Effects of the contacts on shot noise in graphene nano-ribbons
EEffects of the contacts on shot noise in graphene nano-ribbons
A.D. Wiener and M. Kindermann School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
We investigate the shot noise of an impurity-free graphene flake as a function of the chemicalpotential. For large width to length ratios, this noise has been predicted and observed to exhibituniversal characteristics at the Dirac point. Furthermore, a sharp decrease of the shot noise withincreasing carrier density has been predicted. This decrease has also been observed in experiments,but with much smaller slope than predicted. We reconcile this discrepancy between theory andexperiment by including the effects of the contacts to the graphene ribbon.
PACS numbers: 72.80.Vp, 73.23.Ad, 73.50.Td, 73.63.-b
I. INTRODUCTION
Impurity-free graphene near its Dirac point hasbeen predicted to exhibit non equilibrium current fluc-tuations, or shot noise.
These fluctuations are coun-terintuitive, as shot noise vanishes in conductors withoutelectron scattering.
They are due to evanescent wavesthat backscatter electrons, even in the absence of impu-rities. Moreover, Tworzydlo et al. have predicted that theshot noise in a clean sheet of graphene at its Dirac point(zero chemical potential) has universal characteristics;the Fano factor F , defined as the shot noise normalizedby the mean current and expressed in units of the elec-tron charge e , takes the value F = 1 /
3. This predictionhas generated much theoretical and experimental interest in the shot noise of graphene.One goal of the ensuing experimental activity was toconfirm the existence of shot noise due to evanescentwaves in graphene. A Fano factor F ∼ / F = 1 / et al. , the Fano fac-tor reached a peak value F ∼ / As the dependence of F on the chemical potential provides the main experimen-tal evidence for evanescent wave transport in grapheneto date, this deviation from the theoretical prediction isdisturbing.In this article, we show that the measured dependenceof the Fano factor on chemical potential is indeed consis- tent with the assumption that it originates from evanes-cent waves when the effects of the electrical contacts aretaken into account. It has been shown by first-principlescalculations that contacts to graphene have two maineffects on electron transport: doping of the pieces ofgraphene underneath the contacts and electron scatter-ing at the interface from contact to graphene. It turnsout that the Ti-graphene contacts used in the experimentof Ref. 7 are highly transparent. In this article, we there-fore neglect contact scattering and focus on the effects ofdoping through the contacts.While the theory of Ref. 5 assumes a constant elec-tric potential on the graphene ribbon, the local dopingof graphene by the contacts causes that potential to bespace-dependent. We show that this space-dependenceresults in an increase of the voltage scale of F , as ob-served experimentally. This effect can be understoodin terms of an effective reduction of the length of thegraphene ribbon to a region around the potential min-imum. Remarkably, the maximum of the Fano factorremains F ≈ / et al. The reported calculation thus lends ad-ditional support to an interpretation of the experimentof Ref. 7 in terms of evanescent waves in graphene.This article is organized as follows: after a descriptionof the model in Sec. II, we employ a conformal map tech-nique in Sec. III to calculate the electric potential on thegraphene ribbon due to the contacts. We then considerthe effects of screening of this potential by the electronsin the graphene flake and identify a regime where suchscreening is negligible. In that regime, we then calculatethe electron transmission through the graphene ribbonand the resulting Fano factor. We first do this analyt-ically for the semiclassical regime in section Sec. IV. InSec. IV D, we then calculate the transmission from nu-merically obtained wave functions, finding good agree-ment with both the semiclassical results in their regimeof validity and experimental observations. We concludewith a summary in Sec. V. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l (a) (b)(c) FIG. 1: (a) Graphene nano-ribbon geometry consistingof a graphene sheet with highly doped lead regions(dark gray rectangles) separated by an undoped region(white rectangle). An insulating SiO layer of thickness d (not shown) separates the graphene sheet from a Sibackgate (light gray rectangle). (b) An example of asimplified experimental geometry as considered inRef. 7. (c) For the d = 0 case, the conformal map w = f ( ξ ) maps the ( x, iz ) plane to the ( u, iv ) plane.The undoped graphene regions are sent to the real line v = 0 while the doped graphene regions are sent to theline v = π . The height of the doped lead regions shownhere is for illustration purposes and has no physicalmeaning. II. MODEL
In the experiment of Danneau et al. , a graphene nanoribbon (GNR) was mechanically exfoliated, deposited ona Si/SiO substrate as in Fig. 1b, and brought in closeproximity with Ti contacts. An insulating SiO layer ofthickness d separates the GNR from a Si backgate. Weassume that the electronic structure of the sections of theribbon underneath the contacts is unmodified, except fora doping through the contacts. It has been shown usingdensity-functional calculations that this model is a goodapproximation for non-wetting contacts and also for(wetting) Ti contacts at low energies. To find the electric potential U on the GNR, we modelthe system by two metal half-planes separated by a dis-tance 2 L , representing the contacts, and a third metallicplane, representing the Si backgate, at distance d belowthe GNR, as shown in Fig. 1a. All metals are assumedto be perfect, and hence equipotentials, and we choosecoordinates such that the contact edges are along the y -direction and at x = ± L . The z axis is perpendicular to backgate and GNR.In equilibrium, the contact plates are at identical elec-tric potentials U d , while the backgate has a potential U b .Under the reasonable assumption that the distance be-tween the contacts and the GNR is negligible, U d is alsothe electrostatic potential in the graphene regions under-neath the contacts. In the absence of a backgate and inthe limits that we take below, the potential on the rib-bon close to the center of the GNR differs from U d by thework function difference between the GNR and the metalcontacts. The backgate is used to manipulate the poten-tial in this region where, in the same limits, all electronscattering occurs.The low-energy Dirac model of the electron dynamicsin the GNR ( (cid:126) = 1) is, H γ = v σ γ · p + V ( x ) , (1)where σ γ = ( γσ x , σ y ) is a vector of Pauli matrices withthe valley index γ = ± p = − i ∇ is the electron momen-tum and v is the electron velocity. The potential energyof electrons on the GNR is given in terms of the elec-tric potential U by V ( x ) = − eU ( x ). Using translationalinvariance in the ˆ y -direction and choosing boundary con-ditions that do not mix transverse modes, the mo-mentum along the ˆ y -direction is conserved, and the scat-tering problem for wave functions ψ q with ˆ y -momentum q at energy ε becomes one dimensional: v ( − iγσ x ∂ x + qσ y ) ψ q = [ ε − V ( x )] ψ q . (2)We focus on the most interesting regime of the Fanofactor measurement of Danneau et al. , namely an in-terval of chemical potentials − V (0) in the middle of theGNR with width ∆ µ . The Fano factor reaches its max-imum F ≈ / F < .
25 at the boundaries.Existing theory assumes a piecewise constant shape of V , with V = V ∞ (cid:29) v/L underneath the contacts and V = V in between. This potential shape is obtainedfrom our electrostatic model in the limit d →
0, with V (0) = V = − eU b .Such theory predicts oscillations of the Fano factor ofperiod ∆ µ (cid:39) v/L (setting (cid:126) = 1). In the experiment, however, the interval ∆ µ was found to be at least a factorof 2 larger than this prediction. Of course, in the experi-ment d (cid:54) = 0, and the electrostatic model of Ref. 5 is over-simplified. Below, we analyze the effects of nonzero dis-tance d between the GNR and backgate, assuming that d is small in a sense that will be specified. In that case, onestill has V (0) ≈ − eU b , but the potential V is no longerpiecewise constant. We show below that such non-zero d resolves the discrepancy between theory and experiment. III. CALCULATION OF THE POTENTIALA. The conformal map
We start with the calculation of the electrostatic po-tential U ( x ) on the GNR. In a first approximation, weneglect doping of the GNR, effectively assuming it is aperfect insulator. We will justify this approximation forsmall d in the following section. Additionally assumingthat the contacts lie directly on top of the GNR and thatthe distance d to the backgate is small compared to thelength L of the ribbon, d (cid:28) L , we find the potential U onthe GNR to a good approximation (at distance ∆ x (cid:29) d from the contacts) from the electric field E on the ribbonwhen d = 0: U = U b + ˆ z · E d/(cid:15) r , (3)where (cid:15) r = (cid:15)/(cid:15) ≈ . is the relative electric permit-tivity of the SiO layer separating the GNR from thebackgate. Here, (cid:15) is the vacuum permittivity and (cid:15) isthe permittivity of silicon oxide.The electric field E at d = 0 may be found by anappropriate conformal mapping, exploiting translationalinvariance in the direction along the contact edges. Theelectric potential U satisfies the two-dimensional Laplaceequation ∇ U ( x, z ) = 0, along with the boundary condi-tions U ( | x | > L, z = 0) = U d ,U ( | x | < L, z = 0) = U b . (4)The conformal map f ( ξ ) = ln (cid:18) ξ − Lξ + L (cid:19) (5)from the complex plane ξ = ( x, iz ) into the complex plane w = ( u, iv ) maps these boundary conditions onto thoseof a parallel plate capacitor, with a plate separation of π , as shown in Fig. 1c. When z →
0, the map f ( ξ ) sendsthe graphene regions under the contacts at | x | > L in the ξ -complex plane to the real line v = 0 in the w -complexplane. Similarly, for z →
0, the region | x | < L in the ξ -plane is mapped to the line v = π in the w -plane.The potential in the ( u, iv ) plane is thus U ( u, v ) = ( U b − U d ) vπ . (6)Under the map Eq. (5) this gives the electric potential U ( x, z ) = U b − U d π tan − (cid:18) zLx + z − L (cid:19) + U b . (7)in the original ( x, iz ) plane.By differentiation, we now find the electric field E onthe GNR for d = 0 from Eq. (7) and, using Eq. (3), we obtain the electric potential on the ribbon at d (cid:28) L and L − | x | (cid:29) d as U ( x ) = U b + ˆ z · E ( x, d/(cid:15) r = U b + U b − U d (cid:15) r π (1 − x /L ) (cid:18) dL (cid:19) . (8) B. Corrections due to screening by the graphenesheet
Next, we account for the screening of the electric po-tential generated by the contacts arising from the elec-trons in the ribbon itself. Quantifying this screening re-quires a determination of the electron density n ( x ) thataccumulates on the ribbon due to the potential V ( x ).The corresponding charge density − en induces a screen-ing electric field E sc , which in turn modifies the potentialon the ribbon U . Under our assumption d (cid:28) L we have U ( x ) = U b + ˆ z · [ E ( x,
0) + E sc ( x, d/(cid:15) r , and E sc isapproximately given by Gauss’ law:ˆ z · E sc ( x,
0) = − en ( x ) (cid:15) . (9)A determination of the electron density at zero tem-perature and chemical potential n ( x ) = (cid:88) q,ε< | ψ q,ε ( x ) | (10)requires the wave functions ψ q,ε ( x ). Rather than car-rying out the requisite quantum mechanical calculation,we identify a parameter regime where the screening field E sc may be neglected, | E sc | (cid:28) E . The key observationallowing this approximation is that at U b = 0, V ( x ) isof first order in d . Semiclassically, the induced electrondensity is of order V /v ∼ d . One thus expects that E sc = O ( d ), while E = O ( d ), such that | E sc | (cid:28) E at d →
0. Based on this semiclassical reasoning, one thusexpects that for d below a critical distance d c the effectsof screening may be neglected. In appendix VII, we rigor-ously establish the existence of such a critical distance forthe relevant interval ∆ µ of gate voltages and we compute d c . We find d c = (cid:15) r v (cid:126) ( (cid:15) L ) / e − / ( U b − U d ) − / . (11)For the remainder of this article we assume that d (cid:28) d c (cid:28) L . In that regime we have, to a good approxima-tion, V ( x ) = − eU ( x ) and V (0) = − eU b . In the exper-iment of Danneau et al. d c ≈
15 nm, while d ≈
300 nmand L ≈
200 nm. The experiment therefore is not in thelimit that we assume. Our calculation therefore merelyhighlights the qualitative physics of the voltage scale en-hancement observed in that experiment.
IV. TRANSPORT CALCULATIONSA. The transfer matrix
In order to calculate the q -dependent transmissionprobabilities for transport through a GNR as in Fig. 1a,we employ the transfer matrix method. Without re-striction, we fix the valley index γ = +1 in the Hamilto-nian Eq. (2), and we confine our analysis to equilibriumat electrochemical potential µ = 0, such that we requirethe transfer matrix only for electrons with energy ε = 0.The transfer matrix M ( x, x (cid:48) ) for the requisite Diracspinors of a mode with transverse momentum q and en-ergy ε = 0 satisfies the equation i∂ x M ( x, x (cid:48) ) = (cid:20) iqσ z + V ( x ) v σ x (cid:21) M ( x, x (cid:48) ) . (12)Additionally, M satisfies the conditions M ( x, x ) = I , M ( x, x (cid:48) ) = M ( x, x (cid:48)(cid:48) ) M ( x (cid:48)(cid:48) , x (cid:48) ), det M ( x, x (cid:48) ) = 1 and M † ( x, x (cid:48) ) σ x M ( x, x (cid:48) ) = σ x . The latter condition ensurescurrent conservation.In order to extract the transmission probability fromthe transfer matrix, it is necessary to factor out theasymptotic evolution of the electron states at | x | → ∞ .This is accomplished by using matrices A ± ( x ) that sat-isfy Eq. (12) in the regions under the contacts, where thepotential is constant, V = eU d ≡ vk F . The solution ofthis equation gives the A ± matrices as A ± ( x ) = (cid:115) k F p x (cid:18) p x ± iqk F e ∓ ip x x − p x ± iqk F e ± ip x x e ∓ ip x x e ± ip x x (cid:19) , (13)where p x = (cid:112) k − q . The columns of A ± are made ofright and left-moving states that are normalized to carryunit current. The transmission probability is extractedfrom the transfer matrix M ( x, y ) as T = 1 / | α | , where (cid:18) α β ∗ β α ∗ (cid:19) = lim x →∞ A − ( x ) M ( x, − x ) A − ( − x ) . (14)Here, | α | − | β | = 1 due to the current conservationcondition. B. Analytic calculation of transmission probability
In parts of the central graphene region with potentialEq. (8), transport is semiclassical and the transfer matrixcan be found using an adiabatic approximation. This isthe case whenever | qV (cid:48) /V ( V /v − q ) | (cid:28)
1. Looselyspeaking, this condition is met where the potential islarge, such as near the contacts. It has been shown in Ref.26 that no electron scattering takes place in those regions.All shot noise is therefore produced in the regions thatdo not allow such an approximation, near the classicalturning points where V = ± vq . As previously mentioned, we assume that the regionsof interest, where electron scattering occurs, are at | x | (cid:28) L . The precise form of the potential at | x | (cid:39) L is thusirrelevant, and we may approximate the potential Eq. (8)quadratically, V ( x ) = ρx − V b , (15)where ρ = e ( U d − U b ) d/(cid:15) r πL and V b = eU b is set by thebackgate voltage U b . For a dimensional analysis, we firstwrite the Dirac equation (2) with potential (15) at q = 0in terms of the dimensionless variable γ = x/ ˜ x , where˜ x = ( ρ/v ) − / . This results in an energy scale ∆ µ ≈ v/ ˜ x for the interval I D of the first oscillation of F .To make rigorous analytical progress, we assume | V b | (cid:29) v/ ˜ x . While this is not the most relevant limitexperimentally, this calculation will provide physical in-sight into the transport problem. Our analytical ap-proach decomposes the GNR into “adiabatic regions,”where the semiclassical approximation may be applied,and “non-adiabatic regions” near the classical turningpoints V = ± vq where it cannot. One finds that for | V b | (cid:29) v/ ˜ x , each non-adiabatic region is short enoughfor the potential to allow linearization throughout theregion. The transfer matrix for a linear potential hasbeen found exactly in Ref. 26. This, together with theadiabatic solution for the remaining regions and the com-position rule M ( x, x (cid:48) ) = M ( x, x (cid:48)(cid:48) ) M ( x (cid:48)(cid:48) , x (cid:48) ), allows us toconstruct the transfer matrix through the entire GNR.We find the above condition | V b | (cid:29) v/ ˜ x for appli-cability of the described analytical approach by self-consistently assuming that the potential V may be lin-earized in the non adiabatic regions. It has been shown that, in this case, transport through a non-adiabatic re-gion around a pair of classical turning points V = ± vq is exponentially suppressed by a factor exp( − πvq /V (cid:48) ).Thus, only modes with q (cid:46) (cid:112) V (cid:48) /v contribute signifi-cantly to transport, and we may neglect all other modes.The condition quoted above for adiabatic electron dy-namics thus effectively becomes | vV (cid:48) | (cid:28) V . Using theexplicit form Eq. (15) of the potential V , we find thatthis condition is fulfilled everywhere except in regions oflength ∆ x around the points x with V ( x ) = 0 that areshort enough to allow linearization of V , that is ∆ x (cid:28) x .Therefore, in the above limit | V b | (cid:29) v/ ˜ x the transfer ma-trix can indeed be constructed from that of electrons ina linear potential and the one for adiabatic evolution. Inappendix VIII we calculate transport through the GNRin this limit. C. Analytic results
From the transfer matrix Eq. (31) obtained in ap-pendix VIII, one analytically extracts the transmissionprobability using Eq. (14). The resulting transmissionprobability takes the form T = (cid:12)(cid:12)(cid:12) α + ie − iφ ( b ∗ ) (cid:12)(cid:12)(cid:12) − ,φ = (cid:90) x − (cid:96)(cid:96) − x (cid:112) V ( x ) /v − q dx, (16)where b = −√ πe πθ/ θ / − iθ / Γ(1 − iθ ) and V ( ± x ) = 0.The wave function acquires the phase φ from traversingthe central adiabatic region separating the turning pointat x = (cid:96) − x from the one at x = x − (cid:96) .The Fano factor is found from the transmission prob-abilities T n = T ( q n ) of modes with wavenumbers q n ac-cording to Eq. (16) as F = (cid:80) n T n (1 − T n ) (cid:80) n T n . (17)In the continuum limit W (cid:29) L , the sums over the modeindex become integrals over the momentum q . The Fanofactor is plotted as a function of the backgate voltage inunits of v/L in Fig. 2 (solid curve). The curve showsoscillations in gate voltage with a maximum of F ∼ / describing evanescent modetransport in a piecewise constant potential. However,the width of the peaks is broader than predicted by thetheory of Ref. 5, in agreement with the observations ofDanneau et al. Stretching the limits of applicability of our semiclassi-cal approach, Eq. (16) predicts that the first oscillation ofthe Fano factor with backgate voltage has period ∆ µ =(3 π/ / v/ ˜ x . Substituting ρ = e ( U d − U b ) d/(cid:15) r πL , thepredicted oscillation period∆ µ = η ∆ µ | d =0 (18)is enhanced by a factor η = (9 e | U b − U d | d/ (cid:15) r vπ ) / compared to the period ∆ µ | d =0 = πv/L for a piecewiseconstant potential. The enhancement factor for the pa-rameters of the experiment of Ref. 7 takes a value of η ≈
2. The physical reason for this enhancement is nowclear; rather than being confined to the ribbon of length2 L , the electron states in the presence of doping from thecontacts form standing waves between the two non adi-abatic regions that introduce electron scattering around x = ± x . The width ∆ µ of the first oscillation of F as afunction of backgate voltage is thus enhanced.Unfortunately, our above analytic approach breaksdown at | V b | (cid:46) v/ ˜ x , the regime V b ∈ I D of the exper-imentally most interesting first Fano factor oscillation,and the above considerations are not quantitatively cor-rect. Physically, this is the regime where the potentialmay not be linearized in the non adiabatic regions. Inthis case, the two non adiabatic regions merge into one.To quantitatively access that regime, we next performnumerical calculations of the transfer matrix. FIG. 2: Analytic (solid curve) and numerical (circles)results for the Fano factor F as a function of backgatevoltage in units of v/L , along with the constant F = 1 /
3. The curves are generated assuming (cid:126) v = 0 . × − eV · m, L = 500 nm and ρ = 10 − eV / m . The analytic results are calculatedfrom the transmission probability of Eq. (16). Thenumerical result is calculated from the numericallyintegrated wave functions according to Eqs. (2). Theresults differ at low backgate voltages where the turningpoints no longer lie within the linear region of thepotential and the semiclassical approximation used herebreaks down. D. Numerical calculation of the transmissionprobability
In this section, we obtain the transmission probabilityby numerical integration of Eq. (2). Numerical resultsfor the Fano factor are plotted in Fig. 2 (circles). At | V b | (cid:29) v/ ˜ x , where our analytic approach is justified, thecurve agrees with our analytic results, as expected. Ournumerical calculation confirms what we had observed an-alytically for the first and most relevant Fano factor max-imum with | V b | (cid:29) v/ ˜ x . Doping from the contacts in-creases the voltage scale of the Fano factor oscillations.From the full width at half maximum of our numericalresults, we conclude that the width of the first peak ofthe Fano factor is enhanced by a factor η ≈ . F ≈ / F ≈ .
305 is as compat-ible with the experimental value F = 0 .
318 as the value F = 1 / V b in Fig. 2. They do not show in the exper-imental data, Fig. 3 of Ref. 7. This discrepancy couldhave various reasons. We here briefly speculate on two ofthem. First, the gate voltage scale where the oscillationsin Fig. 2 set in is about V b ≈ L = 200nm). This is beyond therange of gate voltages where our quadratic approxima-tion Eq. (15) is justified for the work function difference | U d − U b | ≈ V b L/v >
10 shown in Fig. 2 are out-side the regime of validity of our theory. Second, we notethat our prediction Fig. 2 assumes a perfect geometry ofa ribbon between two perfecty straight and parallel con-tact edges. In reality, of course, this is not the case andthe length of the ribbon varies over its width, in partic-ular in samples with a large
W/L -ratio such as the onesthat were measured for Fig. 3 of Ref. 7. Such variationswill wash out the predicted oscillations. It will suppressthem the more the further the energy is from the Diracpoint.
V. DISCUSSION AND CONCLUSIONS
In conclusion, we have shown that the anomalouslylarge voltage scale observed in the GNR shot noise exper-iments of Danneau et al. is consistent with evanescentwave transport when the effect of doping by the contactsis accounted for. We have identified a regime of smallgraphene-backgate distances where the effects of screen-ing by conduction electrons in the GNR can be neglected.While not well satisfied in the experiment of Ref. 7, thislimit gives insight into the qualitative physics of the volt-age scale enhancement observed by Danneau et al. In this regime, we find the electric potential on theGNR with contacts, and we use it to obtain the Fanofactor as a function of the backgate voltage. We employboth a semiclassical and a numerical approach. The semi-classical approach illuminates the origin of the predictedincreased gate voltage period of the Fano factor. The po-tential due to doping from the leads introduces electronscattering around, generally, two pairs of classical turningpoints of the conduction electrons at distance 2 x < L .The standing waves that form in between cause oscilla-tions of the Fano factor with period ∆ µ (cid:39) v/x , which islarger than the scale ∆ µ (cid:39) v/L of the same oscillationswithout doping by the contacts. Our numerical resultsshow that the contact potential enhances ∆ µ by a factorof η ≈
2, consistent with the experimental observations. Our calculations demonstrate that an interpretation ofthe experiment by Danneau et al. in terms of evanes-cent waves is possible and strongly indicated, despite thediscussed discrepancy with the original theory. VI. ACKNOWLEDGEMENT
We acknowledge support by the NSF under DMR-1055799 and DMR-0820382.
VII. APPENDIX A
In this appendix, we identify a parameter regime wherethe screening field E sc due to doping of the GNR may beneglected in our calculation of the Fano factor F . Self-consistently, we thus assume that V ( x ) = − eU ( x ) forthe argument below. In our limits only the region | x | (cid:28) L contributes to the shot noise (see main text) and wemay approximate the potential V as in Eq. (15). Asexplained in the main text, the central length scale in theproblem then is ˜ x = ( ρ/v ) − / , the typical wavelength ofthe electron states at the Fano factor maximum, whichsets the energy scale v/ ˜ x of the interval ∆ µ of the firstoscillation of F .We first show that transport states with transverse mo-menta q (cid:29) / ˜ x are irrelevant for the determination of F .To see this, we perform an adiabatic expansion of Eq. (2)in the spirit of Ref. 26, but for vq > V ( x ). That expan-sion is valid if | V (cid:48) ( x ) | (cid:28) vq , which is fulfilled at | x | < ˜ x for the transport states (that is, states at the Fermi level,with energy zero) with q (cid:29) / ˜ x in the window of back-gate voltages ∆ µ of interest. For such q , this adiabaticcalculation results in a transmission of electrons throughthe region | x | < ˜ x , which is exponentially suppressed in q ˜ x (cid:29)
1. Consequently T n (cid:28)
1, and states with q (cid:29) / ˜ x do not contribute to the Fano factor, Eq. (17).Moreover, F , Eq. (17), depends only on the transmis-sion eigenvalues T n . These eigenvalues receive no contri-butions from regions in space with semiclassical electrondynamics, which is explicitly evident in section IV B. Ac-cording to Ref. 26, semiclassical dynamics takes place for | qV (cid:48) /V k | (cid:28) V (cid:48) denotes the first derivative of V ), and regions where that condition is met may there-fore be disregarded in our calculation. For the relevantstates in the gate voltage interval ∆ µ which do not satisfy q (cid:29) / ˜ x , the adiabatic condition is satisfied for | x | > s with s = f ˜ x and f (cid:29)
1. The region | x | > s is thus ir-relevant for the determination of the shot noise, and weneed not further consider it. For screening by the conduction electrons to be negli-gible in our calculation of F , the condition | E sc | (cid:28) E therefore needs to hold only at | x | < s . An evaluation ofthis condition requires an upper bound on the inducedelectron density n , Eq. (10), at | x | < s . Our strategy willbe to obtain that density by semiclassical calculations,which are straightforward. For many electronic states at ε <
0, which contribute to n in Eq. (10), the semiclassi-cal approximation at | x | < s holds directly. First, for allstates with | q | > f (cid:48) /s ( f (cid:48) (cid:29)
1) the above condition forsemiclassical dynamics is violated at most in an intervalof length ∆ x (cid:28) s in | x | < s . This has a negligible effect,and we may evaluate the density n lq due to all stateswith | q | > f (cid:48) /s and ε < n v due to all states that have | q | < f (cid:48) /s and ε < − vf (cid:48) /s may be found semiclassically.It then remains to find an upper bound on the chargedensity n − due to states ψ − with energies − f (cid:48) v/s < ε < | q | < f (cid:48) /s . To this end, we apply the Friedelsum rule at chemical potential ε Friedel = vf (cid:48) /s to allstates with momenta | q | < f (cid:48) /s . The Friedel sum rulerelates scattering phase shifts to the number of parti-cles induced by the potential V . The wave functions atthe energy ε = ε Friedel and, correspondingly, the phaseshifts entering the sum rule may be evaluated semiclas-sically. Consequently, the total particle number N Friedel due to all states with ε < ε
Friedel and | q | < f (cid:48) /s canbe calculated semiclassically (even though the semiclas-sical approximation does not hold for all involved statesindividually). We then note that at | x | ≥ √ f (cid:48) s thelocal electron density n Friedel ( x ) due to all states with | q | < f (cid:48) /s at chemical potential ε Friedel can also be ob-tained semiclassically.Since both N Friedel and the electron density n Friedel ( x )for | x | ≥ √ f (cid:48) s may be found semiclassically, we con-clude that also N s Friedel , the total number of electrons at | x | ≤ √ f (cid:48) s for chemical potential ε Friedel and | q | < f (cid:48) /s can be calculated semiclassically: N s Friedel = N Friedel − (cid:82) | x |≥ √ f (cid:48) s dx n Friedel . We next decompose the electrondensity at | q | < f (cid:48) /s as n Friedel = n v + n − + n + , wherewe also introduce the charge density n + of all states with0 < ε < ε Friedel and | q | < f (cid:48) /s . With this notation wehave N s Friedel = (cid:82) √ f (cid:48) s − √ f (cid:48) s dx ( n v + n + + n − ) may be evalu-ated semiclassically. Using now that the density n v dueto all states with ε < − vf (cid:48) /s and | q | < f (cid:48) /s is semiclassi-cal, we conclude that also N s + − = (cid:82) √ f (cid:48) s − √ f (cid:48) s dx ( n + + n − )may be evaluated semiclassically.In order to bound the screening field we need boundsnot only on the integral of the density, but on the elec-tron density itself. We do this by bounding its variation.Squaring Eq. (2), we have (cid:12)(cid:12) ψ † q ( x ) ∂ x ψ q ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ψ † q ( x ) ψ q ( x ) (cid:12)(cid:12) ( | ε − V ( x ) | /v + | q | )(19)for states with transverse momentum q and energy ε .Summing over all involved q and ε , one finds | ∂ x n − ( x ) | ≤ f (cid:48) n − ( x ) /s (20)for | x | < √ f (cid:48) s . We finally integrate Eq. (20) from x = − √ f (cid:48) s to − s < x < s to find n − ( x ) ≤ n − ( − (cid:112) f (cid:48) s ) + (4 f (cid:48) /s ) (cid:90) x − √ f (cid:48) s dx (cid:48) n − ≤ n − ( − (cid:112) f (cid:48) s ) + (4 f (cid:48) /s ) (cid:90) √ f (cid:48) s − √ f (cid:48) s dx (cid:48) ( n − + n + ) ≤ n − ( − (cid:112) f (cid:48) s ) + 4 f (cid:48) N s + − /s, (21)where we have used the non-negativity of n − and n + . Asshown above, both n ( − √ f (cid:48) s ) and N s + − may be calcu-lated semiclassically. Equation (21) therefore allows us to establish an upper bound on the total particle density n = n lq + n v + n − at | x | < s and chemical potential ε = 0 semiclassically. The calculation is straightforwardand using the resulting bound on n in Eqs. (10) and (9),we find that the screening field E sc has negligible effect,that is | E sc | (cid:28) E at | x | < s , if d (cid:28) d c , where d c is thecritical distance given in Eq. (11). VIII. APPENDIX B
In this appendix we derive the transfer matrix of elec-trons through the potential Eq. (15) in the approxima-tions described in section IV B. Exploiting the inversionsymmetry of the potential V , we may restrict our anal-ysis to x > x < P = σ y R , where R denotes reflection on the line x = 0, such that R ψ ( x ) = ψ ( − x ). Therefore, from thetransfer matrix M at x, x (cid:48) >
0, one obtains the one for − x, − x (cid:48) < P M P − . Then, from M ( x,
0) for x > M (0 , y ) = σ y M − ( − y, σ y (22)at y <
0, and accordingly the transfer matrix throughthe entire ribbon is M ( x, y ) = M ( x, M (0 , y )= M ( x, σ y M − ( − y, σ y , (23)with x > y <
1. Adiabatic (semiclassical) regions
Following the work of Cheianov and Falko, the transfer matrix in the adiabatic regions, where | qV (cid:48) /V ( V /v − q ) | (cid:28)
1, is found using the transfor-mation Y ( x ) = 1 V ( x ) (cid:18) iχ iχ ∗ V ( x ) V ( x ) (cid:19) , (24)with χ = v [ q + ik ( x )] and the longitudinal wavenumber k ( x ) = (cid:112) [ V ( x ) /v ] − q . For our form of V , the trans-verse momentum q is negligible compared to k through-out the adiabatic regions with | qV (cid:48) /V k | (cid:28)
1, and Y ( x )simplifies to Y ± = (cid:18) − sgn V sgn V (cid:19) , (25)where sgn V is the sign of the potential V ( x ). The trans-fer matrix for the adiabatic regions M ad ( x, y ) is thengiven by M ad ( x, y ) = Y ( x ) ˜ M ad ( x, y ) Y − ( y ) , (26)where the matrix ˜ M ad ( x, y ) satisfies an equation thatsimplifies in the adiabatic limit (cid:12)(cid:12) qV (cid:48) /V k (cid:12)(cid:12) (cid:28) ∂ x ˜ M ad ( x, y ) = ik ( x ) σ z ˜ M ad ( x, y ) . (27)
2. Non-adiabatic regions
The adiabatic condition | qV (cid:48) /V k | (cid:28) x = x ± , where k ( x ) = 0, andat x , where V ( x ) = 0. We assume that the length ofthe entire non-adiabatic region at x >
0, which includesthe interval ( x − , x + ), is small on the scale x on whichthe potential varies, as discussed above. In this limit, wemay approximate the potential V ( x ) linearly in the nonadiabatic region, V ( x ) ≈ ρx ( x − x ) = 2 (cid:112) V b ρ ( x − x ) , (28)and we have x = x ± (cid:96) .Electron transport through a linear potential ingraphene has an analytic solution. We use here the so-lution formulated in Ref. 26 for potentials that may belinearized in the non adiabatic regions and that reachasymptotic values at | x | → ∞ . To this end, we define anauxiliary potential ˆ V with asymptotic values at | x | → ∞ and a linear region around the turning points which coin-cides with the linear region of the true potential V , as inFig. 3. Due to the assumption | V b | (cid:29) v/ ˜ x , the potential V may be linearized throughout the non adiabatic regionat x >
0, and we may choose ˆ V to coincide with V inthat entire region. The transfer matrix for a GNR withpotential ˆ V is given by Eq. (14) with α = e πθ ,β ∗ = − e πθ/ √ πe iπ/ θ / iθ Γ(1 + iθ ) e iϕ ,ϕ = k F (cid:96) − (cid:90) ∞ (cid:96) (cid:104) ˆ k ( x ) − k F (cid:105) dx,θ = q v √ V b ρ , (29)where ˆ k ( x ) = (cid:113) [ ˆ V ( x ) /v ] − q .
3. Concatenation of adiabatic and non-adiabatic regions
In order to calculate the transfer matrix for transportthrough the entire right side of the potential x >
0, wecombine the transfer matrices through the adiabatic re-gions with the solution Eq. (29) for transport throughthe auxiliary potential ˆ V . Without loss of generality, wechoose the auxiliary potential ˆ V to coincide with the truepotential V not only in the non adiabatic region, but inthe entire region extending from the the potential mini-mum at x = 0 up to the right end of the non-adiabatic region at x (cid:48) . We obtain from Eqs. (13), (14), and (29)the transfer matrix ˆ M ( x (cid:48)∞ , x ∞ ) of the potential ˆ V , where x ∞ < x (cid:48)∞ > V :ˆ M ( x (cid:48)∞ , x ∞ ) = ˆ A + ( x (cid:48)∞ ) (cid:18) α β ∗ β α ∗ (cid:19) ˆ A − − ( x ∞ ) . (30)FIG. 3: Plots of the quadratic potential V in the GNR(solid curve) and the asymptotically constant potentialˆ V considered by Cheianov and Falko (dashed curve).The two potentials coincide in the region between thepoints x = 0 and x (cid:48) (here, primed coordinates refer topoints to the right of the origin). We define x suchthat V ( x ) = 0. The classical turning points for theright side of the potential at x ± = x ± (cid:96) lie within theregion where V can be linearized. The points x ∞ and x (cid:48)∞ are asymptotically far away from the turning points.To construct the transfer matrix M ( x,
0) from x = 0through the non adiabatic region to a point x > x (cid:48) inthe true potential V (see Fig. 3), we first perform an adi-abatic transfer in ˆ V from x = 0 to the point x ∞ < M ( x (cid:48)∞ , x ∞ ) to transport in the potential ˆ V from x ∞ to a point x (cid:48)∞ >
0. Next, we perform anotheradiabatic transfer in ˆ V from x (cid:48)∞ back to x (cid:48) , the right endof the non adiabatic region, where ˆ V and V begin to de-viate. The resulting transfer matrix describes transportfrom 0 to x (cid:48) in the potential ˆ V . Finally, we transportadiabatically in the true potential V from the point x (cid:48) to x , resulting in M ( x,
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0. In reality, Ti contacts n-dopegraphene, such that ρ < The sign of ρ , however, canbe changed by a particle-hole transformation U = σ z , andis thus straightforwardly accounted for in our calculation. In our above electrostatic calculation, the electric field E diverges at the edge of the contacts when d →
0, which onemay suspect could introduce electron scattering. We note,however, that this is an artifact of our approximation thatexpands U to first order in d . Moreover, the effects of thelarge electric fields that do emerge at the edges of metalcontacts to graphene were taken into account in the first-principles calculation of Ref. 21, and they were found toinduce only a negligible amount of electron scattering One checks that Eq. (9) implies semiclassical electrontransport at | x | > s not only for − eU , but also for theself-consistent potential V ( x ), provided that d stays belowthe critical distance d cc