Abstract
Graphene charge carriers behave as massless Dirac fermions, opening the exciting possibility to observe longrange virtual tunneling of electrons in a solid. In granular metals, electron hops arising from series of virtual transitions are predicted to yield observable currents at lowenough temperatures, but to date experimental evidence is lacking. We report on electron transport in granular graphene films selfassembled by hydrogenation of suspended graphene. While the logconductance shows a characteristic T^{−1/2} temperature dependence, cooling the samples below 10 K drives a triple crossover: a slope break in logconductance, simultaneous to a substantial increase in magnetoconductance and onset of large mesoscopic conductance fluctuations. These phenomena are signatures of virtual transitions of electrons between distant localized states, and conductance statistics reveal that the high crossovertemperature is due to the Dirac nature of granular graphene charge carriers.
Introduction
Shortlived particles allowed by Heisenberg's uncertainty principle are called “virtual”, but the effects they induce are very real. They lead to vacuum fluctuations and mediate fundamental forces^{1}, explain the Lamb shift of atomic levels^{2}, the Casimir effect^{3} and possible Hawking radiations^{4}. Though unobservable, virtual particles are key ingredients of modern quantum electrodynamics. Considerable research efforts have thus been devoted to measuring the most direct consequences of their existence, such as virtualtoreal photon conversion during dynamic Casimir effect^{5}. Quantum dot nanostructures are excellent test beds too, as electron tunneling through virtual states^{6,7,8} generates background currents observable below ~100 mK^{9}. Interestingly, higherorder currents between distant localized states are predicted to arise from multiple transitions to virtual states in macroscopic granular metals^{10,11,12}, but this phenomenon coined multiple elastic cotunneling (MEC) has not been observed yet. From this perspective, graphene is a particularly promising material, in which virtual excitations of charges play a special role. They are theoretically predicted to induce Dirac fermions' jittery motion called zitterbewegung^{13}, and give a minimum conductivity to pristine graphene. In granular form, graphene dots' linear density of states (DOS)^{14} means that highenergy virtual states should contribute significantly to elastic cotunneling currents despite shorter lifetimes, unlike ordinary granular metals (GM). This not only makes granular graphene the ideal platform for the first observation of MEC, but also provides a rare opportunity to measure longrange effects mediated by highenergy virtual states.
Results
Recently, chemical functionalization of graphene^{15} was reported to be a viable method to produce granular graphene^{16}, due to the tendency of adatoms to form electrically insulating clusters. At high enough adatoms concentrations, such clusters merge into percolative pathways, effectively partitioning the graphene sheet into weaklycoupled graphene dot arrays, or GM. To conveniently fabricate graphene GMs, we exposed suspended CVD graphene sheets^{17} to hydrogen plasma^{18,19}, thereby allowing adsorption of sufficiently high concentrations of hydrogen atoms on both sides of the graphene scaffold. We then stamped asproduced doublyhydrogenated graphene (DHG) films on 90 nmSiO_{2} chips. Au/Cr contacts in two and fourprobe geometries were then fabricated. Typical devices are shown in inset of figure 1(a). To observe the chargeneutrality point (CNP) at gate voltages V_{g} ~ 0 V, the samples were then vacuumdried^{20} insitu at 10^{−6} Torr for a day before cooling below 0°C. We then measured electron transport in 8 devices from room temperature down to 2.4 K. Figure 1(a) shows the roomtemperature resistance R in units of resistance quantum h/e^{2}, against chargecarrier density n for a typical sample (D0). The R(n) curve has a characteristic graphenelike shape, but is broad and R(n) ≫ 1 throughout the whole range of measured densities, indicating strong localization.
Next, we measure the conductance G for different bias voltages V_{sd} and temperatures T to extract the typical localization length ξ of our samples. The inset of figure 1(b) shows G against V_{sd} at different temperatures T between 2.3 K and 20 K for device D1. For consistency, all subsequent data shown in main text correspond to the same device D1. Data taken for other devices are reported in Supplementary Material, section A. We observe that above V_{sd} ~ 0.1 V, G increases with V_{sd} while below, G is biasindependent, indicating an Ohmic behavior. Crucially, G(V_{sd}, T) data can be used to extract ξ without assuming any particular transport mechanism. Since an electron hopping against the sourcedrain electric field E = V_{sd}/L over a distance d increases its energy by an amount eEd, it was shown that charge carriers experience an effective temperature where L = 4 μm is the channel length, and α ≈ 0.67 is a constant^{21}. Importantly, T_{eff} uniquely determines G, which implies that constantconductance domains of (V_{sd}^{2}, T^{2})  space are straight lines of slope −(αeξ/k_{B}L)^{2}. Figure 1(b) shows such domains extracted from G(V_{sd},T) data at 1, 2, 4, 10 and 20 nS. As expected, they are wellfitted by straight lines of slope ~ −1.4 × 10^{−4} V^{2}K^{−2}, giving ξ ≈ 45 nm.
We now show that our DHG samples have a GM structure by complementing our electric transport measurements with Raman spectroscopy data, presented in figure 1(c). The spectrum shown in figure 1(c) exhibits a prominent D peak as well as broad 2D and D + D′ peaks. The D peak partially overlaps with the G peak. The D′ peak at 1614 cm^{−1} almost completely merges with the G peak at ~ 1588 cm^{−1} but can still be resolved as shown by the inset of figure 1(c). These are characteristics of strongly sp^{3}hybridized graphene^{22}. The Dpeak of sp^{3}hybridized graphene probes the distribution of sp^{3}bonds on the graphene lattice^{23}. More precisely, the ratio I_{D}/I_{G} of integrated intensities of the D and G peaks is related to the typical adatom cluster size r_{s} and the mean distance L_{D} between nearest cluster centers by the following formula^{24,25}:
where C_{A} and C_{S} are constants, r_{A} = r_{S} + δ, and δ is the average distance laserexcited electrons travel before recombining with holes. Experimentally, it was shown that C_{A} ~ 4, C_{S} ~ 0.9 and δ ~ 2 nm for a laser of 2.4 eV^{26,27}. By fitting Fano lineshapes^{28} to the Raman spectrum peaks, we calculate the peaks integrated intensities and I_{D}/I_{G} ~ 1.8. This value is clearly inconsistent with a random distribution of isolated adatoms, which would yield L_{D} ~ δ ≪ ξ. Conversely, hydrogen adatoms form clusters of typical radius r_{s}^{29}. Simple geometric considerations discussed in section B of Supplementary Material impose that L_{D}, r_{s} and ξ typically follow the relation . Since ξ ≈ 45 nm, we find r_{s} ~ 30 nm by solving F(r_{s}, L_{D}) ≈ 1.8, leading to L_{D} ≈ 2 r_{s}. In other words, clusters tend to merge, isolating graphene dots of size ξ, and our DHG samples have a GM structure. This agrees with previous studies on graphene quantum dots of size ~ ξ^{30} yielding comparable Raman spectra due to edge scattering^{26,31}.
Next, we focus on identifying the dominant charge transport mechanisms by analyzing the temperaturedependence of the conductance G(T). More precisely, one expects G to follow a G = G_{0}exp(−(T_{0}/T)^{γ}) law characteristic of hopping transport, where γ and T_{0} depend on the exact hopping mechanism^{32}. We thus measured G around CNP for different T between 2.4 K and 300 K. G(T) is systematically measured in the lowbias Ohmic regime, where both electricfielddriven electron hopping and Joule heating are negligible. As shown in section C of Supplementary Material, we observed a reduced activation energy^{33} β = dlnG/dlnT linear in lnG with slope ~−1/2 both for lnG < −21.5 and lnG > −19. Therefore, we plotted G against T^{−1/2} in figure 2(a). Strikingly, between 300 K and ~12 K, and between ~8 K and 2.4 K, but with a much smaller slope. A distinct slope break is thus identified around T_{cross} = 10 K. Graphene being atomically thin, this phenomenon certainly does not reflect a decrease in effective sample dimensionality from three to two dimensions^{34}. Besides, our measured samples have a channel width W ≈ 6–7 μm systematically larger than the length L ≤ 5 μm to avoid any possible 2D to 1D crossover upon lowering the temperature. A G_{0} exp(−(T_{0}/T)^{γ}) fit both above and below T_{cross} respectively gives γ = 0.56 ± 0.04 and γ = 0.495 ± 0.05. In these regimes, G(T) is thus neither of the Arrhenius type (γ = 1) nor of the Mott's 2D variablerange hopping (VRH) type (γ = 1/3). However, γ = 1/2 suggests an EfrosShklovskii (ES) VRH behavior, the 1D Mott VRH being excluded due to the geometry of our devices. Such behavior contrasts with the result of several earlier studies, in particular^{18}, performed on strongly localized hydrogenated graphene samples, where the conduction is attributed to 2D Mott VRH. However, these results were obtained for less hydrogenated graphene samples, fabricated by exposing a single graphene face to hydrogen plasma. Conversely, a γ = 1/2 behavior was already reported in heavily oxidized graphene films^{16} with a GM structure. However, the presence of both a gamma1/2 behavior and a slope break as observed in figure 2(a) around 10 K has never been observed before, to the best of our knowledge. While a crossover between two γ = 1/2 regimes with different slopes is not expected in standard VRH theories^{32}, it is predicted for granular systems^{10}, each regime reflecting a distinct transport mechanism illustrated in figure 2(b): multiple inelastic cotunneling (MIC) at high temperatures and multiple elastic cotunneling (MEC) at low temperatures. During MIC, multiple electrons simultaneously tunnel from the Fermi sea of a grain to an excited state of a neighboring grain, along a string of grains ultimately left in an excited state, thus requiring a finite temperature or electric field. During MEC, charge carriers tunnel between distant grains by transiting to intermediate virtual states, without absorbing or emitting phonons. To verify that our G(T) data result from a MICtoMEC crossover, we fit a function of the form G(T) = G_{MIC}(T) + G_{MEC}(T), where and are MIC and MEC conductance terms. The best fit is obtained for G_{in} ≈ 6 μS, T_{in} ≈ 700 K, G_{el} ≈ 1.5 nS, and T_{el} ≈ 6 K. We now show that these values match theoretical expectations for cotunneling in granular graphene. We start with G_{in}, which is the total conductance in the T ≫ T_{in} limit and is related to the nearestgrain tunneling conductance g by g ~ LG_{in}/W, where W ≈ 7 μm is the sample width. This leads to g ~ 0.1 ≪ 1 in units of e^{2}/h, which is a characteristic of insulating GM. The interdot tunneling conductance g should also be compared to the intradot conductance g_{dot}^{35}. Since before hydrogenation, our graphene samples have a typical mobility μ ≈ 5000 cm^{2}/V.s, the intradot meanfree path is where . A graphene grain of size ~ξ with at least one Dirac fermion, corresponding to an areal density of ~5 × 10^{10}/cm^{2}, thus has l > 13 nm and a Thouless energy^{36} larger than ~10 meV. Therefore, E_{Th} typically exceeds the dots mean energy level spacing^{37} , so that is at least a few e^{2}/h. g_{dot} ≫ g is thus satisfied throughout the experimentally relevant range of densities. This is again perfectly consistent with an insulating granular metal behavior, and contrasts sharply with the case of homogeneously disordered systems^{10}, where . Next, we notice that T_{in} ≈ 700 K is significantly larger than the charging energy of a grain , where ε ≈ 3.5 is the dielectric constant of the intergranular medium^{16}. This agrees with MIC theory^{10} which predicts T_{in} = χ_{in}T_{c}, where χ_{in} is a logarithmically Tdependent coefficient. As T ≪ T_{c}, where is the rate of inelastic tunneling between two neighboring grains^{12}, giving χ_{in} ≈ 8 at T = 10–20 K, and hence T_{in, theory} ≈ 800 K, close to the experimental value. We finally focus on T_{el}, whose value extracted from figure 2(a) is one order of magnitude smaller than T_{c}. This remains true at all V_{g} despite fluctuations, and T_{el} averages to θ_{el} ≈ 10 K. More details on the statistics of T_{el} are reported in Supplementary Material, section D. Theoretically, , where p is the probability of virtual transition to a neighboring grain^{10,11}. For conventional twodimensional GMs, the DOS in each grain is constant and the level spacing is small compared to the charging energy, leading to^{10} and T_{el} of the order of few T_{c}. Therefore, Schrödinger fermions cannot account for the observed T_{el} ≪ T_{c} behavior. Unlike MIC for which only states within k_{B}T_{eff} around the Fermi level contribute to G, virtual transitions to highenergy states contribute to the MEC conductance and the band structure plays a key role. In the Dirac fermions case, the short lifetime of highenergy fermions is compensated by a DOS which increases linearly with energy E, making the contribution of highenergy virtual states significant. It remains true for chaotic dots with edges of random shape^{14,37}. Following refs. 11 and 38, we calculated in Supplementary Material (section E) a lowerbound for the virtual transition probability of graphene's Dirac Fermions: , where Γ is the energy bandwidth. This is in accord with experimental statistics on p_{el} shown in inset of figure 2(a) and implies that p_{in} ~ p_{el} just above T_{cross}, between 20 K and 40 K, thus providing strong evidence for MICtoMEC crossover.
A powerful way to gain further insight is to analyze the variations of G with V_{g} and V_{sd}, as each transport mechanism leaves its own mesoscopic fluctuations footprints^{38,39,40,41}. Figure 3(a) shows G measured at 3 K for −7.5 V ≤ V_{g} ≤ 7.5 V and 0 ≤ V_{sd} ≤ 200 mV and exhibits vertical stripes of width ΔV_{g} ~ 200 mV, corresponding to peaks and valleys in conductance reproduced at all measured V_{sd}. This is highlighted by figure 3(b), which shows four different traces extracted from figure 3(a) at V_{sd} = 10 mV, 80 mV, 120 mV and 175 mV. While overall the conductance increases with V_{sd}, the different fixedV_{sd} G(V_{g}) curves show reproducible peaks and valleys. To analyze these fluctuations quantitatively, we systematically extracted the standard deviation of logconductance σ_{lnG} at fixed sourcedrain bias from figure 3(a) and three other data sets shown in Supplementary Material, section F. Figure 4 shows σ_{lnG} plotted against T_{eff} from 3 K up to 80 K, in doublelog scale. Two different regimes can be clearly distinguished. Below T_{eff} = 10 K, σ_{lnG} is weakly temperaturedependent while above 10 K, σ_{lnG} decreases rapidly with T_{eff}. Quite remarkably, these two regimes coincide with the two distinct G(T) regimes observed in figure 2(a). We start by analyzing the highT_{eff} regime, where σ_{lnG} is very well described by the powerlaw . This behavior is clearly incompatible with Mott or ES VRH as such phenomena would lead to with a < 1^{41}. We now compare the observed fluctuations above 10 K to expected MICinduced fluctuations. From a hopping percolation viewpoint^{42}, the fluctuations in logconductance σ_{LnG} are related to the standard deviation s of logconductance at the scale of a hopping distance r_{hop}, by where , and ν ≈ 4/3 is the critical exponent in two dimensions. Since MIC is a fundamentally phaseincoherent process involving a different charge carrier for each intermediate transition (see figure 2(b)), MICinduced fluctuations in conductance do not originate from quantum interferences between electron wave functions. It must rather arise from fluctuations in interdot conductances accompanying changes in percolation network as gate voltage is tuned. From this perspective, we find  based on probabilistic arguments developed in Supplementary Material, section G1 and G3  that where^{10} and N_{th} is the number of energy levels accessible in a neighboring grain by emitting or absorbing a phonon. For a 2D GM with parabolicband grains, and hence . This corresponds to , which does not satisfactorily fit the data. However, in graphene grains, energy levels are not evenly spaced. The n^{th} energy level from neutrality point has an expected value of energy where . Therefore, and (see Supplementary Material, section G2). This yields , which agrees very well with the data. In other words, the unusually strong T_{eff} – dependence of σ_{lnG} above 10 K can only be understood in terms of inelastic cotunneling of Dirac fermions.
We now discuss the subT_{cross} regime, where σ_{lnG} ~ 0.5 is weakly Tdependent, clearly ruling out MIC as dominant transport mechanism below 10 K. This is perfectly consistent with figure 2(a) which indicates a crossover to MEC below 10 K. Since MEC is a phasepreserving process, it is tempting to assign this behavior to quantum interferences between distinct phasecoherent chargecarrier paths^{43,44}, a phenomenon known to produce almost T_{eff}independent conductance fluctuations of large magnitude. This view is supported by magnetotransport experiments carried out at 2.4 K for −500 mV ≤ V_{sd} ≤ 500 mV. The inset of figure 4 shows the relative magnetoconductance MC = G(8T)/G(0T) against T_{eff} from 2.4 K up to 40 K. MC is almost constant and close to 1 above T_{cross} where MIC dominates, reflecting its phaseincoherent nature, whereas MC rapidly increases to ~2 below T_{cross}, a manifestation of quantum interferences^{11,44,45} attributable to MEC. Moreover, figure 4 shows that the subT_{cross} data are welldescribed by with η ~ 0.16 or less, suggesting conduction is limited by a strongly resistive portion of the granular graphene film of size ~ r_{hop}, or “bottleneck”. Such a situation systematically occurs in granular media at sufficiently low T_{eff} when σ_{LnG} approaches unity^{41,46}, which is the case around T_{cross}. Since quantum interferences within a bottleneck must give the main contribution to σ_{lnG}, the observed plausibly approximates the law for systems of size close to the phasecoherence length^{47,48,49}. In summary, both the significant increase in magnetoconductance and weak temperaturedependence of σ_{lnG} below 10 K indicate the appearance of quantum interferences. The presence of such strong interferences only below T_{cross} is consistent with the existence of a crossover from phaseincoherent MIC above 10 K to MEC below 10 K.
Discussion
Our results show that doublyhydrogenated graphene is not amorphous but has a granular structure instead. This finding is consistent with the tendency of hydrogen atoms adsorbed on graphene to phaseseparate^{50} and form electrically insulating clusters. This observation is particularly interesting as graphane phases – or fully hydrogenated graphene regions – are predicted to be hightemperature superconductors^{51}. From this perspective, hydrogenation of suspended graphene sheets appears as a viable route towards the synthesis of novel granular superconductors^{10,52}. The granularity of our doublyhydrogenated graphene samples is reflected by the presence of Dirac fermions in this material, which can exist in graphene grains or dots, but not in amorphous media. Both the large crossover temperature from inelastic to elastic cotunneling and the temperature dependence of the logconductance fluctuations with gate voltage are signatures of the presence of Dirac fermions in our samples.
In conclusion, we observed multiple elastic cotunneling for the first time in a granular metal. In our granular graphene samples, both multiple inelastic and elastic cotunneling mechanisms showed signatures of Dirac fermions. The presence of large highorder elastic cotunneling currents in granular graphene establishes granular Dirac materials as ideal platforms for the study of vacuum fluctuations and quantum noise^{53}.
Methods
Device fabrication
CVD graphene was grown on copper foil using the technique described in ref. 17. The copper foil was etched away using an ammonium persulfate solution and transferred on a Quantifoil S7/2 square mesh holey carbon TEM grids, following a polymerfree transfer technique similar to ref. 54. Our freestanding graphene sheets were subsequently exposed to hydrogen plasma following the technique used in ref. 19. The resulting samples were deposited on 90 nmSiO_{2} chips immediately after plasma exposure, by gently stamping the TEMgridsupported graphene sheet onto the substrate. This procedure generally left a large number of 7 × 7 μm^{2} squareshaped hydrogenated graphene flakes on SiO_{2}, corresponding to the TEM grid mesh geometry. Devices were then fabricated by standard electronbeam lithography technique followed by thermal evaporation of Au/Cr electrodes.
Electron transport measurement
After fabrication, samples were loaded in a variable temperature insert coupled to a 9 T superconducting magnet. A pressure of ~10^{−6} Torr was maintained during the experiments. Electron transport measurements were carried out with a Keithley 6517B Electrometer/HighResistance Meter. To eliminate possible DC noise, we used the following procedure: for each bias V_{sd}, the sourcedrain current I_{sd} was measured 10 times at +V_{sd} within ~1 s, then 10 times at –V_{sd} within ~1 s. The resulting noisefiltered current was then systematically calculated as , where corresponds to the 10point average. V_{sd} was sourced by Keithley 6517B or Keithley 6430. Keithley 6430 was systematically used to source the gate voltage. When needed, a fixed I_{sd} was sourced with a Keithley 6221, while V_{sd} was measured with Keithley 6517 B.
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Acknowledgements
We thank A.H. Castro Neto, V. Pereira, M.A. Cazalilla and J. Martin for helpful discussions. This work was supported by the Singapore Millennium FoundationNUS Research Horizons award (R144001271592; R144001271646), and NUSYIA (R144000283101).
Author information
Author notes
 Manu Jaiswal
Current address: Department of Physics, India Institute of Technology Madras, Chennai 600036 India.
Affiliations
Department of Physics, National University of Singapore, Singapore 117542, Singapore
 Alexandre Pachoud
 , Manu Jaiswal
 & Barbaros Özyilmaz
NUS Graduate School for Integrative Sciences and Engineering (NGS), Singapore 117456, Singapore
 Alexandre Pachoud
 & Barbaros Özyilmaz
Graphene Research Center, National University of Singapore, Singapore 117542, Singapore
 Alexandre Pachoud
 , Manu Jaiswal
 , Yu Wang
 , Kian Ping Loh
 & Barbaros Özyilmaz
Department of Chemistry, National University of Singapore, Singapore 117543, Singapore
 Manu Jaiswal
 , Yu Wang
 & Kian Ping Loh
Department of Chemistry, Seoul National University, Seoul 152742, Korea
 ByungHee Hong
School of Electrical & Electronic Engineering, Yonsei University, Seoul, 120749, Korea
 JongHyun Ahn
Nanocore, 4 Engineering Drive 3, National University of Singapore 117576, Singapore
 Barbaros Özyilmaz
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A.P. and M.J. designed and performed the experiments. A.P. and M.J. analyzed the data. A.P. wrote the paper. B.H.H. and J.H.A. prepared graphene by chemical vapor deposition. Y.W. and K.P.L. hydrogenated graphene. All authors discussed the data and reviewed the manuscript. B.Ö. devised and supervised the project.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Barbaros Özyilmaz.
Supplementary information
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Supplementary Information
Ozyilmaz  Supplementary Material  Multiple Virtual Tunneling of Dirac fermions in Granular Graphene
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