain dependences of the conductance are found for | V g | < V Cg and for | V g | > V Cg , respectively.
500 3 (a) G ( / h ) Vg=0,N=100 M=101 M=201 M=401 G /M (b) t /t t /t Vg=0,M=101 N=100 N=200 N=400
FIG. 9: (Color online.) The conductance versus t /t at V g =0 with t = 0 .
96 fixed. (a) for N = 100 and M = 101, 201and 401; and (b) for M = 101 and N = 100, 200 and 400.Inset in (a) for the conductance scaled by the width M . For ZGNRs, the topology of structure not only pro-tect the e-h symmetry of the electronic transport butalso stimulates the analysis of more general features ofthe conductance in a whole range of t /t , which mightbe beyond the values given by the relations Eqs. (3).In principle, one has actually two limits: t /t ≫ t /t ≪
1. For the former case, the system possesses anordered and insulating ground state consisting of dimer-ized t -bonds, which already emerges actually with a zeroconductance at t /t ≈ . M = 101 and N & M asdemonstrated in Fig. 9(a) and (b). When t /t ≪
1, thehoneycomb lattice becomes M weakly coupled metallic(zigzag) chains and the conductance reaches its max-imal value M G as seen in Fig. 9(a) where the con-ductance is shown as a function of t /t for N = 100with M = 101 ,
201 and 401. One can see that G is en-hanced by decreasing t /t and indeed proportional to M as being well renormalized by M for t /t . .
05 inthe inset. Fig. 9(b) shows that the conductance is alsoreduced by increasing the length of ribbons, implyinga non-ballistic transport. In addition, the conductancefluctuations around G show up for t /t & .
10 100 10001E-61E-51E-41E-30.010.11 10 100 1000 ( b ) N M=10, =0 S=0 S=0.1 S=0.2 S=0.3 M=11, =0 S=0 S=0.1 S=0.2 S=0.3 G ( / h ) ( a ) N M=10, = = /2 S=0 S=0.1 S=0.2 S=0.3 M=11, = = /2 S=0 S=0.1 S=0.2 S=0.3
FIG. 10: (Color online.) Odd-even M effects on the conduc-tance for narrow ZGNRs for θ = 0 in (a) and θ = π/ Figure 10 shows an even-odd M effect on G for ZGNRs,which is relevant for either experiments or designingnano-devices with narrow and short ribbons. This ef-fect diminishes for sufficiently wide and long ribbons sothat our above discussions for Figs. 8 and 9 are given justfor odd but sufficiently large M . For S = 0 and V g = 0,one finds that the conductance of narrow ZGNRs showstwo different scaling behaviors according to the parity of M as N → ∞ [32, 33]. The conductance is a constantaround G for odd M to indicate metallic nature, while G ∼ N − for even M to present a semiconducting fea-ture [30]. In the presence of strains, one can still find twotypes of scaling behaviors for the conductance at V g = 0as shown in Figs. 10(a) and (b). When M is odd, the con-ductance changes a little with both N and S . When M is even, although the conductance decreases in a powerlaw N − , G is suppressed for θ = 0 and enhanced for θ = π/ S if N is fixed. V. EFFECTS OF EDGE RELAXATION
In realistic situations, there may exist passivation andspin poarlization on the ribbon edges, which affect theband structure of narrow ribbons [28]. However, these edge modes only affect the edge and nearby carbon atomsfor wide enough ribbons, in contrast, the strain can affectall the carbon-carbon bonds in the ribbon. As shownin Fig.3, the band gaps oscillate with the strain for alltypes of AGNRs and is around 0 . t for M = 10 (i.e., N a = 20), while the edge relaxation induces one onlyaround 0 . t as shown in Ref.28. This indicates theedge relaxation has much smaller effect on the band gapthan the strain effect, which even becomes smaller andsmaller as the ribbon width increases.Comparing with the strain effect, edge relaxation onthe transport properties of GNRs is negligible in rela-tively wide graphene ribbons. In Fig. 11, the black solidlines are the conductance of pristine GNRs and the othersare the data with edge relaxation. For AGNRs, we set thehopping integral at the edge te to be 1 . t as suggestedin Ref. 28 and the on-site energies ǫ e being 0 . t and0 . t . It is clearly seen that the effects of edge relaxationson electronic transport are negligible as anticipated forarmchair ribbons wider than 10nm (i.e., M ≥ t e = 1 . t and theon-site energies ǫ e being 0 . t according to [34]. It isclearly seen that the effects of edge relaxations on elec-tronic transport are negligible for zigzag ribbons widerthan 10nm (i.e., M ≥ VI. SUMMARY AND CONCLUSIONS
In this article, we have investigated the electronictransport of graphene nanoribbons under various tensilestrains with connections to the normal metallic leads.For this purpose, we first calculated the band structuresof strained GNRs with both zigzag and armchair edges.The direction of the uniaxial tension, which is taken to beeither parallel( θ = 0) or perpendicular( θ = π/
2) to theribbon axis, has a crucial effect on the band structure.In the strained armchair GNRs with θ = 0, the bandgap oscillates with the strain in a zigzag pattern, leadingto the transitions between metal and semiconductor. The0 -3 -2 -1 0 1 2 3010203040 -3 -2 -1 0 1 2 30306090-3 -2 -1 0 1 2 301020304050 -3 -2 -1 0 1 2 30306090Vg (c) ZGNRs,N=100,M=50 t e = t , = 0 t e =1.05 t , =0.3 t Vg (d) ZGNRs,N=100,M=100 t e = t , = 0 t e =1.05 t , =0.3 t G ( / h ) Vg (a) AGNRs,N=100,M=50 t e = t , = 0 t e =1.12 t , =0.3 t Vg (b) AGNRs,N=100,M=100 t e = t , = 0 t e =1.12 t , =0.3 t FIG. 11: (Color online.) Effects of edge relaxation on theconductance of AGNRs (a,b) and ZGNRs (c,d) with varioussizes. oscillatory amplitude is almost unchanged as S increases.This kind of band gap is mainly a finite width effect, sinceit vanishes as M goes to infinity. If θ = π/
2, similar oscil-latory gap also appears, but only for the strains smallerthan a critical value S c . As S approaches S c , the oscil-latory amplitude goes to zero, unlike the case for θ = 0.Once S > S c , the other kind of band gap opens which islinear in S − S c and hardly affected by the ribbon width.In fact as the strain with θ = π/ S = S c to separate a liquidphase from a solid phase where the bonds perpendicularto the strain are dimerized.In the zigzag GNRs, the most intriguing phenomenonis the appearance of the flat band in a region of momen-tum [ k s , π − k s ]. As the strain with θ = 0 increases, k s decreases to zero until S = S c , then the flat band extendsthroughout the full Brillouin zone, and the conductionand valence bands are separated. On the contrary, withincreasing S , k s moves towards π and the region of the flat band shrinks into a point for θ = π/ V g = 0 as the strainvaries. The peak in the plot of the conductance versus V g is compatible with that in the DOS plot. Note that not allthe modes with energy V g contribute to the conductance,but only those satisfying the boundary conditions are re-sponsible for the electronic transport, therefore, it is notnecessary to have a one-to-one correspondence betweenthe peaks of the conductance and those of the DOS. Fur-thermore, by measuring the strain dependence of the con-ductance of AGNRs at V g = 0, one can also detect thequantum phase transition induced by the tension perpen-dicular to a C-C bond and determine the critical strainas well.Since we connect the GNRs with square lattices asthe metallic electrodes, it is worth mentioning the fun-damental effect of the topology of the heterojunctionson the conductance of GNRs. In particular, due to thenon-bipartite feature of the electrode-AGNR interfaces,the conductance data of AGNRs is not e-h symmetric,while this kind of symmetry can still be found in thatof ZGNRs, since the electrode-ZGNR interfaces do notbreak the bipartite structure of the whole system. Thisphenomenon has no counterpart in the band structuresobtained with periodic boundary condition, yet it maybe important for designing the nano-size devices. VII. ACKNOWLEDGEMENTS
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