Phys. Status Solidi B 247, Nos. 11–12, 2983–2987 (2010) / DOI 10.1002/pssb.201000307 p s sb

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Resistance and mesoscopic fluctuationsin graphene

Keoni K. Mahelona1, Alan B. Kaiser1, and Viera Skakalova*,2

basic solid state physicswww.pss-b.comp

1MacDiarmid Institute for Advanced Materials and Nanotechnology, SCPS, Victoria University of Wellington, P.O. Box 600,

Wellington, New Zealand2Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany

Received 5 May 2010, revised 24 June 2010, accepted 14 August 2010

Published online 28 September 2010

Keywords autocorrelation, conductance, graphene, mesoscopic fluctuations, power spectrum

*Corresponding author: e-mail v.skakalova@fkf.mpg.de, Phone: þ49 689 1434, Fax: þ49 689 1010

We investigate the mesoscopic resistance fluctuations (MRFs)

that, as we showed previously, cause the exponentially

decaying low-temperature resistance anomaly in graphene

monolayers. We determine autocorrelation functions and

power spectra for the MRFs, finding that the fluctuations have

shorter periods (as a function of gate voltage) near the neutrality

point. We also investigate the origin of the sharp increase of

the resistance of graphene on a Si/SiO2 substrate at higher

temperatures and at higher charge carrier densities. We show

that a very good description of this increase is given by

scattering by the experimentally observed graphene zone-

boundary phonons of energy 160meV (that cause current

saturation in carbon nanotubes) and by lower energy phonons

(70meV) that make a smaller contribution.

An example of mesoscopic resistance fluctuations as a function

of gate voltage measured from the neutrality point (with

resistance at temperature 51K subtracted).

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1 Introduction Electronic conduction in monolayergraphene has unusual features owing to its unique bandstructure [1, 2]. The linear electron dispersion relation nearthe K-point, leading to conical electron energy surfaces,means that the carriers change from electron-like to hole-likeas the Fermi energy is decreased through the chargeneutrality point (NP) by application of a gate voltage. Inthe vicinity of the NP the charge carriers appear to form‘‘puddles’’ of electrons and holes [3] associatedwith chargedimpurities.

Extraordinarily highmobilities up to 200 000 cm2/s havebeen measured at low temperatures for suspended grapheneflakes [4, 5], but the best achieved for graphene on substrates(as required for many devices) is far lower [6, 7]. The originof the resistance and its temperature dependence ingraphene, in particular the role played by atomic-scaledefects [8], nanoscale ripples and associated flexuralphonons in graphene sheets [6], substrate phonons [7],

charged impurities [9], andmesoscopic fluctuations [10–15],are not yet fully understood. In this paper, we investigate thecharacteristics of the mesoscopic resistance fluctuations(MRFs) (Fig. 1) that we showed cause low-temperatureanomalies in the low-temperature resistance [13].Surprisingly, these anomalies decayed exponentially astemperature increased, in contrast to theoretical expectations[16] of weak inverse power laws. Very recently, Branchaudet al. [14] have confirmed our exponential decay result fortheir mesoscopic fluctuation data in graphene.

We also examine the temperature dependence of theresistance of graphene monolayers on Si/SiO2 substrates,analyzing our own data and that of others [6, 7] to helpidentify which phonons could give rise to the strongsuperlinear increase of resistance at higher temperatures.

2 Mesoscopic resistance fluctuations We show inFig. 1 an example of our measurements of the gate voltage

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2984 K. K. Mahelona et al.: Resistance and mesoscopic fluctuations in graphenep

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Figure 1 Resistance of monolayer graphene on Si/SiO2 as a func-tion of gate voltage (a) and subtracted resistance fluctuations DR(b) at 4.2K. These data are for an electrode separation of 900 nm. Figure 2 (online color at: www.pss-b.com) Autocorrelation

function and power spectrum (arbitrary units) for resistance fluctua-tionsas a functionofgatevoltageVg forelectrode separation900 nm,determined separately for gate voltages near the NP(�5 eV<Vg< 5 eV), on the electron side (5 eV<Vg< 15 eV),and on the hole side (�15 eV<Vg<�5 eV).

dependence of the resistance at 4.2K of a graphenemonolayer on a Si/SiO2 substrate [13].

Using DC measuring techniques, the dominant featuresare the MRFs which can reach an amplitude comparablewith the value of resistance itself at low temperatures. Adifference between the MRFs at large carrier densities andthose near the NP is that the period as a function of gatevoltage appears to be smaller near the NP, as suggested bythe example shown in Fig. 1.

To investigate this feature more fully, we havecalculated the autocorrelation function and power spectrumfor our data on MRFs as a function of gate voltage in Fig. 2.

We do not have evidence for particular periods that areprevalent in MRFs, but it is clear from the power spectrumthat near the NP the dominant periods as a function of gatevoltage are below 4 eV. In contrast, for the electron and holeregions away from the NP, fluctuations with periods greaterthan 4 eV dominate. This difference is clearly reflected in thefaster oscillatory behavior of the autocorrelation functionnear the NP. One likely cause is the smaller carrier density inthis region, which means that changes in carrier densityinduced by changes in gate voltage will be fractionallygreater and so have more effect on the interference pattern(just as the magnitude of the averaged resistance itself showsmore sensitivity to changes in gate voltage). We note,however, that it is the frequency rather than the amplitude ofthe MRFs that appears to show the most change near the NP(because of the large resistance near the NP, this means thatthe fluctuations in conductance decrease in amplitude nearthe NP, as noted earlier [10]).

Another factor causing shorter fluctuation periods closeto the NP is the relatively small size of electron and holepuddles (apparently caused by charge-donating impurities

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[9]) with an intrinsic disorder length scale of approximately30 nm in graphene [3]. Very recently, Branchaud et al. [14]found that conductance fluctuations as a function ofmagnetic field were visible at very low fields as a precursorto the quantum Hall effect. They also linked the narrowerfrequency range of their graphene conductance fluctuationsat the NP to the presence of electron and hole puddles ofsmall size. The increase of one type of carrier (and a decreasein the other) moving away from the NP would lead to agreater spread of puddle sizes, reflected in a greater range ofconductance fluctuation frequencies.

We also mention that Kechedzhi et al. [17] calculatedautocorrelation as a function of the Fermi energy (controlledby the gate voltage) for their measurements of conductancefluctuations. However, they averaged over a range ofmagnetic field values to obtain a correlation functionsmoothly decreasing with Fermi energy shift, from whichthe electron temperature could be determined.

3 Effect of MRFs on T-dependence of resistanceWe show in Fig. 3 an example of our measurements of thetemperature dependence of the resistance at a gate voltage set12.6V away from the charge NP. The most striking featuresare the large anomaly at low temperatures and the strongsuperlinear increase at high temperatures.

We look first at the behavior below 60K. This anomalychanges from a decrease to an increase in resistance and backagain with small changes in the gate voltage. This strongsensitivity to small changes shows that the anomaly is

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Figure 3 (online color at: www.pss-b.com) Resistance of ourgraphene on Si/SiO2 (1450 nm wide, measured between electrodes900 nm apart) as a function of temperature for gate voltageVg¼ 12.6V, i.e., far from the NP. The fits are to Eq. (1) with Rhp

given by Eq. (2) with phonon energies E1¼ 160meV andE2¼ 70meV (black line), by Eq. (3) (violet line, almost identicalto thepreviousfit),andbyEq. (2)usingSiO2phononenergies [7] (redline), as discussed in the text.

associated with mesoscopic fluctuations rather than, forexample, weak localization effects that would be systematic.More specifically, when the gate voltage is held at aminimum of the MRF pattern at low temperature, theresistance increases with temperature as the fluctuation iswashed out by thermal effects (as in the Abstract figure). Incontrast, measuring R(T) at the maximum of a fluctuationyields a corresponding decrease as temperature increasesuntil the fluctuations are suppressed (typically around 50–70K). In this way, we decisively demonstrated that this low-temperature anomaly is due to MRFs, its sign reflectingthe peaks and troughs of the MRFs as a function of gatevoltage [13].

We find that our data for the T-dependent resistance inFig. 1 (and for other cases for high carrier densities i.e., awayfrom the NP) are well described by the expression:

www

R ¼ R0 þ Rf exp � T

Tf

� �þ aT þ Rhp; (1)

where the first term is the residual resistance, theexponential decay gives good fits to the low-temperatureanomaly [13], and the last two terms account for scatteringby acoustic phonons (the linear resistance term aT) and byhigh-energy phonons (the term Rhp that increases sharplyabove 150K, as discussed in the next section).

We expected that the resistance fluctuationswould decaywith temperature following an inverse power law, aspredicted theoretically for universal conductance

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fluctuations due to dephasing by inelastic scattering or byenergy averaging [16], but as we showed earlier [13] theexponential decay gives a significantly better fit. Theexponential decay constant (Tf) of resistance R(T) withtemperature, determined for 22 separate data sets at differentfixed values of gate voltage Vg on several samples, was (withstandard deviation) Tf¼ 15� 3K.

It is interesting to compare this decay constant withthat derived by averaging resistance measurements as afunction of gate voltage (Tf¼ 16� 1K). In this case, we areevaluating the root-mean-square amplitude of a very largenumber of fluctuations as we change the fluctuation patternby changing the gate voltage. The fact that these two valuesfor decay constant of the MRFs are so similar is additionalevidence for our conclusion that MRFs cause the low-temperature resistance anomaly.

Our fitting expression near the NP corresponding toEq. (1) involved [13] an activated contribution to theconductance since the resistance decreases above 150K, butthe low-temperature anomaly described by the exponentialterm was still present in a similar form.

4 Effect of phonon scattering on T-dependenceof resistance Scattering of charge carriers in metals bythermally excited acoustic phonons typically gives aresistance linear in temperature in metals above somefraction of the Debye temperature. A linear term is observedin graphene on Si/SiO2 substrates [6, 7], and has also beenseen in suspended graphene [5]. Our data also suggest smalllinear terms in the resistance far from the NP, but the maineffect (as in other measurements for graphene on substrates[6, 7]) is a strong nonlinear increase in resistance above150K.

As shown in Fig. 3,we find that this increase above 150Kis consistent with scattering by the graphene zone-boundary(K-point) phonons with energies E1¼ 160meV andE2¼ 70meV [18]:

Rhp ¼ R1 expE1

kT

� ��1

� ��1

þ R2 expE2

kT

� ��1

� ��1

; (2)

where the temperature dependences are given by the Bose–Einstein function [7] and the coefficients R1 and R2

determine the magnitudes. The dominant contributioncomes from the higher energy phonon (160meV) that isbelieved [19] to be responsible for limiting the current insingle-wall carbon nanotubes at high electric potentialssufficient to allow emission of high-energy axial 2kFphonons that couple particularly strongly to electrons[20, 21]. The reduction in the conductance often seen inthick single-wall carbon nanotube networks above 250K isalso ascribed to these 2kF phonons [22].

We note that the exponential terms in Eq. (2) areextremely small below 100K, and so do not affect the fit to

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the fluctuation term exp (�T/Tf) term which is very smallabove 100K, i.e., the fluctuation decay fitting parameter Tf isessentially independent of the fitting parameters R1 and R2

that determine the fits to the increase of resistance at hightemperature.

Morozov et al. [6] found that the superlinear componentRhp in their resistance data followed a T5 behavior

� 20

Rhp ¼ bT5; (3)

Figure 4 (onlinecolorat:www.pss-b.com) (a)Resistance termRhp

of graphenemonolayers away from theNPasmeasured byMorozovet al. [6] and Chen et al. [7], fitted to Eq. (3) (top lines at the highesttemperature), Eq. (2) with experimental values E1¼ 160meV andE2¼ 70meV forK-point zone-boundary phonons in graphene (cen-tral lines at highT), andEq. (2)withE1¼ 155meVandE2¼ 59meVas for SiO2 interface phonons [7] (bottom lines at highest T). (b)Separation of Rhp into contributions from the 160 and 70meVphonons in our fit to the data of Chen et al. in (a) involving thesephonons. The data are shifted vertically for clarity (by definitionRhp! 0 as T! 0).

which may represent a transition to scattering by high-energy phonons such as those associated with ripples in thegraphene sheet (arising, for example, from deposition on asubstrate). This T5 law also gives a very good description ofour data (almost identical to our fit described above, asshown in Fig. 3). This confirms that a T5 law can simulatethe effect of higher energy phonons in this case.

Chen et al. [7] described the strong high-temperatureresistance increase that they observed by an expressioninvolving a sum of two Bose–Einstein factors as in Eq. (2)where the two termswere taken as arising fromSiO2 phononsat the graphene/substrate interface, with energies 155 and59 eV, and magnitude ratio R1/R2¼ 6.5 as calculated byFratini and Guinea [23]. These SiO2 phonons, however, givea significantly poorer fit to our data (the curvature at hightemperatures is too shallow, as shown in Fig. 3).

To explore the situation further, we have made detailedfits to the available data comparing the three models forphonon scattering described above. The results are shown inFig. 4a.

We find that for the data sets of both Morozov et al. [6]and ofChen et al. [7] (using their data up to 500K), ourmodelusing the experimental K-point graphene phonons gave aslightly better fit than the others (it has two magnitude fittingparametersR1 andR2 while the others have one). For the dataof Chen et al. [7], the values of the goodness of fit R2 were98.9% for our 160/70meV phonon model, 97.3% for the T5

term, and 97.1% for the SiO2 phonon model. For the data ofMorozov et al., R2¼ 89.9% for the 160/70meV phononmodel, 89.3% for theT5 term, and 86.9% for the SiO2 phononmodel. The SiO2 phonon fit would improve if the theoreticalrelative coupling ratio R1/R2 were varied.

For the data of Chen et al., we show in Fig. 4b theseparation of the fit line values into components arising fromthe 160 and 70meV phonons, respectively. It is clearly the160meV phonon that plays the key role in producing thesharp increase in resistance at high temperatures.

5 Conclusions The period of fluctuations as a functionof gate voltage varies considerably, without evidence ofdominance by particular periods, but is generally less than4 eV near the NP in our configuration of graphene on a300 nm layer of SiO2 on Si and electrodes 900 nm apart. Onthe other hand, the fluctuation periods tend to be larger atgate voltages further from the NP. We conclude that thefluctuations on average show faster fluctuations (as afunction of gate voltage) near the NP than in the higher

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conductance regions on both the electron and hole sides.Near the NP, where the resistance is high and thecharge density is small, the fluctuations would be expectedto be more sensitive to the carrier density changes as thegate voltage is altered.

Another contribution to our observation of shorter periodfluctuations near the charge NP is likely to be the narrowingof the range of sizes of electron and hole puddles near theNP,as proposed by Branchaud et al. [14] to interpret theirfluctuation observations (we note, however, that their dataare not directly comparable to ours as they made AC ratherthan DC measurements).

Turning to the origin of the strong increase of resistanceat temperatures above 150K in graphene on substrates,we note that Morozov et al. found that using a PMMAsubstrate did not make a significant difference to theresistance at high temperature, suggesting that the SiO2

interface phonons may not play a key role in the scattering.

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This is supported by our result that for our data and for otherdata sets [6, 7], the SiO2 phonons do not give the best fit. A T5

term gives a good overall description of the high-T anomaly,but the best fit, for each data set, was given by a modelinvolving the 160meV graphene K-point phonon, the samephonon that plays a key role [19, 22] in limiting or reducingthe conductance of single-wall carbon nanotubes (rolled-upgraphene). Our results suggest that these graphene phononsof energy 160meV (together with a smaller contributionfrom lower energy phonons) are able to account for thestrong increase in resistance above 150K seen in monolayergraphene on Si/SiO2 substrates.

Acknowledgements We are very grateful to ourcollaborators Jai Seung Yoo and Siegmar Roth at MPI Stuttgart.V. S. acknowledges the Center of Excellence CENAMOST (SlovakResearch and Development Agency Contract No. VVCE-0049–07)for support of project APVV-0628-06.

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