Transport in Graphene: Ballistic or Diffusive?

Mario F. Borunda,1, 2, ∗ H. Hennig,2 and Eric J. Heller2, 3

1Department of Physics, Oklahoma State University, Stillwater, OK 74078, USA2Department of Physics, Harvard University, Cambridge, MA 02138, USA.

3Department of Chemistry and Chemical Biology,Harvard University, Cambridge, MA 02138, USA.

(Dated: September 25, 2021)

We investigate the transport of electrons in disordered and pristine graphene devices. Fano shotnoise, a standard metric to assess the mechanism for electronic transport in mesoscopic devices, hasbeen shown to produce almost the same magnitude (≈ 1/3) in ballistic and diffusive graphene devicesand is therefore of limited applicability. We consider a two-terminal geometry where the grapheneflake is contacted by narrow metallic leads. We propose that the dependence of the conductanceon the position of one of the leads, a conductance profile, can give us insight into the charge flow,which can in turn be used to analyze the transport mechanism. Moreover, we simulate scanningprobe microscopy (SPM) measurements for the same devices, which can visualize the flow of chargeinside the device, thus complementing the transport calculations. From our simulations, we findthat both the conductance profile and SPM measurements are excellent tools to assess the transportmechanism differentiating ballistic and diffusive graphene systems.

PACS numbers: 72.80.Vp, 73.23.Ad, 72.10.Fk

I. INTRODUCTION

As predicted by early band theory studies,1 charge car-riers in graphene are chiral quasiparticles that have a lin-ear Dirac-like dispersion relation resulting in fascinatingelectronic2 and transport3 properties. As a result of thislinear energy-momentum relationship, the valence andconduction bands intersect at the Dirac points locatedat the Brillouin zone corners. For neutral graphene, thechemical potential lies exactly at the Dirac points. Useof a gate potential can shift the chemical potential awayfrom the Dirac point and thus tune the charge density ofthe graphene device over a large range (as compared toother electronic materials). The high degree of controlover the charge density and the linear dispersion rela-tion are features that distinguish graphene from the two-dimensional electron gas in semiconductor heterostruc-tures and other metallic conductors. Understanding themechanism for charge carrier transport in graphene in thelow carrier density regime (the linear dispersion regime)and in the presence of disorder has received considerableattention.2–5

The wide variety of fabrication techniques6–11 has re-sulted in graphene devices that operate in different dis-order regimes. The strength of the disorder charge car-riers encounter can be quantified via the electron mobil-ity. The standard fabrication method of mechanical ex-foliation and deposition of graphene flakes on SiO2 sub-strates6 can result in good quality devices with mobilitiesapproaching a few tens of thousands cm2V −1s−1, valuescomparable to those found in conventional semiconductordevices. Yet, the substrate provides a source of impuri-ties12 and corrugations13,14 for the electrons to scatter,limiting the mean free path l to about a hundred nanome-ters. Etching away the substrate is routinely done to pro-duce suspended graphene devices, removing some sources

of scattering. Etching followed by current annealing in-creases the carrier mobility in suspended samples to val-ues exceeding 200,000 cm2V −1s−1, resulting in mean freepaths comparable to the size of the device.9,10 Similarsample quality (80,000 cm2V −1s−1 carrier mobility) isobtained in devices where the substrate is single-crystalhexagonal boron nitride.11

It is apparent that replacing oxide based substrates byhigh-quality crystal substrates or suspended graphene re-sults in devices with improved mobility and reduced elec-tron scattering events.9–11 For transport near or at theDirac point, where the density of states vanishes, the con-ductivity has a minimum value (4e2/πh), even for idealpristine graphene.15 Experimentally, the phase coherenttransport in graphene devices with short-and-wide ge-ometries has been characterized as ballistic16–18 indicat-ing evanescent wave transport. In an ideal (no impuritiesor defects) graphene device, the mean free path is longerthan the size of the device resulting in ballistic conduc-tivity. Thus, transport in graphene qualitatively changesfrom metallic diffusive in SiO2 substrates to what hasbeen characterized as ballistic in suspended systems andwhen using crystalline boron nitride substrates.

Our paper is organized as follows. In Sec. II we dis-cuss the usual transport measures, emphasizing the nullresults obtained from the Fano shot noise, and presenttwo alternative methods that can also estimate the trans-port mechanism. In Sec. III we present the computa-tional method used to model transport in two-terminalgraphene devices. In Sec. IV we report results of extract-ing the transport mechanisms from modeled transportmeasurements. We present a summary and conclusion inSec. V.

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II. TRANSPORT MEASURES

The possibility of several transport mechanisms makesit essential to develop measures that clearly distinguishand classify the different transport regimes. In princi-ple, Fano shot noise is the metric of choice to assess themechanism for electronic transport in mesoscopic devices.However, the Fano shot noise is of limited applicability,with the same value for the Fano factor found in disor-dered or pristine devices.

A. Shot noise and Fano factor

Shot noise measurements can determine the statisticsrelevant for transport in mesoscopic conductors.19 Shotnoise is a consequence of charge quantization and can beassessed by the Fano factor defined as the ratio of noisepower and mean current,

F =

∑Nn=1 Tn(1− Tn)∑N

n=1 Tn. (1)

From the above definition, the Fano factor should be zerofor conventional ballistic systems, i.e. if perfect transmis-sion is present. It has also been established that for Pois-son processes F = 1 and for diffusive metallic conductorsF = 1/3.20

However, the Fano factor calculated for pristinegraphene,15,21,22 disordered systems,23,24 samples withsubstrate roughness,25 and for transport along an n-p junction26 all have similar magnitude (≈ 1/3) tothat found in diffusive metallic conductors, making theFano shot noise a cumbersome measure of the trans-port mechanism in graphene devices.3 Calculations inpristine graphene predict that the Fano factor at theDirac point is identically 1/3 when (1) a wide device(where the aspect ratio W/L is above 3) is contactedby doped graphitic leads,15 (2) when the leads are quan-tum wires creating an effective-contact model simulat-ing the metallic lead/graphene interface,21 and (3) forgraphene junctions with realistic metal contacts.22 Awayfrom the Dirac point, the Fano factor also indicates dif-fusive metallic transport.21,22 A slightly smaller yet uni-versal value (F = 0.295) was found numerically for dis-ordered systems.23,24 When disorder originates mainlyfrom roughness in the substrate, the Fano factor liesslightly above the 1/3 value and has been shown nu-merically to increase slightly with disorder.25 Transportalong an n-p (electron rich/hole rich) junction is selec-tive of those quasiparticles that approach the n-p inter-face almost perpendicularly and results in a Fano factorof F = 1 −

√1/2.26 Again, in all of these different situ-

ations, the Fano factor measurement results in 1/3 or avalue numerically close to this number.

Experiments have found similar results. The Fano fac-tor measured in clean devices18 is close to the value of

1/3 found analytically for ballistic samples.15 Measure-ments in disordered devices17 follow the numerical trendsof Ref. 25. These similar shot noise values warrant de-veloping an alternative quantitative understanding of thecarrier dynamics crucial for testing the transport regimeof graphene devices.

B. Conductance profiles

Our quest for a better measure to assess transportmechanisms begins with a simple question: How does theconductance change with respect to the vertical displace-ment of one of the leads? By calculating the conductancebetween two metallic contacts, we model transport intwo-terminal devices. We extract the conductance profilefrom the dependence of the conductance on the positionof one of the leads, and use this to distinguish the mech-anism of transport. In particular, we study the transportproperties of clean and disordered graphene devices con-tacted by narrow metallic leads where one of the twoleads can be moved along the edge of the device. More-over, we simulate scanning probe microscopy (SPM)27

measurements for the same devices, which sheds light onthe charge flow inside the device.28

Diffusive transport in disordered systems is based onthe charge carriers scattering multiple times off impu-rities or boundaries as they traverse the system. Thismechanism can be described classically by a randomwalk. In ballistic transport, the charge carriers traversethe system with minimal scattering. Ballistic transport isexpected in an ideal graphene strip, given that the crystallattice has no defects and no impurities are present. Yetfor ideal graphene, the dynamics of the electrons producethe same shot noise as that found in classical diffusion.15

Determining the transport mechanism in graphene de-vices is important given that transport experiments arepossible in the quasi-ballistic limit, that is, where themean free path is of the order of the size of the system.

Here, we argue that the transition from diffusive toballistic transport in graphene, along with the limitingcases, can be quantified by alternative methods based onmeasuring the flow of charge in the system. First, theconductance profile, that is, the conductance as a func-tion of the vertical displacement of one of the leads, cangive a measure of the flow inside the device. For the caseof ballistic transport, scattering does not impede the flowof charge carriers and the conductance profiles are welldescribed by a linear (triangular) fit due to the convolu-tion of the square windows created by the two leads. Inthe case of diffusive transport, the charge carriers expe-rience several scattering events and the transport is thenwell described by Brownian motion with a drift. Thus,charge flow will be a Gaussian function characteristic ofdiffusion; for this reason, the conductance profile can befitted to a Gaussian function.

3

FIG. 1: (Color online) Schematic setup for the transportcalculation. A rectangular graphene flake, depicted by redand blue atomic sites arranged in a honeycomb lattice, iscontacted between two narrow metallic contacts, depicted bythe green atomic sites arranged in a square lattice, form-ing a normal-conductor/graphene/normal-conductor junc-tion. This schematic is not to scale; simulated devices wereconsiderably bigger with ∼ 700 carbon atoms along the zigzagedge.

III. COMPUTATIONAL METHOD

We use the nonequilibrium Green’s function (NEGF)formalism30 to calculate the transport properties of anormal-conductor/graphene/normal-conductor junction,given that in experiments the electronic contacts are usu-ally metallic. The metallic lead/graphene interface wehave investigated follows that of Robinson and Schome-rus.31 A schematic of the geometry of the devices in-vestigated is shown in Fig. 1, where rectangular (widthW and length L) graphene flakes have armchair bound-aries along their longitudinal direction and narrow metal-lic leads connected along the zigzag boundary. The twometallic terminals of width WL are modeled by semi-infinite square lattice regions that have a quadratic dis-persion relation.

The tight-binding model Hamiltonian of the device isgiven by

H =∑i

εic†i ci −

∑〈i,j〉

γijc†i cj , (2)

where εi is the on-site potential of the lattice site i, ci(c†i )

is the annihilation (creation) operator acting on site i,the second sum is over nearest neighbors 〈i, j〉, and γijdenotes the hopping matrix elements. The on-site po-tential changes due to contributions from impurities andthe gate voltage applied to the device, εi = εimp + εgate.The graphene section consists of atomic sites placed inthe hexagonal lattice with lattice constant a. As out-lined in Ref. 31, metallic contacts are modeled as a regionwith a square lattice arrangement with lattice constant

aL =√

3a, matching the A(B)-atom in the zigzag inter-face at the left(right) of the graphene flake.

Disorder is introduced in two different ways. To gen-erate edge-disordered samples, we randomly remove 30%of carbon atoms from sites located in the three outeratomic layers of the device.23 This type of disorderededge without disorder puddles in the bulk of the systemis a reasonable model for suspended graphene devices.

Bulk disordered in graphene devices is linked to thepresence of charged impurities in the substrate.32 To gen-erate such disorder potentials, Nimp lattice sites are se-lected at random from the total number of atomic sites(Ntot) in the device. The position Ri of each of the Nimp

lattice sites has an on-site potential amplitude Vi chosenrandomly from the interval (−δ, δ) and smoothed out overa range ξ by convolution with a Gaussian,

εimp(Ri) =

Nimp∑i=1

Vi exp

(−|r −Ri|2

2ξ2

). (3)

The range of the convolution is important given that inthe extreme case of ξ << a the atomic scale disorderpotential would break the A-B symmetry caused by hav-ing two atoms in the unit cell. For our calculations weassume that ξ = 2a resulting in a short-range potential(when compared to other length scales in the system)that varies smoothly on the atomic scale and suppressesthe effect of intervalley scattering. The parameters Nimp

and δ determine the mean free path l of the disorderedsystem. Using the Born approximation, the mean freepath can be quantified as32,33

l =4

kFK0, (4)

where kF is the Fermi wave vector and K0 is the dimen-sionless correlator given by

K0 =LW

(~vNtot)2

Ntot∑i=1

Ntot∑j=1

〈εimp(Ri)εimp(Rj)〉. (5)

For the type of bulk disordered systems considered here(ξ << L,W ), Rycersz et al.32 have found that

K0 =

√3

9

δ2

γ2Nimp

Ntotκ, (6)

where

κ =1

Nimp

Nimp∑i=1

Ntot∑j=1

exp

(−|r −Ri|2

2ξ2

). (7)

The NEGF formalism30,34 is a sophisticated frameworkfor obtaining the transmission and other quantities inrealistic devices. The retarded Green’s function in theatomic orbital basis is given by

Gr(E) = [E −H − Σ(E)]−1, (8)

4

where the non-Hermitian self-energy matrix

Σ(E) = ΣL(E) + ΣR(E), (9)

introduces the effect of attaching leads to the left andright ends of the device. The energy E tunes the Fermilevel from the Dirac point (E = 0) to any charge den-sity induced by the gate voltage in the device. The self-energies determine the escape rates of electrons from thedevice into the semi-infinite ideal leads. Using the Lan-dauer formula35, it is possible to obtain the transmissionfunction

T (E, Vds) = Tr [ΓR(E, Vds)GrΓL(E, Vds)G

a] , (10)

where Ga(E) = (Gr(E))† is the advanced Green’s func-tion. The matrices

Γp(E, Vds) = i

[Σp

(E − eVds

2

)− Σ†p

(E +

eVds2

)](11)

introduce a level broadening due to the coupling of theleads and a source-to-drain voltage given by Vds. TheFano factor [Eq. (1)] for a particular energy and Vds iscalculated using the following expression,

F = 1− Tr [ΓRGrΓLG

aΓRGrΓLG

a]

T. (12)

Finally, the current in two-terminal devices can be ob-tained from the Landauer formula

I(Vsd) =2e

h

∫ ∞−∞

dE T (E) [f(E − µL)− f(E − µR)]

(13)where the energy window is defined from the difference ofthe Fermi functions of the macroscopic reservoirs wherethe leads terminate. In our calculations we assume thelinear response limit (Vds → 0), where the relationshipbetween conductanceG and current is given by I = GVds.

IV. RESULTS AND DISCUSSION

A. Fano shot noise

We investigate rectangular graphene devices with atwo-terminal geometry where the source and drain leadsare assumed to be perfect ballistic and metallic con-ductors. This geometry has been investigated previ-ously in the context of quantum scars in graphene36 andfor the geometry-dependent conductance fluctuations ingraphene quantum dots.37 For all results presented hereand as illustrated in Fig. 1, the edges of the graphenealong the transport direction are in the armchair config-uration.

We use Eq. 12 to calculate the Fano factor for pristine,edge disordered, and bulk disordered graphene devices.As presented in Fig. 2, our results are similar to thosefound in the literature.15,21–25 The Fano factors found are

Pris

tine

D. E

dge

D. B

ulk0

0.1

0.2

0.3

0.4

F

L = 85 nmL = 44 nmL = 23 nmL = 11 nm

FIG. 2: (Color online) Fano factor at a Fermi energy EF =0.5γ (with hopping matrix element γ). The limit F = 1/3as W/L → ∞ is shown as a dotted line. The data points ofthe disordered systems have error bars showing the varianceof the value over several realizations. Each of the data pointscorresponds to different system lengths (L = 11, 23, 44, and85 nm) while the widths of the graphene device and of theleads are kept constant at W = 74 nm and WL = 10 nm,respectively. The mean free path is l = 27 nm in the systemwith bulk disorder.

near the theoretical value F = 1/3 that applies to pris-tine graphene as W/L → ∞ near the Dirac point limit.As Towrzydlo et. al. reported,15 the F = 1/3 value isa theoretical maximum for armchair edge graphene de-vices, increasing the charge density reduces the value ofthe Fano factor.

For pristine devices, the Fano factor does show a trendas a function of the system size. The longer device calcu-lated (83 nm) presented the lowest value of F while thelargest value corresponds to the 44 nm graphene sample.Likewise, the Fano factor for the graphene devices withdisordered edges do not exhibit a trend in the sizes stud-ied. However, the fluctuations in F from the mean value,illustrated with the standard deviation (error bars), showthat as the devices get longer, the deviation from themean increases. In contrast, for diffusive graphene, wefind that the value of the Fano factor does show a mono-tonic dependence as a function of the W/L ratio, ap-proaching the F = 1/3 value in the larger devices. Yet,given that the shot noise metric is not significantly differ-ent from the value found in classical diffusive transport,even in the narrow-leads geometry, the Fano factor is aninadequate metric for predicting the transport mecha-nism.

5

B. Transport and the profile of the conductance

In order to visualize the charge flow at the edge ofthe sample, we calculate the profile of the conductanceG(∆y), that is, the conductance as a function of the po-sition of the drain lead with respect to the source leadthat remains fixed. In Fig. 3, we present the conductiv-ity as a function of the displacement of the drain lead∆y for several system lengths (L=11, 21, 43 nm) and forthe three models considered here. Both the width of thedevice and the width of the leads are kept constant atW = 74 nm and WL = 10 nm, respectively. Thus, theschematic shown in Fig. 1 is not representative of thesize of the devices considered in this work as our calcula-tion are considerably larger with about 700 atomic sitesalong the zigzag edge (width of system). Numerical cal-culations of these system sizes require the use of efficientrecursive Green’s function methods.

While the displacement of one of the leads from thecenter of the device is a theoretical construct that canbe easily investigated numerically, the experimental im-plementation is not trivial. Our proposal of “moving theleads” could be realized by fabricating several sampleswhere one of the leads is attached at a different locationin each device. Another possibility is to replace one of themetallic leads by the tip of a scanning tunneling micro-scope. When the tip is brought close enough to the edge,the tunneling current could be measured as a function ofthe position of the scanning tunneling microscope.

Figure 3(a) corresponds to pristine graphene devices,Fig. 3(b) corresponds to realizations with edge disorder,and Fig. 3(c) to systems where the presence of bulk disor-der would form electron-rich and hole-rich puddles38 andthe mean free path is l = 27 nm, as estimated from theBorn approximation. Figure 3(a) shows that the maxi-mum conductance is associated with the two leads beingcollinear (∆y ≈ 0). For a short device the maximumconductance occurs at ∆y = 0 and as the length of thesystem in increased there are local maxima near ∆y = 0.Increasing the length of the device reduces the conduc-tance peak due to the increased number of reflections atthe boundary of the sample. Yet, we find an envelopeof maximum conductance when the vertical positions ofthe two leads overlap with each other. Similar featuresare seen in the conductance profiles for the disorderedsamples. We find that for all three systems the widthof the central peak is approximately 2WL, corroboratingthe idea that the peak in the displacement conductanceis due to the overlap of the two leads.

Although our interest is to extract the transport mech-anisms from the conductance profiles, there are cer-tain features visible in the conductance curves of Fig. 3that are worth explaining such as the fluctuations nearthe edges of the conductance curves. We carefullychecked that these fluctuations are not only present forindividual conductance curves and remain after aver-aging over an energy window. Thus, the fluctuationsare consistent with universal conductance fluctuations

(a)

(b)

(c)

FIG. 3: (Color online) Profile of the conductance for pristine,edge disordered, and bulk disordered systems. The conduc-tance versus displacement of one of the leads from the centerof the device G(∆y) is calculated at a Fermi energy EF = 0.5γ(with hopping matrix element γ). Each of the curves corre-sponds to different system lengths (L = 11, 21, and 43 nm)while the width of the graphene device and of the leads arekept constant at W = 74 nm and WL = 10 nm, respectively.(a) Clean pristine system. (b) System with edge disorder. (c)System with bulk disorder where the mean free path is l = 27nm.

(UCFs),39,40 which we therefore review.

In a disordered mesoscopic conductor where the systemis of comparable size or smaller than the phase coherencelength of the charge carriers but large compared to theaverage impurity spacing, the transmission of carriers isaffected by interference of many different paths through

6

−10 −5 0 5 102

3

4

5

6

7

∆ y [nm]

G[2e2/h]

−30 −20 −10 0 10 20 301

2

3

4

5

6

7

∆ y [nm]

G[2e2/h]

E F = 0.499t

−30 −20 −10 0 10 20 301

2

3

4

5

6

7

∆ y [nm]

G[2e2/h]

E F = 0.499t

−10 −5 0 5 102

3

4

5

6

7

∆ y [nm]

G[2e2/h]

−30 −20 −10 0 10 20 301

2

3

4

5

6

7

∆ y [nm]

G[2e2/h]

E F = 0.499t

−10 −5 0 5 102

3

4

5

6

7

∆ y [nm]

G[2e2/h]

(a) (b)

(c) (d)

(e) (f)

FIG. 4: (Color online) The transport mechanism is classified according to the shape of the central peak (of width 2WL centeredaround ∆y=0) of the conductance profile curves. G(∆y) for the pristine system (a,b) is better fit by a triangular fit indicative ofballistic transport. In systems with edge disorder (c,d) and bulk disorder (e,f), the central peak is well described by a Gaussianenvelope pointing to diffusive transport. The dashed lines indicate the region where the two leads have an overlap. Parametersfor all calculations are EF = 0.499γ, L = 11 nm, W = 74 nm and WL = 10 nm.

the system. As these paths are typically long comparedto the wavelength of the charge carriers, the accumulatedphase along the paths changes randomly in response tovariation in an external parameter (e.g., the magneticfield or gate voltage). This results in a random interfer-ence pattern and reproducible fluctuations in the conduc-tance of a universal magnitude on the order of 2e2/h.39,40

UCFs can also be created by the displacement of a singlescatterer.41–43 But how can UCFs occur in a ballistic de-vice? The role of disorder in providing a distribution ofrandom phases can as well be taken by chaos. Thus, bal-listic mesoscopic cavities like quantum dots in high mo-bility two-dimensional electron gases that form chaoticbilliards show the same universal fluctuations.44–46 Re-

7

markably, experiments in graphene quantum dots foundstrong indications of chaos in billiard systems.47

In Fig. 4 we presents our conductance profile analysisfor the 11 nm devices. A striking feature of these con-ductance profiles is the apparent fit of the peaks in thedisplacement conductance curves to either triangular orGaussian functions. The conductance profile for a pris-tine device of length L = 11 nm is shown in Fig. 4(a) andthe peak is presented in Fig. 4(b). The envelope of thepeak in the conductance is well described by a triangularfit (red line). In the case of conventional ballistic trans-port, the classical expectation for the charge flow at theedge is a triangular shape of width 2WL. This triangu-lar shape is the result of the convolution of two squarewindows, where the window sizes are given by the leadwidth WL. Gaussian fits for the conductance profile (notshown) differ significantly particularly near the cusp ofthe curve.

Diffusive transport is based on the multiple scatteringpaths taken by charge carriers as they transverse a de-vice. In Fig. 4(c) we present the conductance profile foran edge disordered system of length L = 11 nm and aclose-up of the peak with a fit shown in Fig. 4(d). In thiscase, and in contrast with the wide and ballistic device,the data for the edge-disordered device is best fitted bya Gaussian curve, where the envelope of the conductanceprofile is the result of the spatial overlap of the leads.Similarly, a Gaussian curve describes well the conduc-tance profile of a diffusive system, as seen in Figs. 4(e)and 4(f). We note that Barthelemy et al.29 used a sim-ilar approach to distinguish normal diffusion (reflectedby a Gaussian profile) from the anomalous transport as-sociated with Levy transport. While we have been ableto account for the features of the envelope of the centralpeak of width 2WL in a quasi-classical way, the conduc-tance profile curves exhibit several quantum interferenceeffects that need a different interpretation. Along withUCFs, Fabry-Perot conductance interferences show upas regularly spaced fluctuations seen in the conductanceprofiles of the pristine systems.

C. Charge flow

Scanning probe microscopy (SPM)27 has recently beenused to image mesoscopic transport effects such as uni-versal conductance fluctuations48 and weak localization49

in graphene devices. SPM has also been used to imagecurrent transport in a graphene quantum dot connectedto leads through two small constrictions, imaging con-ductance resonances of the quantum dot and observinglocalized states.50 The tip of the cryogenic SPM capaci-tively couples to the graphene device inducing a movablescatterer. Conductance maps created via SPM have beenproposed as a way to probe the chiral nature of the chargecarriers in graphene.51 The work carried out by Bere-zovsky et al.48,49 and Schnez et al.50 probed the regimeof coherent diffusive transport in graphene. In light of

the above, it is interesting to consider the possibility ofusing SPM to gain insight into the transport mechanismby imaging the flow of charge carriers – particularly, theballistic to diffusive crossover regime in graphene devices.

We simulate the effect of the capacitively-coupled tipof the SPM on the graphene device as a point chargeq located a height a above the substrate. The chargeinduces a local charge density perturbation given by

n(ρ) =−qa

2π(ρ2 + a2)3/2(14)

where ρ is the in-plane radial coordinate away from theposition of the point charge. The charge q is chosen toyield an rms charge density n ≈ 4 × 1011 e cm−2, inagreement with the observed charge puddles in graphene.The tip height a controls the width of the induced chargedensity; we have chosen a = 10 nm, a routinely used tip-to-substrate distance in SPM experiments which resultsin a half-width for the induced density puddle of about20 nm.

During SPM measurements, a transport measurementis carried out while the tip of the microscope scans theover the device. The resulting conductance maps aregenerated by calculating the conductance as a functionof SPM tip position G(x, y) and rastering the tip positionin a plane above the device. The effect of the SPM tipis included by adding to the on-site potential of eachof the lattice sites the energy due to the charge densityperturbation created by the tip. The total on-site energyis obtained from the contributions of the perturbationfrom the tip, the effect from the gate voltage, and theimpurities present, εi = εimp + εgate + εtip.

In the previous section we extracted conductance pro-files that allowed the classification of transport as ballisticor diffusive. To explain these observations and to obtainan intuitive picture of the transport mechanisms, we ex-plore charge flow as a function of the position of one ofthe leads. As in the conductance profile calculations, theinjection lead remains fixed in the left edge of the sample.The two terminals are identical. As seen in the panels ofFig. 5, the SPM conductance maps exhibit a significantchange in conductance when the charge flow between theleads is obstructed or enhanced by the presence of theperturbation induced by the scanning probe.

The conductance map shown in Fig. 5(a) was obtainedwhen the right lead is positioned near the bottom edge ofthe device. As the scanning probe rasters along the tophalf of the device, the change in conductance is minimalas reflected by the mostly uniform conductance in thatregion of the image. However, as the tip scans the bottomhalf of the device, the conductance map shows a region oflower conductance that connects the left lead to the rightlead, as indicated by the large blue feature across the bot-tom of the image. In this case, the trans-conductance isreduced when the local perturbation induced by the tip ofthe SPM is over a region of considerable charge flow. InFig. 5(b), the right lead contacts the device at a slightlyhigher position. Consequently, the blue feature corre-

8

FIG. 5: (Color online) Charge flow in pristine graphene obtained from the SPM conductance map calculations. The maps ofconductance as a function of SPM tip position, G(x, y), show the charge flow in the graphene device. Each snapshot presentsthe conductance map for different position of the right lead, starting from left at the bottom portion of the device, towardsthe right where the right lead is located at the same height of the left lead. We can see that the majority of the flow is overa path connecting the left and right lead. When the two leads are collinear, there are several bands of enhanced conductancedue to the interference caused by paths that scatter from the tip and those that go from one lead straight to the other lead.The Fermi energy is set to EF = 0.5γ.

sponding to the region of charge flow is now at a higherposition, “following” the position of the right lead. Fur-thermore, there are two regions of higher conductance inthe image, corresponding to an increase of the conduc-tance due to the presence of the local perturbation. Thehigher conductance (red) bands are a consequence of theredirection of the charge flow that without the tip beingpresent would not have contributed to the conductance.Thus, the tip redirects charge into the drain lead thatotherwise would not have exited the device.

Similarly, placing the right lead closer to the centralregion of the device (Fig. 5(c)), we find that the posi-tion of conductance features are correlated with the pathstarting at the left lead and ending in the right lead. Inall images, the effect of narrow lead constrictions is seenfrom the size of the band of near constant conductance.Surprisingly, the feature does not become significantlywider as the charge flows away from the source lead. Allof the panels present conductance fluctuations of order∼ ±2e2/h.

The conductance map presented in Fig. 5(d) corre-sponds to a device where the leads are collinear (∆y = 0).Here, the conductance is higher than in the ones to theleft. Along with the central band of conductance drop,Fig. 5(d) also shows regions of constructive interferencealong bands parallel to the transport. The limit to thesize of the features present in the conductance maps isproportional to the Fermi wavelength of the system27,28,which for graphene is inversely proportional to the Fermienergy given by

λF = 2πvFEF

. (15)

The Fermi wavelength in the system presented in Fig. 5is λF = 2.7 nm. Figure 6 presents the simulated conduc-tance maps for the same configuration of the leads forEF = 0.1 γ and EF = 0.9 γ, corresponding to Fermiwavelengths of 13 nm and 1.5 nm. The resolution re-solved in the images, i.e., the size of the pixels in each of

FIG. 6: (Color online) Conductance maps for two Fermi en-ergies, (left) EF = 0.1γ and (right) EF = 0.9γ, correspondingto Fermi wavelengths of 13 nm and 1.5 nm, respectively.

the maps, is also 1.5 nm.

Previous analytical results of Braun et al.51 considerballistic trajectories between two constrictions in thepresence of the tip scattering potential. To first order,as treated in Ref. 51, there are only two trajectories thatinterfere with each other: the source-to-drain trajectoryand the source-tip potential-drain trajectory, revealingan interference pattern in the conductance maps. In par-ticular, the first order calculation predicts that the in-terference pattern depends on the Fermi wavelength ofthe system. Comparing the conductance maps for threedifferent energies (Fig. 5(d) and those in Fig. 6), the in-terference pattern are very similar. Increasing the Fermienergy of the system does not reveal the Fermi wave-length of the particles but rather results in better resolu-tion of the interference patterns. In contrast to analyticalresults,51 our numerical calculation treats all possible tra-jectories including reflections from the edge of the deviceand multiple scattering events. We believe that the pat-tern found is due to the redirection of the charge carriersas the tip scans the device and does not dependent on thewavelength but rather on the geometry of the system andthe interference of several trajectories the charge carrierscan take as they traverse the device.

9

FIG. 7: (Color online) SPM conductance maps simulations for disordered systems. (Left) Graphene sample with disorderededges and (middle, right) devices with bulk disorder. While the interference pattern survives in devices with edge disorder, itis not present in systems with bulk disorder. At a mean free path of l = 108 nm, larger than the system size, L = 65 nm, themobility of the disordered devices presented in the middle panel is comparable to that of devices in crystalline substrates. Thesimulation presented in (c) is for a system with l = 24 nm. Other parameters are as in Fig. 5.

In what follows we study the effect of edge and bulkdisorder on the simulated conductance maps. In Fig. 7,we present conductance maps for graphene devices withedge disorder (left panel) and bulk disorder (middle andright panels). As seen in the left panel, the interferencepattern is present in edge-disordered graphene devices.The effect of disorder at the edges of the device inducesrandom scattering, modifying previously interfering tra-jectories and thus reducing the strength of the pattern.This is more noticeable near the edge of the device. Still,the result for edge-disordered systems is a comparablepattern found for pristine graphene (Fig. 5(d)).

Our calculations show an important difference for bulk-disordered systems. Surprisingly, the SPM simulationsindicate diffusive rather than ballistic transport (in themesoscopic systems considered) even on the regime wherethe mean free path (estimated from the Born approxima-tion) is longer than the system size. This result does notquestion the validity of the Born approximation; rather,it shows the sensitivity of the SPM method for the geom-etry studied. Even when the size of the system is smallerthan l, for this geometry, the oath traveled by the chargecarrier from initially entering the device to exiting at thedrain lead can be significantly longer. Our calculationstake into account all possible trajectories that the chargecarriers can take, including the possibility of several re-flections from the leads and multiple scattering eventsfrom the walls of the device. The presence of randomdisorder due to charged impurities in the substrate re-sults in conductance maps with no interference patterns,(Fig. 7, middle and right panels). The middle panel ofFig. 7 is the conductance map for a system where themean free path (l = 107.5 nm) is larger than the size ofthe device (L = 65 nm). This value of l is attainablein graphene devices with a crystalline BN-substrate, re-sulting in relatively high mobility11. The mean free pathof the system pictured in the right panel is l = 24 nm,comparable to high quality graphene devices in a siliconoxide substrate.

Ballistic transport is expected when the mean freepath is larger than the size of the system. In contrast,in graphene devices with weak bulk disorder such thatl > L, we find diffusive transport signatures. This indi-cates that for the system and sizes considered, multiplereflections of the Dirac electrons from the edges domi-nate. This mesoscopic effect needs to be carefully con-sidered in both theoretical and experimental studies onmesoscopic graphene flakes. As a result, SPM imagingcould be used to explore the ballistic to diffusive tran-sition in graphene devices. The visible interference pat-tern in the conductance maps would be robust againstthe presence of edge disorder, inevitable in most exper-imental realizations, but is sensitive to the presence ofdisorder due to bulk disorder in the system. Finally, wewould like to comment on possible experimental realiza-tions of SPM measurements in graphene. Several SPMmeasurements have been carried out for graphene deviceson a SiO2 substrate48–50 and can be readily carried out insuspended graphene, where the ballistic transport regimecan be reached, in graphene membranes,52 and in deviceswith crystalline substrate, where the ballistic to diffusivecross-over should be observable.

V. CONCLUSION

We have proposed a method to study the mechanismsof electronic transport based on displacement conduc-tance, that is, the conductance as a function of the po-sition of the drain lead in a two-terminal device. Themethod extracts the conductance profile of the chargeflow at the edge of the device and it can be applied todiscern the mechanism of transport from ballistic to dif-fusive. It is worth noting that this technique might becapable of uncovering signatures of anomalous transportin mesoscopic systems. Several quantum interference ef-fects (Fabry-Perot resonances and UCFs) are also foundin the resulting conductance curves. The method that

10

we applied to graphene is general, resolves the transportmechanism in graphene which cannot be accomplishedvia Fano shot noise, and can be used as well to studyother systems such as semiconductor heterostructures.

In this article, we simulated SPM measurements forthe same devices considered in the conductance profilestudy. Probe imaging is important as we are able to vi-sualize the flow of charge and do not have to rely ontransport metrics. For the graphene devices considered,the SPM simulations suggest diffusive rather than ballis-tic transport even in the “clean” regime where the meanfree path is larger than the systems size. Our numericalresults suggest that in the process of escaping the de-vice, charge carriers perform multiple reflections. Fromour simulations, we expect that SPM measurements arewell suited to study the crossover between ballistic anddiffusive transport in graphene devices.

VI. ACKNOWLEDGMENTS

We are grateful to S. Barraza-Lopez, S. Bhandari,J. Berezovsky, and R. M. Westervelt for valuable dis-cussions. M.F.B. and E.J.H. were supported by theDepartment of Energy, Office of Basic Science (GrantNo. DE-FG02-08ER46513), and H.H. acknowledges fund-ing through the German Research Foundation (GrantNo. HE 6312/1-1). While most of the calculations wererun on the Odyssey cluster supported by the FAS at Har-vard University, the rest were performed on the Vaquerocluster supported by Oklahoma State University and out-fitted with a Tesla K20 GPU donated by the NVIDIACorporation.

∗ Electronic address: mario.borunda@okstate.edu1 P. R. Wallace, Phys. Rev. 71, 622 (1947).2 A. H. Castro Neto, F. Guinea, N. M. R. Peres,

K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81,109 (2009).

3 S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.Mod. Phys. 83, 407 (2011).

4 A. K. Geim, Science 324, 1530 (2009).5 A. V. Rozhkov, G. Giavaras, Y. P. Bliokh, V. Freilikher,

and F. Nori, Phys. Rep. 503, 77 (2011).6 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.

Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,Science 306, 666 (2004).

7 C. Berger, et al., Science 312, 1191 (2006).8 J. C. Meyer, A. K. Geim, M. I. Katsnelson,

K. S. Novoselov, T. J. Booth, and S. Roth, Nature446, 60 (2007).

9 K. I. Bolotin, K. J. Sikes, Z. Jiang, G. Fudenberg, J. Hone,P. Kim, and H. L. Stormer, Solid State Commun. 146, 351(2008)

10 X. Du, I. Skachko, A. Barker, and E. Y. Andrei, NatureNano. 3, 491 (2008).

11 C. R. Dean, et al., Nature Nano. 5, 722 (2010).12 S. Adam, E. H. Hwang, V. Galitski, and S. Das Sarma,

Proc. Natl. Acad. Sci. 104, 18392 (2007).13 M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A

366, 195 (2007).14 M. Gibertini, A. Tomadin, M. Polini, A. Fasolino, and

M. I. Katsnelson, Phys. Rev. B 81, 125437 (2010).15 J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, and

C. W. J. Beenakker, Phys. Rev. Lett. 96, 246802 (2006).16 F. Miao, S. Wijeratne, Y. Zhang, U. C. Coskun, W. Bao,

and C. N. Lau, Science 317, 1530 (2007).17 L. DiCarlo, J. R. Williams, Y. Zhang, D. T. McClure, and

C. M. Marcus, Phys. Rev. Lett. 100, 156801 (2008).18 R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi,

J. Salmilehto, A. F. Morpurgo, and P. J. Hakonen, Phys.Rev. Lett. 100, 196802 (2008).

19 Y. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000).20 C. W. J. Beenakker and M. Buttiker, Phys. Rev. B 46,

1889 (1992).

21 H. Schomerus, Phys. Rev. B 76, 045433 (2007).22 S. Barraza-Lopez, M. Kindermann, and M. Y. Chou, Nano

Lett. Article ASAP, (2012).23 E. Louis, J. A. Verges, F. Guinea, and G. Chiappe, Phys.

Rev. B 75, 085440 (2007).24 P. San-Jose, E. Prada, and D. S. Golubev, Phys. Rev. B

76, 195445 (2007).25 C. H. Lewenkopf, E. R. Mucciolo, and A. H. CastroNeto,

Phys. Rev. B 77, 081410(R) (2008).26 V. V. Cheianov and V. I. Fal’ko, Phys. Rev. B 74,

041403(R) (2006).27 M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller,

R. M. Westervelt, K. D. Maranowski, and A. C. Gossard,Science 289, 2323 (2000).

28 M. A. Topinka, B. J. LeRoy, R. M. Westervelt,S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Mara-nowski, and A. C. Gossard, Nature 410, 183 (2001).

29 P. Barthelemy, J. Bertolotti, and D. S. Wiersma, Nature453, 495 (2008).

30 H. Haug and A.-P. Jaujo, Quantum Kinetics in Transportand Optics of Semiconductors, 2nd Ed. (Springer, Berlin,2008).

31 J. P. Robinson and H. Schomerus, Phys. Rev. B 76, 115430(2007).

32 A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, EPL79, 57003 (2007).

33 H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603(2002).

34 B. K. Nicolic, L. P. Zarbo, and S. Souma, Spin cur-rents in Semiconductor nanostructures: a nonequilibriumGreen-function approach, In A. V. Narlikar and Y. Y. Fu(Eds.), The Oxford Handbook on Nanoscience and Tech-nology Frontiers and Advances, (Oxford University Press,Oxford, 2010).

35 R. Landauer, IBM J. Res. Develop. 1, 233 (1957).36 L. Huang, Y.-C. Lai, D. K. Ferry, S. M. Goodnick, and R.

Akis, Phys. Rev. Lett., 103, 054101 (2009).37 L. Huang, R. Yang, and Y.-C. Lai, EPL, 94, 58003 (2011).38 J. Martin, N. Akerman, G. Ulbricht, T. Lohmann,

J. H. Smet, K. Von Klitzing, and A. Yacoby, Nature. Phys.4, 144 (2008).

11

39 P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622(1985).

40 B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42,359 (1985).

41 S. Feng, P. A. Lee and A. D. Stone, Phys. Rev. Lett. 56,1960 (1986).

42 B. L. Altshuler and B. Z. Spivak, JETP Lett. 42, 447(1985).

43 M. F. Borunda, J. Berezovsky, R. M. Westervelt, and E. J.Heller, ACS Nano 5, 3622 (2011).

44 C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hop-kins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992).

45 R. A. Jalabert, H. U. Baranger, and A. D. Stone, Phys.Rev. Lett. 65, 2442 (1990).

46 H. Baranger, R. Jalabert, and A. Stone, Chaos 3, 665

(1993).47 L. Ponomarenko, F. Schedin, M. Katsnelson, R. Yang,

E. Hill, K. Novoselov, and A. Geim, Science 320, 356(2008).

48 J. Berezovsky, M. F. Borunda, E. J. Heller, and R. M.Westervelt, Nanotechnology 21, 274013 (2010).

49 J. Berezovsky and R. M. Westervelt, Nanotechnology 21,274014 (2010).

50 S. Schnez, J. Guttinger, M. Huefner, C. Stampfer, K. En-sslin, and T. Ihn, Phys. Rev. B 82, 165445 (2010).

51 M. Braun, L. Chirolli, and G. Burkard, Phys. Rev. B 77,115433 (2008).

52 W. L. Wang, S. Bhandari, W. Yi, D. C. Bell, R. Westervelt,and E. Kaxiras, Nano Lett. 12, 2278 (2012).